A Study on Groupoids, Ideals and Congruences via Cubic Sets

Abstract

The inclusion, the intersection and the union between cubic sets are each defined in two ways. From this point of view, we introduce the concepts of cubic subgroupoids, cubic ideals, cubic subgroups, and cubic congruences as two types, respectively, and discuss their various properties. In particular, we give a relationship between the set of all cubic normal subgroups of a group and all cubic congruences on the group.

1. Introduction

The concepts of fuzzy sets as the generalization of crisp sets and interval-valued fuzzy sets as the generalization of fuzzy sets was introduced by Zadeh [1] in 1965 and [2] in 1971. In 2012, Jun et al. [3] proposed the notion of cubic sets combined with interval-valued fuzzy sets and fuzzy sets. After that, Chinnadurai et al. [4] studied further properties of cubic sets, and Kang and Kim [5] dealt with some properties for a mapping of cubic set. Zeb et al. [6] and Rashid et al. [7] applied cubic sets to topological structures and decision-making problems, respectively. Additionally, many researchers [8,9,10,11,12,13] investigated ideal structures based on cubic sets in various algebras. Now, let us take a look at recent research trends based on cubic sets. Lee et al. [14] formed the concrete category of cubic relations and morphisms between them, and studied some of its categorical structures in the sense of the topological universe. Loganayaki and Jayanthi [15] defined P-cubic and R-cubic continuous mappings, and dealt with some of their properties. Muhiuddin et al. [16] discussed various properties of cubic graphs and gave an example for a traffic flow problem as an application of cubic graphs. Alhazaymeh et al. [17] introduced the notion of cubic vague sets and applied it to decision-making problems. Riaz et al. [18] proposed the concept of correlation coefficients for cubic bipolar fuzzy sets and applied it to pattern recognition.

Jun et al. [19] defined a cubic subgroup and obtained its various properties. Jun and Khan [20] studied ideal structures based on cubic sets in semigroups. Recently, Muhiuddin [21] defined a cubic interior ideal in semigroups and obtained some of its properties. Amalanila and Jayalakshmi [22] discussed cubic near-ring structures. Gaketem and Iampan [23] introduced the notion of cubic filters in semigroups and investigated its various properties.

In many books (for example, Rosenfeld [24] and Howie [25]), we can see that a complete description of the congruences on a group in terms of its normal subgroups was proposed as follows:

$$\mathrm{There}\mathrm{exists}\mathsf{a}\mathrm{one}-\mathrm{to}-\mathrm{one}\mathrm{mapping}\mathrm{from}N\left(G\right)\mathrm{onto}C\left(G\right)$$

(1)

where $N\left(G\right)$ and $C\left(G\right)$ denote the set of all normal subgroups of a group G and all congruences on G.

The order between cubic numbers is defined by two types called P-order and R-order. The purpose of our study is to define cubic subgroupoid, cubic ideal, cubic subgroup and congruence in two ways and discuss the properties of each. To do this, first, we introduce the concepts of P-cubic subgroupoids and P-cubic ideals [respectively, R-cubic subgroupoids and R-cubic ideals] and obtain some of their properties. Furthermore, we define a level set of a cubic set and we give a necessary and sufficient condition for it to be subgroupoid (resp., ideal) (See Theorem 2). Next, we define a cubic subgroup (resp., a cubic normal subgroup) and find a necessary and sufficient condition for the level set of a cubic set to be subgroup (resp., normal subgroup) (See Theorem 4 [resp., Theorem 5]). Finally, we define a cubic congruence and provide the necessary and sufficient condition that the product of two cubic congruences is also a cubic congruence (See Theorem 10 and 11). Furthermore, we give the extension of (1) already mentioned in the introduction via cubic sets (See Proposition 35).

2. Preliminaries

In this section, we list some basic notions needed in the next sections. Throughout this paper, let $I=[0,1]$, let $X,Y,Z$ denote nonempty sets, let $P\left(X\right)$ denote the power set of X and let ${\chi }_{{}_A}$ be the characteristic function of $A\in P\left(X\right)$.

A mapping $A:X\to I$ is called a fuzzy set in X (See [1]). $\mathbf{0}$ [resp., $\mathbf{1}$] is called the empty [resp., whole] fuzzy set in X and defined by $\mathbf{0}\left(x\right)=0[resp.\mathbf{1}(x)=1]$ for each $x\in X$. $I^X$ denotes the set of all fuzzy sets in X.

Each member of the set $\left[I\right]=\{\tilde{a}=[a^{-},a^{+}]\in I:0\le a^{-}\le a^{+}\le 1\}$ is called an interval-valued fuzzy number. Refer to [3] for the definitions of order ≤, the equality =, the complement ${}^c$, the infimum ⋀ and the supremum ⋁ on $\left[I\right]$.

A mapping $\tilde{A}:X\to \left[I\right]$ is called an interval-valued fuzzy set (briefly, an IVFS) in X (See [26,27,28]). $\tilde{\mathbf{0}}$ [resp., $\tilde{\mathbf{1}}$] is called the empty [resp., whole] interval-valued fuzzy set in X and defined by $\tilde{\mathbf{0}}\left(x\right)=[0,0][\mathrm{resp}.\tilde{\mathbf{1}}\left(x\right)=[1,1]]$ for each $x\in X$. We denote the set of all IVFSs in X as $IVFS\left(X\right)$. See [26,27,28] for the definitions of the inclusion ⊂, the equality =, the complement ${}^c$, the intersection ∩ and the union ∪ on $IVFS\left(X\right)$.

Members of the set $\left[I\right]\times I$ are called cubic numbers and are written as $\tilde{\tilde{a}}=〈\tilde{a},a〉,\tilde{\tilde{b}}=〈\tilde{b},b〉,\tilde{\tilde{c}}=〈\tilde{c},c〉,\cdots .$ Refer to [29] for the definitions of two types of orders between two cubic numbers, and two types of infimums and supremums of any collection of cubic numbers.

Definition 1

([3]). A mapping $\mathcal{A}=〈\tilde{A},A〉:X\to \left[I\right]\times I$ is called a cubic set in X. There are four special cubic sets in X, defined as follows:

$$\ddot{0}=〈\tilde{\mathbf{0}},\mathbf{1}〉,\ddot{1}=〈\tilde{\mathbf{1}},\mathbf{0}〉,\widehat{0}=〈\tilde{\mathbf{0}},\mathbf{0}〉,\widehat{1}=〈\tilde{\mathbf{1}},\mathbf{1}〉.$$

Then $\widehat{0}$ [resp., $\widehat{1}$] is called the empty [resp., whole] cubic set in X and $CS\left(X\right)$ denotes the set of all cubic sets in X.

Definition 2

([3]). For a nonempty set X, let $\tilde{\tilde{a}}$ be a cubic number and let $\mathcal{A}\in CS\left(X\right)$. Then, $\mathcal{A}$ is called a constant cubic set with $\tilde{\tilde{a}}$, denoted by ${\mathcal{C}}_{\tilde{\tilde{a}}}$, if ${\mathcal{C}}_{\tilde{\tilde{a}}}\left(x\right)=\tilde{\tilde{a}}$ for each $x\in X$.

Definition 3

([3]). Let $\mathcal{A},\mathcal{B}\in CS\left(X\right)$. Then, the equality and the inclusion between $\mathcal{A}$ and $\mathcal{B}$ are defined as follows:

(i) (Equality) $\mathcal{A}=\mathcal{B}\iff \tilde{A}=\tilde{B}andA=B$.

(ii) (P-inclusion) $\mathcal{A}⊏\mathcal{B}\iff \tilde{A}\subset \tilde{B}andA\subset B$.

(iii) (R-inclusion) $\mathcal{A}⋐\mathcal{B}\iff \tilde{A}\subset \tilde{B}andA\supset B.$

Refer to [3] for the definitions of the compliment (${}^c$) of a cubic set, two types of unions (⊔, ⋓) and two types of intersection (⊓, ⋒) between two cubic sets.

It is obvious that $\widehat{0}⊏\mathcal{A}⊏\widehat{1}$ and $\ddot{0}⋐\mathcal{A}⋐\ddot{1}$ for each $\mathcal{A}\in CS\left(X\right)$. Moreover, $\widehat{0}⊏\ddot{0},\ddot{1}⊏\widehat{1}$ and $\ddot{0}⋐\widehat{0},\widehat{1}⋐\ddot{1}.$

Definition 4

([5,29]). Let $f:X\to Y$ and let $\mathcal{A}\in CS\left(X\right),\mathcal{B}\in CS\left(Y\right)$.

(i) The pre-image of $\mathcal{B}$ under f, denoted by $f^{-1}(\mathcal{B})=〈f^{-1}(\tilde{B}),f^{-1}\left(B\right)〉$, is a cubic set in X defined as follows: for each $x\in X$,

$$f^{-1}\left(\mathcal{B}\right)\left(x\right)=〈f^{-1}(\tilde{B})\left(x\right),f^{-1}\left(B\right)\left(x\right)〉=〈\tilde{B}\left(f\left(x\right)\right),B\left(f\left(x\right)\right)〉.$$

(ii) The P-image and the R-image of $\mathcal{A}$ under f, denoted by $f^P\left(\mathcal{A}\right)$ and $f^R\left(\mathcal{A}\right)$, are cubic sets in Y, respectively defined as follows: for each $y\in Y$,

$$f^P\left(\mathcal{A}\right)\left(y\right)=\left\{\begin{array}{c}〈\underset{x\in f^{-1}\left(y\right)}{\bigvee }\tilde{A}\left(x\right),\underset{x\in f^{-1}\left(y\right)}{\bigvee }A\left(x\right)〉if{f}^{-1}\left(y\right)\ne \varnothing \hfill \\ 〈[0,0],0〉otherwise,\hfill \end{array}\right.$$

$$f^R\left(\mathcal{A}\right)\left(y\right)=\left\{\begin{array}{c}〈\underset{x\in f^{-1}\left(y\right)}{\bigvee }\tilde{A}\left(x\right),\underset{x\in f^{-1}\left(y\right)}{\bigwedge }A\left(x\right)〉if{f}^{-1}\left(y\right)\ne \varnothing \hfill \\ 〈[0,0],1〉otherwise.\hfill \end{array}\right.$$

In fact, $f^P\left(\mathcal{A}\right)=〈f(\tilde{A}),f\left(A\right)〉$ and $f^R\left(\mathcal{A}\right)=〈f(\tilde{A}),f^R\left(A\right)〉$, where $f(\tilde{A})$ and $f\left(A\right)$ denote the image of $\tilde{A}$ and A under f, respectively, and $f^R\left(A\right)$ is a fuzzy set in X defined as the second component of $f^R\left(\mathcal{A}\right)$ (see [19] for the definition of $f^R\left(\mathcal{A}\right)$).

3. Cubic Subgroupoids and Cubic Ideals

In this section, we introduce the concepts of cubic groupoids and cubic ideals of a groupoid, and obtain their various properties and give some examples related to them. We define the level set of a cubic set of a groupoid and we give the relationship between a cubic subgroupoid of a groupoid and its level set. Additionally, we obtain the relationship between a cubic left ideal (resp., right ideal and ideal) and its level set. Moreover, we provide a sufficient condition that the image of a cubic subgroupoid (resp., left ideal, right ideal and ideal) under a groupoid homomorphism is also a cubic subgroupoid (resp., left ideal, right ideal and ideal).

Definition 5

([30]). For a groupoid $(X,·)$, let $A,B\in I^X$. Then, the product $A{\circ }_{{}_F}B$ of A and B is a fuzzy set in X defined as follows: for each $x\in X$,

$$\left(A{\circ }_{{}_F}B\right)\left(x\right)=\left\{\begin{array}{c}\underset{yz=x}{\bigvee }[A\left(y\right)\wedge B\left(z\right)]if yz=x\hfill \\ 0otherwise.\hfill \end{array}\right.$$

Definition 6

([31]). For a groupoid $(X,·)$, let $\tilde{A},\tilde{B}\in IVFS\left(X\right)$. Then, the product $\tilde{A}{\circ }_{{}_{IVF}}\tilde{B}$ of $\tilde{A}$ and $\tilde{B}$ is an IVFS in X, defined as follows: for each $x\in X$,

$$(\tilde{A}{\circ }_{{}_{IVF}}\tilde{B})\left(x\right)=\left\{\begin{array}{c}\underset{yz=x}{\bigvee }[A^{-}\left(y\right)\wedge B^{-}\left(z\right)],\underset{yz=x}{\bigvee }[A^{+}\left(y\right)\wedge B^{+}\left(z\right)]if yz=x\hfill \\ \left[0,0\right]otherwise.\hfill \end{array}\right.$$

By using the definitions of the inf and the sup of cubic numbers, we can define the product of two cubic sets as follows.

Definition 7.

For a groupoid $(X,·)$, let $\mathcal{A},\mathcal{B}\in CS\left(X\right)$. Then, two types product of $\mathcal{A}$ and $\mathcal{B}$, denoted by $\mathcal{A}{\circ }_{{}_P}\mathcal{B}$ and $\mathcal{A}{\circ }_{{}_R}\mathcal{B}$, are cubic sets in X defined as follows: for each $x\in X$,

$$\left(\mathcal{A}{\circ }_{{}_P}\mathcal{B}\right)\left(x\right)=\left\{\begin{array}{c}\underset{yz=x}{\overset{P}{\bigvee }}\left[\mathcal{A}\left(y\right){\wedge }^P\mathcal{B}\left(z\right)\right]if yz=x\hfill \\ 〈[0,0],0〉otherwise,\hfill \end{array}\right.$$

$$\left(\mathcal{A}{\circ }_{{}_R}\mathcal{B}\right)\left(x\right)=\left\{\begin{array}{c}\underset{yz=x}{\overset{R}{\bigvee }}\left[\mathcal{A}\left(y\right){\wedge }^R\mathcal{B}\left(z\right)\right]if yz=x\hfill \\ 〈[0,0],1〉otherwise.\hfill \end{array}\right.$$

Remark 1.

From Definitions 5–7, we can easily see that the following holds:

(1) $\mathcal{A}{\circ }_{{}_P}\mathcal{B}=〈\tilde{A}{\circ }_{{}_{IVF}}\tilde{B},A{\circ }_{{}_F}B〉,$

(2) $\mathcal{A}{\circ }_{{}_R}\mathcal{B}=〈\tilde{A}{\circ }_{{}_{IVF}}\tilde{B},A{\circ }_{{}_R}B〉,$ where

$$\left(A{\circ }_{{}_R}B\right)\left(x\right)=\left\{\begin{array}{c}\underset{yz=x}{\bigwedge }[A\left(y\right)\vee B\left(z\right)]if yz=x\hfill \\ 1otherwise.\hfill \end{array}\right.$$

Example 1.

Let $X=\{a,b,c\}$ be the groupoid with the following Cayley table (see

Table 1. The Cayley table of the groupoid X.

·	a	b	c
a	a	a	a
b	b	a	b
c	c	c	a

):

Consider two octahedron sets $\mathcal{A}$ and $\mathcal{B}$ in X, respectively given by:

$$\mathcal{A}\left(a\right)=〈[0.3,0.6],0.5〉,\mathcal{A}\left(b\right)=〈[0.2,0.4],0.7〉,\mathcal{A}\left(c\right)=\left([0.4,0.7],0.3\right.〉,$$

$$\mathcal{B}\left(a\right)=〈[0.2,0.6],0.7〉,\mathcal{B}\left(b\right)=〈[0.3,0.5],0.6〉,\mathcal{B}\left(c\right)=〈[0.4,0.7],0.8〉.$$

Then, we can easily calculate $\mathcal{A}{\circ }_{{}_P}\mathcal{B}$ and $\mathcal{A}{\circ }_{{}_R}\mathcal{B}$ (see

Table 2. The products of $\mathcal{A}$ and $\mathcal{B}$.

	$\left(\mathcal{A}{\circ }_{{}_{\mathit{P}}}\mathcal{B}\right)\left(\mathit{t}\right)$	$\left(\mathcal{A}{\circ }_{{}_{\mathit{R}}}\mathcal{B}\right)\left(\mathit{t}\right)$
a	$〈[0.4,0.7],0.6〉$	$〈[0.4,0.7],0.6〉$
b	$〈[0.2,0.4],0.7〉$	$〈[0.2,0.4],0.7〉$
c	$〈[0.3,0.6],0.3〉$	$〈[0.3,0.6],0.7〉$

):

$x_{{}_a}\in I^X$ is called a fuzzy point with the support $x\in X$ and the value $a\in I$ with $a>0$ (See [32]), if for each $y\in X$,

$$x_{{}_a}\left(y\right)=\left\{\begin{array}{c}a\mathrm{if} y=x\hfill \\ 0\mathrm{otherwise}.\hfill \end{array}\right.$$

We denote the set of all fuzzy points in X by $F_P\left(X\right)$.

It is well known [33] that $A={\bigcup }_{x_{{}_a}\in A}{x}_{{}_a}$ for each $A\in I^X$.

$x_{{}_{\tilde{a}}}\in IVFS\left(X\right)$ is called an interval-valued fuzzy point (briefly, IVFP) with the support $x\in X$ and the value $\tilde{a}\in \left[I\right]$ with $a^{-}>0$ (see [28]), if for each $y\in X$,

$$x_{{}_{\tilde{a}}}\left(y\right)=\left\{\begin{array}{c}\tilde{a}\mathrm{if} y=x\hfill \\ \left[0,0\right]\mathrm{otherwise}.\hfill \end{array}\right.$$

We denote the set of all IVFPs in X by $IV{F}_P\left(X\right)$.

It is well known [28] that $\tilde{A}={\bigcup }_{x_{{}_{\tilde{a}}}\in \tilde{A}}{x}_{{}_{\tilde{a}}}$ for each $\tilde{A}\in {\left[I\right]}^X$.

Definition 8

([34]). Let $\mathcal{A}\in CS\left(X\right)$ and let $\tilde{\tilde{a}}=〈\tilde{a},a〉$ be any cubic number such that $a^{-}>0$ and $a>0$. Then, $\mathcal{A}$ is called a cubic point in X with the support $x\in X$ and the value $\tilde{\tilde{a}}$, denoted by $x_{{}_{\tilde{\tilde{a}}}}$, if for each $y\in X$,

$$x_{{}_{\tilde{\tilde{a}}}}\left(y\right)=\left\{\begin{array}{c}\tilde{\tilde{a}}if y=x\hfill \\ 〈[0,0],0〉otherwise.\hfill \end{array}\right.$$

The set of all cubic points in X is denoted by $C_P\left(X\right)$. It is clear that $x_{{}_{\tilde{\tilde{a}}}}=〈x_{{}_{\tilde{a}}},x_{{}_a}〉$.

Definition 9

([34]). Let $x_{{}_{\tilde{\tilde{a}}}}\in C_P\left(X\right)$ and let $\mathcal{A}\in CS\left(X\right)$.

(i) $x_{{}_{\tilde{\tilde{a}}}}$ is said to P-belong to $\mathcal{A}$, denoted by $x_{{}_{\tilde{\tilde{a}}}}{\in }_P\mathcal{A}$, if

$\tilde{a}\le \tilde{A}\left(x\right)$ and $a\le A\left(x\right)$, i.e., $x_{{}_{\tilde{a}}}\in \tilde{A}$ and $x_{{}_a}\in A$.

(ii) $x_{{}_{\tilde{\tilde{a}}}}$ is said to R-belong to $\mathcal{A}$, denoted by $x_{{}_{\tilde{\tilde{a}}}}{\in }_R\mathcal{A}$, if

$\tilde{a}\le \tilde{A}\left(x\right)$ and $a\ge A\left(x\right)$, i.e., $x_{{}_{\tilde{a}}}\in \tilde{A}$ and $x_{{}_{1-a}}\in A^c$.

It is obvious that $\mathcal{A}={\bigsqcup }_{{}_{x_{{}_{\tilde{\tilde{a}}}}{\in }_P\mathcal{A}}}{x}_{{}_{\tilde{\tilde{a}}}}$ and $\mathcal{A}={⋓}_{{}_{x_{{}_{\tilde{\tilde{a}}}}{\in }_R\mathcal{A}}}{x}_{{}_{\tilde{\tilde{a}}}}$ for each $\mathcal{A}\in CS\left(X\right)$

Proposition 1.

For a groupoid $(X,·)$, let $\mathcal{A},\mathcal{B}\in CS\left(X\right)$ and let $x_{\tilde{\tilde{a}}},y_{\tilde{\tilde{b}}}\in C_P\left(X\right)$. Then, we have

(1) $x_{\tilde{\tilde{a}}}{\circ }_{{}_P}{y}_{\tilde{\tilde{b}}}=〈{\left(xy\right)}_{\tilde{a}\wedge \tilde{b}},{\left(xy\right)}_{a\wedge b}〉,x_{\tilde{\tilde{a}}}{\circ }_{{}_R}{y}_{\tilde{\tilde{b}}}=〈{\left(xy\right)}_{\tilde{a}\wedge \tilde{b}},{\left(xy\right)}_{a\vee b}〉,$

(2) $\mathcal{A}{\circ }_{{}_P}\mathcal{B}={\bigsqcup }_{{}_{x_{\tilde{\tilde{a}}}{\in }_P\mathcal{A},y_{\tilde{\tilde{b}}}{\in }_P\mathcal{B}}}{x}_{\tilde{\tilde{a}}}{\circ }_P{y}_{\tilde{\tilde{b}}}$, $\mathcal{A}{\circ }_{{}_R}\mathcal{B}={⋓}_{{}_{x_{\tilde{\tilde{a}}}{\in }_R\mathcal{A},y_{\tilde{\tilde{b}}}{\in }_R\mathcal{B}}}{x}_{\tilde{\tilde{a}}}{\circ }_R{y}_{\tilde{\tilde{b}}}$.

Proof. 

(1) The proofs are straightforward from Definition 7, Remark 1, Proposition 3.2 (a) in [31] and Proposition 1.1 (i) in [30].

(2) The proof of the first part follows from Definition 7, Remark 1, Proposition 3.2 (b) in [31] and Proposition 1.1 (ii) in [30]. Then, we prove only the second part. From Remark 1 (2), it is sufficient to show that $A{\circ }_{{}_R}B={\bigcap }_{x_{{}_a}{\in }_{R}A,y_{{}_b}{\in }_{R}B}{x}_{{}_a}{\circ }_{{}_R}{y}_{{}_b}.$

Let $C={\bigcap }_{x_{{}_a}{\in }_{R}A,y_{{}_b}{\in }_{R}B}{x}_{{}_a}{\circ }_{{}_R}{y}_{{}_b}$. For each $z\in X$, we may suppose that there are $u,v\in X$, such that $uv=z,$$x_{{}_a}\ne \mathbf{1}$ and $y_{{}_b}\ne \mathbf{1}$ without loss of generality. Then, we can easily check that

$$\left(A{\circ }_{{}_R}B\right)\left(z\right)=\underset{z=uv}{\bigwedge }[A\left(u\right)\vee B\left(v\right)]\le (\bigcap _{x_{{}_a}{\in }_{R}A,y_{{}_b}{\in }_{R}B}{x}_{{}_a}{\circ }_{{}_R}{y}_{{}_b})\left(z\right)=C\left(z\right).$$

Since $u_{{}_{A\left(u\right)}}{\in }_{R}A$ and $v_{{}_{B\left(v\right)}}{\in }_{R}B$, we get

$$C\left(z\right)\left(z\right)=\underset{x_{{}_a}{\in }_{R}A,y_{{}_b}{\in }_{R}B}{\bigwedge }\underset{z=uv}{\bigwedge }[x_{{}_a}\left(u\right)\vee y_{{}_b}\left(v\right)]\le \underset{z=uv}{\bigwedge }[u_{{}_{A\left(u\right)}}\vee v_{{}_{B\left(v\right)}}]=\left(A{\circ }_{{}_R}B\right)\left(z\right).$$

Thus, $\left(A{\circ }_{{}_R}B\right)\left(z\right)=C\left(z\right).$ So $\mathcal{A}{\circ }_{{}_R}\mathcal{B}={⋓}_{{}_{x_{\tilde{\tilde{a}}}{\in }_R\mathcal{A},y_{\tilde{\tilde{b}}}{\in }_R\mathcal{B}}}{x}_{\tilde{\tilde{a}}}{\circ }_R{y}_{\tilde{\tilde{b}}}$. □

The following are the immediate consequences of Definition 7.

Proposition 2.

Let $(X,·)$ be a groupoid.

(1) If “·" is associative (resp., commutative) in X, then so are “${\circ }_{{}_P}$" and “${\circ }_{{}_R}$" in $CS\left(X\right)$.

(2) If “·" has an identity $e\in X$, then we have

  ($2_a$) $e_{{}_{\widehat{1}}}\in CP\left(X\right)$ is an identity of “${\circ }_{{}_P}$" in $CS\left(X\right)$, i.e.,

$\mathcal{A}{\circ }_{{}_P}{e}_{{}_{\widehat{1}}}=e_{{}_{\widehat{1}}}{\circ }_{{}_P}\mathcal{A}=\mathcal{A}$ for each $\mathcal{A}\in CS\left(X\right)$,

  ($2_b$) $e_{{}_{\ddot{1}}}\in IVI{O}_P\left(X\right)$ is an identity of “${\circ }_{{}_R}$" in $CS\left(X\right)$, i.e.,

$\mathcal{A}{\circ }_{{}_R}{e}_{{}_{\ddot{1}}}=e_{{}_{\ddot{1}}}{\circ }_{{}_R}\mathcal{A}=\mathcal{A}$ for each $\mathcal{A}\in CS\left(X\right)$.

Definition 10.

For a groupoid $(X,·)$, let $\mathcal{A}\in CS\left(X\right)$. Then,

(i) $\widehat{0}\ne \mathcal{A}$ is called a P-cubic subgroupoid in X, if $\mathcal{A}{\circ }_{{}_P}\mathcal{A}⊏\mathcal{A}$, i.e.,

$$\tilde{A}{\circ }_{{}_{IVF}}\tilde{A}\subset \tilde{A},A{\circ }_{{}_F}A\subset A,$$

(ii) $\ddot{0}\ne \mathcal{A}$ is called an R-cubic subgroupoid in X, if $\mathcal{A}{\circ }_{{}_R}\mathcal{A}⋐\mathcal{A}$, i.e.,

$$\tilde{A}{\circ }_{{}_{IVF}}\tilde{A}\subset \tilde{A},A{\circ }_{{}_R}A\supset A.$$

$PCGP\left(X\right)$ [resp., $RCGP\left(X\right)$] denotes the set of all P-[resp., R-] cubic subgroipoids in X. In (ii), if $A{\circ }_{{}_R}A\supset A$, then A is called a fuzzy anti-subgroupoid in X and $FAGP\left(X\right)$ denotes the set of all fuzzy anti-subgroupoids in X.

Let $FGP\left(X\right)$ [resp., $IVFGP\left(X\right)$] denote the set of all fuzzy [resp., interval-valued fuzzy] subgroupoids in a groupoid X in the sense of Liu [30] [resp., Kang and Hur [31]].

Remark 2.

(1) For a groupoid X, let $\mathcal{A}\in CS\left(X\right).$ Then,

  (1${}_a$) $\mathcal{A}\in PCGP\left(X\right)$ if and only if $\tilde{A}\in IVFGP\left(X\right),A\in FGP\left(X\right)$,

  (1${}_b$) $\mathcal{A}\in RCGP\left(X\right)$ if and only if $\tilde{A}\in IVFGP\left(X\right),A\in FAGP\left(X\right)$.

(2) If A is a subgroupoid of a groupoid X, then we can easily see that

$${{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_P}}=〈[{\chi }_{{}_A},{\chi }_{{}_A}],{\chi }_{{}_A}〉\in PCGP\left(X\right)and{{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_R}}=〈[{\chi }_{{}_A},{\chi }_{{}_A}],{\chi }_{{}_{A^c}}〉\in RCGP\left(X\right).$$

(3) For a groupoid X, let $A\in I^X$. If $A\in FGP\left(X\right)$, then clearly,

$$〈[A,A],A〉\in PCGP\left(X\right)and〈[A,A],A^c〉\in RCGP\left(X\right).$$

(4) For a groupoid X, let $\tilde{A}\in IVFS\left(X\right)$. If $\tilde{A}\in IVFGP\left(X\right)$, then

$$〈\tilde{A},A^{-}〉,〈\tilde{A},A^{+}〉\in PCGP\left(X\right)and〈\tilde{A},{\left(A^{-}\right)}^c〉,〈\tilde{A},{\left(A^{+}\right)}^c〉\in RCGP\left(X\right).$$

(5) For a groupoid X, it is clear that for each $\mathcal{A}\in PCGP\left(X\right)$ [resp., $RCGP\left(X\right)$] and each $x\in X$,

$$\mathcal{A}\left(x^n\right){\ge }_P\mathcal{A}\left(x\right)[\mathrm{resp}.\mathcal{A}\left(x^n\right){\ge }_R\mathcal{A}\left(x\right)],$$

where $x^n$ is any composite of x’s.

(6) For a groupoid X, let $\tilde{\tilde{a}}$ be any cubic number. Then, ${\mathcal{C}}_{\tilde{\tilde{a}}}\in PCGP\left(X\right)$ and ${\mathcal{C}}_{\tilde{\tilde{a}}}\in RCGP\left(X\right).$

Example 2.(1) Let $(X,·)$ be the groupoid and $\mathcal{A}$ be the cubic set in X given in Example 1. Then, we can easily calculate that

$$\left(\tilde{A}{\circ }_{{}_{IVF}}\tilde{A}\right)\left(a\right)=[0.4,0.7]\nleqq [0.3,0.6]=\tilde{A}\left(a\right),$$

$$\left(A{\circ }_{{}_R}B\right)\left(a\right)=0.3\ngeqq 0.5=A\left(a\right).$$

Thus, $\mathcal{A}\notin PCGP\left(X\right)$ and $\mathcal{A}\notin RCGP\left(X\right)$.

(2) Let $X=\{a,b,c\}$ be the groupoid with the following Cayley table (see

Table 3. The Cayley table of the groupoid X.

·	a	b	c
a	a	a	a
b	b	a	b
c	c	c	c

):

Consider the cubic sets $\mathcal{A},\mathcal{B}$ in X defined as follows:

$$\mathcal{A}\left(a\right)=〈[0.3,0.6],0.8〉,\mathcal{A}\left(b\right)=〈[0.2,0.4],0.7〉,\mathcal{A}\left(c\right)=〈[0.4,0.7],0.6〉,$$

$$\mathcal{B}\left(a\right)=〈[0.3,0.6],0.6〉,\mathcal{B}\left(b\right)=〈[0.2,0.4],0.7〉,\mathcal{B}\left(c\right)=〈[0.4,0.7],0.8〉.$$

Then, we can easily check that $\mathcal{A}\in PCGP\left(X\right)$ and $\mathcal{B}\in RCGP\left(X\right)$.

From Definitions 7 and 10, Proposition 2 (1) and Remark 2 (1${}_a$), we obtain the following consequence.

Proposition 3.

For a groupoid $(X,·)$, let $\widehat{0}\ne \mathcal{A}\in CS\left(X\right)$. Then, the following are equivalent:

(1) $\mathcal{A}\in PCGP\left(X\right)$,

(2) for any $x_{{}_{\tilde{\tilde{a}}}},y_{{}_{\tilde{\tilde{a}}}}{\in }_P\mathcal{A}$, $x_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_P}{y}_{{}_{\tilde{\tilde{a}}}}{\in }_P\mathcal{A},$ i.e., $(\mathcal{A},{\circ }_{{}_P})$ is a groupoid,

(3) for any $x,y\in X$, $\mathcal{A}\left(xy\right){\ge }_P\mathcal{A}\left(x\right){\wedge }^P\mathcal{A}\left(y\right),$ i.e.,

  (i) $A^{-}\left(xy\right)\ge A^{-}\left(x\right)\wedge A^{-}\left(y\right),A^{+}\left(xy\right)\ge A^{+}\left(x\right)\wedge A^{+}\left(y\right),$

  (ii) $A\left(xy\right)\ge A\left(x\right)\wedge B\left(y\right).$

From Remark 2 (1${}_a$) and the above Proposition, it is obvious that $(\mathcal{A},{\circ }_{{}_P})$ is a groupoid if and only if $(\tilde{A},{\circ }_{{}_{IVF}})$ and $(A,{\circ }_{{}_F})$ are groupoids.

Proposition 4.

For a groupoid $(X,·)$, let $\mathcal{A}\in PCGP\left(X\right)$.

(1) If “·” is associative in X, then so is “${\circ }_{{}_P}$” in $\mathcal{A}$, i.e., for any $x_{{}_{\tilde{\tilde{a}}}},y_{{}_{\tilde{\tilde{a}}}},z_{{}_{\tilde{\tilde{a}}}}{\in }_P\mathcal{A},$

$$\left(x_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_P}{y}_{{}_{\tilde{\tilde{a}}}}\right){\circ }_{{}_P}{z}_{{}_{\tilde{\tilde{a}}}}=x_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_P}\left(y_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_P}{z}_{{}_{\tilde{\tilde{a}}}}\right),$$

(2) If “·" is commutative in X, then so is “${\circ }_{{}_P}$" in $\mathcal{A}$, i.e., for any $x_{{}_{\tilde{\tilde{a}}}},y_{{}_{\tilde{\tilde{a}}}}{\in }_P\mathcal{A},$

$$x_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_P}{y}_{{}_{\tilde{\tilde{a}}}}=y_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_P}{x}_{{}_{\tilde{\tilde{a}}}},$$

(3) If “·" has an identity $e\in X$, then for each $x_{{}_{\tilde{\tilde{a}}}}{\in }_P\mathcal{A},$

$$e_{{}_{\widehat{1}}}{\circ }_{{}_P}{x}_{{}_{\tilde{\tilde{a}}}}=x_{{}_{\tilde{\tilde{a}}}}=x_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_P}{e}_{{}_{\widehat{1}}}.$$

From Definitions 7 and 10, Proposition 2 (1) and Remark 2 (1${}_b$), we have the following.

Proposition 5.

For a groupoid $(X,·)$, let $\ddot{0}\ne \mathcal{A}\in CS\left(X\right)$. Then, the following are equivalent:

(1) $\mathcal{A}\in RCGP\left(X\right)$,

(2) for any $x_{{}_{\tilde{\tilde{a}}}},y_{{}_{\tilde{\tilde{a}}}}{\in }_R\mathcal{A}$, $x_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_R}{y}_{{}_{\tilde{\tilde{a}}}}{\in }_R\mathcal{A},$ i.e., $(\mathcal{A},{\circ }_{{}_R})$ is a groupoid,

(3) for any $x,y\in X$, $\mathcal{A}\left(xy\right){\ge }_R\mathcal{A}\left(x\right){\wedge }^R\mathcal{A}\left(y\right),$ i.e.,

(i) $A^{-}\left(xy\right)\ge A^{-}\left(x\right)\wedge A^{-}\left(y\right),A^{+}\left(xy\right)\ge A^{+}\left(x\right)\wedge A^{+}\left(y\right),$

(ii) $A\left(xy\right)\le A\left(x\right)\vee B\left(y\right).$

Proposition 6.

For a groupoid $(X,·)$, let $\mathcal{A}\in RCGP\left(X\right)$.

(1) If “·" is associative in X, then so is “${\circ }_{{}_R}$" in $\mathcal{A}$, i.e., for any $x_{{}_{\tilde{\tilde{a}}}},y_{{}_{\tilde{\tilde{a}}}},z_{{}_{\tilde{\tilde{a}}}}{\in }_R\mathcal{A},$

$$\left(x_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_R}{y}_{{}_{\tilde{\tilde{a}}}}\right){\circ }_{{}_R}{z}_{{}_{\tilde{\tilde{a}}}}=x_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_R}\left(y_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_R}{z}_{{}_{\tilde{\tilde{a}}}}\right),$$

(2) If “·" is commutative in X, then so is “${\circ }_{{}_R}$" in $\mathcal{A}$, i.e., for any $x_{{}_{\tilde{\tilde{a}}}},y_{{}_{\tilde{\tilde{a}}}}{\in }_R\mathcal{A},$

$$x_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_R}{y}_{{}_{\tilde{\tilde{a}}}}=y_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_R}{x}_{{}_{\tilde{\tilde{a}}}},$$

(3) If “·" has an identity $e\in X$, then for each $x_{{}_{\tilde{\tilde{a}}}}{\in }_R\mathcal{A},$

$$e_{{}_{\ddot{1}}}{\circ }_{{}_R}{x}_{{}_{\tilde{\tilde{a}}}}=x_{{}_{\tilde{\tilde{a}}}}=x_{{}_{\tilde{\tilde{a}}}}{\circ }_{{}_R}{e}_{{}_{\ddot{1}}}.$$

Definition 11.

For a groupoid $(X,·)$, let $\mathcal{A}\in CS\left(X\right)$. Then, $\mathcal{A}$ is called a:

(i) P-cubic left ideal (briefly, PCLI) of X, if for any $x,y\in X$,

$$\mathcal{A}\left(xy\right){\ge }_P\mathcal{A}\left(y\right),$$

(ii) P-cubic right ideal (briefly, PCRI) of X, if for any $x,y\in X$,

$$\mathcal{A}\left(xy\right){\ge }_P\mathcal{A}\left(x\right),$$

(iii) P-cubic ideal (briefly, PCI) of X, if it is both a PCLI and a PCRI of X,

(iv) R-cubic left ideal (briefly, RCLI) of X, if for any $x,y\in X$,

$$\mathcal{A}\left(xy\right){\ge }_R\mathcal{A}\left(y\right),$$

(v) R-cubic right ideal (briefly, RCRI) of X, if for any $x,y\in X$,

$$\mathcal{A}\left(xy\right){\ge }_R\mathcal{A}\left(x\right),$$

(vi) R-cubic ideal (briefly, RCI) of X, if it is both an RCLI and an RCRI of X.

We will denote the set of all PCIs [resp., PCLIs, PCRIs, RCIs, RCLIs and RCRIs] of X as $PCI\left(X\right)$ [resp., $PCLI\left(X\right)$, $PCRI\left(X\right)$, $RCI\left(X\right)$, $RCLI\left(X\right)$ and $RCRI\left(X\right)$].

For a groupoid X, let $FI\left(X\right)$ [resp., $FLI\left(X\right)$ and $FRL\left(X\right)$] and $IVFI\left(X\right)$ [resp., $IVFLI\left(X\right)$ and $IVFRL\left(X\right)$] denote the set of all fuzzy ideals [resp., left ideals and right ideals] (See [35]) and the set of all IVFIs [resp., IVLIs, IVRIs] (See [31]) of X.

Remark 3.

From Definition 11, we have the following (a)–(f).

(a) $\mathcal{A}\in PCLI\left(X\right)⟺$ for any $x,y\in X$,

$$\tilde{A}\left(xy\right)\ge \tilde{A}\left(y\right),A\left(xy\right)\ge A\left(y\right),i.e.,$$

$$A^{-}\left(xy\right)\ge A^{-}\left(y\right),A^{+}\left(xy\right)\ge A^{+}\left(y\right),A\left(xy\right)\ge A\left(y\right).$$

(2)

Consequently, $\mathcal{A}\in PCLI\left(X\right)⟺\tilde{A}\in IVFLI\left(X\right),A\in FLI\left(x\right)$.

(b) $\mathcal{A}\in RCLI\left(X\right)⟺\tilde{A}\in IVFLI\left(X\right)$ and A satisfies the condition (3),

$$A\left(xy\right)\le A\left(y\right)\mathsf{f}\mathsf{o}\mathsf{r}\mathsf{a}\mathsf{n}\mathsf{y}x,y\in X,$$

(3)

The fuzzy set A satisfying the condition (3) will be called a fuzzy left anti-ideal in X, and we denote the set of all fuzzy left anti-ideals in X as $FLAI\left(X\right).$

(c) $\mathcal{A}\in PCRI\left(X\right)⟺$ for any $x,y\in X$,

$$\tilde{A}\left(xy\right)\ge \tilde{A}\left(x\right),A\left(xy\right)\ge A\left(x\right),i.e.,$$

$$A^{-}\left(xy\right)\ge A^{-}\left(x\right),A^{+}\left(xy\right)\ge A^{+}\left(x\right),A\left(xy\right)\ge A\left(x\right).$$

(4)

Consequently, $\mathcal{A}\in PCRI\left(X\right)⟺\tilde{A}\in IVFRI\left(X\right),A\in FRI\left(X\right).$

(d) $\mathcal{A}\in RCRI\left(X\right)⟺\tilde{A}\in IVFRI\left(X\right)$ and A satisfies the condition (5),

$$A\left(xy\right)\le A\left(x\right)foranyx,y\in X.$$

(5)

The fuzzy set A satisfying the condition (5) will be called a fuzzy right anti-ideal in X and we denote the set of all fuzzy anti-right ideals in X as $FRAI\left(X\right).$

(e) $\mathcal{A}\in PCI\left(X\right)⟺$ for any $x,y\in X$,

$$\tilde{A}\left(xy\right)\ge \tilde{A}\left(x\right)\vee \tilde{A}\left(y\right),A\left(xy\right)\ge A\left(x\right)\vee A\left(y\right),\tilde{A}i.e.,$$

$$A^{-}\left(xy\right)\ge A^{-}\left(x\right)\vee A^{-}\left(y\right),A^{+}\left(xy\right)\ge A^{+}\left(x\right)\vee A^{+}\left(y\right),A\left(xy\right)\ge A\left(x\right)\vee A\left(y\right).$$

(6)

consequently, $\mathcal{A}\in RCI\left(X\right)⟺\tilde{A}\in IVFI\left(X\right),A\in FI\left(X\right)$.

(f) $\mathcal{A}\in RCI\left(X\right)⟺\tilde{A}\in IVFI\left(X\right)$ and A satisfies the condition (7),

$$A\left(xy\right)\le A\left(x\right)\wedge A\left(y\right).$$

(7)

The fuzzy set A satisfying the condition (7) will be called a fuzzy anti-ideal in X, and we denote the set of all fuzzy anti-ideals in X as $FAI\left(X\right).$

(g) For a groupoid X, let $A\in I^X$. If $A\in FLI\left(X\right)$ [resp., $FRI\left(X\right)$ and $FI\left(X\right)$], then $〈[A,A],A〉\in PCLI\left(X\right)$ [resp., $PCRI\left(X\right)$ and $PCI\left(X\right)$] and $〈[A,A],A^c〉\in RCLI\left(X\right)$ [resp., $RCRI\left(X\right)$ and $RCI\left(X\right)$].

(h) For a groupoid X, let $A\in I^X$. If $A\in FLAI\left(X\right)$ [resp., $FRAI\left(X\right)$ and $FAI\left(X\right)$], then $〈[A^c,A^c],A〉\in RCLI\left(X\right)$ [resp., $RCRI\left(X\right)$ and $RCI\left(X\right)$] and $〈[A^c,A^c],A^c〉\in PCLI\left(X\right)$ [resp., $PCRI\left(X\right)$ and $PCI\left(X\right)$].

(i) For a groupoid X, let $\tilde{A}\in IVFS\left(X\right)$. If $\tilde{A}\in IVFLI\left(X\right)$ [resp., $IVFRI\left(X\right)$ and $IVFI\left(X\right)$], then $〈\tilde{A},A^{-}〉,$$〈\tilde{A},A^{+}〉\in PCLI\left(X\right)$ [resp., $PCRI\left(X\right)$ and $PCI\left(X\right)$] and $〈\tilde{A},{\left(A^{-}\right)}^c〉,$$〈\tilde{A},{\left(A^{+}\right)}^c〉$$\in RCLI\left(X\right)$ [resp., $RCRI\left(X\right)$ and $RCI\left(X\right)$]

Remark 4.

(1) For a groupoid X, let $\tilde{\tilde{a}}$ be any cubic number. Then, ${\mathcal{C}}_{\tilde{\tilde{a}}}\in PCLI\left(X\right)$

$[\mathrm{resp}.\mathrm{PCRI}(\mathsf{X}),\mathsf{P}\mathsf{C}\mathsf{I}(\mathsf{X}\left)\right]$ and ${\mathcal{C}}_{\tilde{\tilde{a}}}\in RCLIP\left(X\right)[\mathsf{r}\mathsf{e}\mathsf{s}\mathsf{p}.\mathsf{R}\mathsf{C}\mathsf{R}\mathsf{I}\left(X\right),\mathsf{R}\mathsf{C}\mathsf{I}\left(\mathsf{X}\right)].$

(2) A PCLI [resp., PCRI, PCI, RCLI, RCRI and RCI] in a semigroup S, a group G and a ring R is defined as Definition 11.

(3) It is obvious that $\mathcal{A}\in PCGP\left(X\right)$ for each $\mathcal{A}\in PCI\left(X\right)$ [resp., $PCLI\left(X\right)$ and $PCRI\left(X\right)$] and $\mathcal{A}\in RCGP\left(X\right)$ for each $\mathcal{A}\in RCI\left(X\right)$ [resp., $RCLI\left(X\right)$ and $RCRI\left(X\right)$] but the converses are not true in general (see Example 3 (1)).

Example 3.

(1) Let $(X,·)$ be the groupoid and $\mathcal{A}\in PCGP\left(X\right)$ given in Example 2 (2). Then, clearly, $A\left(ab\right)=0.5\ngeqq 0.7=A\left(b\right)$. Thus, $A\notin FI\left(X\right)$. So $\mathcal{A}\notin PCLI\left(X\right)$.

(2) Let $X=\{a,b,c\}$ be the groupoid with the following Cayley table (see

Table 4. The Cayley table of the groupoid X.

·	a	b	c
a	a	a	a
b	a	a	c
c	a	b	c

):

Consider two cubic sets $\mathcal{A}$ and $\mathcal{B}$ in X given by:

$$\mathcal{A}\left(a\right)=〈[0.4,0.8],0.8〉,\mathcal{A}\left(b\right)=〈[0.3,0.7],0.7〉,\mathcal{A}\left(c\right)=〈[0.2,0.6],0.5〉,$$

$$\mathcal{B}\left(a\right)=〈[0.4,0.8],0.5〉,\mathcal{B}\left(b\right)=〈[0.3,0.7],0.7〉,\mathcal{B}\left(c\right)=〈[0.2,0.6],0.8〉.$$

Then, we can easily calculate that $\mathcal{A}\in PCLI\left(X\right)$ and $\mathcal{B}\in RCLI\left(X\right)$. But $A^{-}\left(bc\right)=B^{-}\left(bc\right)=0.2\ngeqq 0.3=A^{-}\left(b\right)=B^{-}\left(b\right)$. Thus, $\tilde{A},\tilde{B}\notin IVFRI\left(X\right)$. So $\mathcal{A}\notin PCRI\left(X\right)$ and $\mathcal{B}\notin RCRI\left(X\right)$.

(3) Let $X=\{a,b,c\}$ be the groupoid with the following Cayley table (see

Table 5. The Cayley table of the groupoid X.

·	a	b	c
a	a	a	a
b	b	b	a
c	c	a	c

):

Consider two cubic sets $\mathcal{A}$ and $\mathcal{B}$ in X given by:

$$\mathcal{A}\left(a\right)=〈[0.4,0.8],0.9〉,\mathcal{A}\left(b\right)=〈[0.3,0.7],0.7〉,\mathcal{A}\left(c\right)=〈[0.2,0.6],0.8〉,$$

$$\mathcal{B}\left(a\right)=〈[0.4,0.8],0.9〉,\mathcal{B}\left(b\right)=〈[0.3,0.7],0.7〉,\mathcal{B}\left(c\right)=〈[0.2,0.6],0.8〉.$$

Then, we can easily check that $\mathcal{A}\in PCRI\left(X\right)$ and $\mathcal{B}\in RCRI\left(X\right)$. however, $A^{-}\left(ba\right)=B^{-}\left(ba\right)=0.3\ngeqq 0.4=A^{-}\left(a\right)=B^{-}\left(a\right)$. Thus, $\tilde{A},\tilde{B}\notin IVFLI\left(X\right)$. So $\mathcal{A}\notin PCLI\left(X\right)$ and $\mathcal{B}\notin RCLI\left(X\right)$.

Theorem 1.

For a groupoid X, let $A\in 2^X$.

(1) ${{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_P}}\in PCLI\left(X\right)$ [resp., $PCRI\left(X\right)$ and $PCI\left(X\right)$] if and only if A is a left ideal [resp., a right ideal and an ideal] of X.

(2) ${{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_R}}\in PCLI\left(X\right)$ [resp., $PCRI\left(X\right)$ and $PCI\left(X\right)$] if and only if A is a left ideal [resp., a right ideal and an ideal] of X.

Proof. 

(1) The proof follows from Propositions 3.7, 6.6 in [31] and 3.2 in [35].

(2) The proof is similar to (1). □

Definition 12.

Let $\tilde{\tilde{a}}$ be any cubic number and let $\mathcal{A}\in CS\left(X\right)$. Then, four subsets ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}$, ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}^{*}$, ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$ and ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}^{*}$ of X are defined as follows:

$${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}=\{x\in X:\mathcal{A}{\ge }_P\tilde{\tilde{a}}\}=\{x\in X:\tilde{A}\left(x\right)\ge \tilde{a},A\left(x\right)\ge a\},$$

$${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}^{*}=\{x\in X:\mathcal{A}{>}_P\tilde{\tilde{a}}\}=\{x\in X:\tilde{A}\left(x\right)>\tilde{a},A\left(x\right)>a\},$$

$${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}=\{x\in X:\mathcal{A}{\ge }_R\tilde{\tilde{a}}\}=\{x\in X:\tilde{A}\left(x\right)\ge \tilde{a},A\left(x\right)\le a\},$$

$${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}^{*}=\{x\in X:\mathcal{A}{>}_R\tilde{\tilde{a}}\}=\{x\in X:\tilde{A}\left(x\right)>\tilde{a},A\left(x\right)<a\}.$$

In this case, ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}$ [resp., ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}^{*}$, ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$ and ${\left[\mathcal{A}\right]}_{\tilde{\tilde{a}}}^{*}$] is called a P-$\tilde{\tilde{a}}$- [resp., a strong P-$\tilde{\tilde{a}}$-, an R-$\tilde{\tilde{a}}$- and a strong R-$\tilde{\tilde{a}}$-] level set of $\mathcal{A}$. In particular, we will denote the subset $\{x\in X:A(x)\le a\}$ of X as ${\left[A\right]}_{R,a}$.

Jun et al. [19] called the subset ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$ of X a cubic level set of $\mathcal{A}$.

Proposition 7.

Let $\tilde{\tilde{a}},\tilde{\tilde{b}}$ be any cubic numbers and let $\mathcal{A}\in CS\left(X\right)$.

(1) If $\tilde{\tilde{a}}{\le }_P\tilde{\tilde{b}}$, then ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{b}}}\subset {\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}$ and ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{b}}}^{*}\subset {\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}^{*}$.

(2) If $\tilde{\tilde{a}}{\le }_R\tilde{\tilde{b}}$, then ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{b}}}\subset {\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$ and ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{b}}}^{*}\subset {\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}^{*}$.

Proof. 

The proofs are straightforward from Definitions 2 and 12. □

Theorem 2.

For a groupoid X, let $\mathcal{A}\in CS\left(X\right)$ and let ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}\ne \varnothing$ or ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}\ne \varnothing$ for each cubic number $\tilde{\tilde{a}}$.

(1) $\mathcal{A}\in PCGP\left(X\right)$ if and only if ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}$ is a subgroupoid.

(2) $\mathcal{A}\in PCLI\left(X\right)$[resp., $PCRI\left(X\right)$ and $PCI\left(X\right)$] if and only if ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}$ is a left ideal[resp., a right ideal and an ideal].

(3) $\mathcal{A}\in RCGP\left(X\right)$ if and only if ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$ is a subgroupoid.

(4) $\mathcal{A}\in PCLI\left(X\right)$[resp., $PCRI\left(X\right)$ and $PCI\left(X\right)$]if and only if ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$ is a left ideal[resp., a right ideal and an ideal].

Proof. 

(1) Suppose $\mathcal{A}\in PCGP\left(X\right)$ and let $x,y\in {\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}$. Then, clearly, $\mathcal{A}\left(x\right){\ge }_P\tilde{\tilde{a}}$ and $\mathcal{A}\left(y\right){\ge }_P\tilde{\tilde{a}}$. Thus, by the hypothesis, $\mathcal{A}\left(xy\right){\ge }_P\mathcal{A}\left(x\right){\wedge }^P\mathcal{A}\left(y\right){\ge }_P\tilde{\tilde{a}}.$ So $xy\in {\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}$. Hence, ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}$ is a subgroupoid.

Conversely, suppose ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}$ is a subgroupoid and let $\mathcal{A}\left(x\right)=\tilde{\tilde{a}},\mathcal{A}\left(y\right)=\tilde{\tilde{b}}$, say $\tilde{\tilde{a}}{\le }_P\tilde{\tilde{b}}.$ Then clearly, $x\in {\left[\mathcal{A}\right]}_{\tilde{\tilde{a}}}$ and $y\in {\left[\mathcal{A}\right]}_{\tilde{\tilde{b}}}$. Moreover, by Proposition 7 (1), ${\left[\mathcal{A}\right]}_{\tilde{\tilde{b}}}\subset {\left[\mathcal{A}\right]}_{\tilde{\tilde{a}}}.$ Thus, $x,y\in {\left[\mathcal{A}\right]}_{\tilde{\tilde{a}}}.$ By the hypothesis, $xy\in {\left[\mathcal{A}\right]}_{\tilde{\tilde{a}}}.$ So, we have

$$\mathcal{A}\left(xy\right){\ge }_P\tilde{\tilde{a}}=\tilde{\tilde{a}}{\wedge }^P\tilde{\tilde{b}}=\mathcal{A}\left(x\right){\wedge }^P\mathcal{A}\left(y\right).$$

Hence, $\mathcal{A}\in PCGP\left(X\right)$.

(2) The proof is similar to (1).

(3) Suppose $\mathcal{A}\in RCGP\left(X\right)$ and let $x,y\in {\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$. Then, clearly, $\mathcal{A}\left(x\right){\ge }_R\tilde{\tilde{a}}$ and $\mathcal{A}\left(y\right){\ge }_R\tilde{\tilde{a}}$. Thus, we have

$$\tilde{A}\left(x\right)\ge \tilde{a},A\left(x\right)\le a,and\tilde{A}\left(y\right)\ge \tilde{a},A\left(y\right)\le a.$$

According to the hypothesis, we get $\tilde{A}\left(xy\right)\ge \tilde{A}\left(x\right)\wedge \tilde{A}\left(y\right)\ge \tilde{a}andA\left(xy\right)\le A\left(x\right)\vee A\left(y\right)\le a.$ So $xy\in {\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$. Hence, ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$ is a subgroupoid.

Conversely, suppose ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$ is a subgroupoid and let $\mathcal{A}\left(x\right)=\tilde{\tilde{a}},\mathcal{A}\left(y\right)=\tilde{\tilde{b}}$, say $\tilde{\tilde{a}}{\le }_R\tilde{\tilde{b}}.$ Then, clearly, $x\in {\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$ and $y\in {\left[\mathcal{A}\right]}_{R,\tilde{\tilde{b}}}$. Moreover, by Proposition 7 (2), ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{b}}}\subset {\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}.$ Thus, $x,y\in {\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}.$ According to the hypothesis, $xy\in {\left[\mathcal{A}\right]}_{\tilde{\tilde{a}}}.$ So, we have

$$\mathcal{A}\left(xy\right){\ge }_R\tilde{\tilde{a}}=\tilde{\tilde{a}}{\wedge }^R\tilde{\tilde{b}}=\mathcal{A}\left(x\right){\wedge }^R\mathcal{A}\left(y\right).$$

Hence, $\mathcal{A}\in RCGP\left(X\right)$.

(4) The proof is similar to (3). □

Proposition 8.

For a groupoid X, the following holds:

(1) If ${\left({\mathcal{A}}_j\right)}_{j\in J}={\left(〈{\tilde{A}}_j,A_j〉\right)}_{j\in J}\subset PCGP\left(X\right)$, then ${\sqcap }_{j\in J}{\mathcal{A}}_j\in PCGP\left(X\right)$.

(2) If ${\left({\mathcal{A}}_j\right)}_{j\in J}={\left(〈{\tilde{A}}_j,A_j〉\right)}_{j\in J}\subset RCGP\left(X\right)$, then ${⋒}_{j\in J}{\mathcal{A}}_j\in RCGP\left(X\right)$.

Proof. 

(1) The proof is obvious from Propositions 3.7 in [31] and 3.1 in [35].

(2) Suppose ${\left({\mathcal{A}}_j\right)}_{j\in J}={\left(〈{\tilde{A}}_j,A_j〉\right)}_{j\in J}\subset RCGP\left(X\right)$. According to Proposition 3.7 in [31], ${\bigcap }_{j\in J}{\tilde{A}}_{j\in J}$$\in IVFGP\left(X\right).$ It is sufficient to show that for any $x,y\in X$,

$$\left(\underset{j\in J}{\bigvee }{A}_j\right)\left(xy\right)\le [\left(\underset{j\in J}{\bigvee }{A}_j\right)\left(x\right)\vee \left(\underset{j\in J}{\bigvee }{A}_j\right)\left(y\right)].$$

Since ${\mathcal{A}}_j\in RCGP\left(X\right)$ for each $j\in J$, by Proposition 5 (3),

$$A_j\left(xy\right)\le A_j\left(x\right)\vee A_j\left(y\right)foreachj\in J.$$

Thus, ${\bigvee }_{j\in J}{A}_j\left(xy\right)\le {\bigvee }_{j\in J}[A_j\left(x\right)\vee A_j\left(y\right)]={\bigvee }_{j\in J}{A}_j\left(x\right)\vee {\bigvee }_{j\in J}{A}_j\left(y\right).$ So, according to Proposition 5, ${⋒}_{j\in J}{\mathcal{A}}_j\in RCGP\left(X\right).$ □

Remark 5.

For any $\mathcal{A},\mathcal{B}\in PCGP\left(X\right)$ [resp., $RCGP\left(X\right)$], $\mathcal{A}\bigsqcup \mathcal{B}\notin PCGP\left(X\right)$ [resp., $\mathcal{A}⋓\mathcal{B}\notin RCGP\left(X\right)$] in general.

Example 4.

Let $(X,·)$ be the groupoid and $\mathcal{A}\in PCGP\left(X\right),\mathcal{B}\in RCGP\left(X\right)$ given in Example 3 (1). Consider two cubic sets $\mathcal{A}$ and $\mathcal{B}$ in X defined as follows: for each $x\in X$,

$$\mathcal{A}\left(a\right)=〈[0.2,0.5],0.6〉,\mathcal{A}\left(b\right)=\mathcal{A}\left(c\right)=〈[0.4,0.8],0.7〉,$$

$$\mathcal{B}\left(a\right)=〈[0.3,0.4],0.7〉,\mathcal{B}\left(b\right)=\mathcal{B}\left(c\right)=〈[0.5,0.7],0.8〉.$$

Then, we can easily see that $\mathcal{A},\mathcal{B}\in PCGP\left(X\right)$. Moreover, we have

$$(\tilde{A}\cup \tilde{B})\left(bb\right)=[0.3,0.5)\neg \ge [0.5,0.8]=(\tilde{A}\cup \tilde{B})\left(b\right)=(\tilde{A}\cup \tilde{B})\left(b\right)\wedge (\tilde{A}\cup \tilde{B})\left(b\right).$$

Thus, $\tilde{A}\cup \tilde{B}\notin IVFGP\left(X\right).$ So $\mathcal{A}\cup \mathcal{B}\notin PCGP\left(X\right).$

Remark 6.

For a groupoid X, let ${\left({\mathcal{A}}_j\right)}_{j\in J}\subset PCGP\left(X\right)$ [resp., $RCGP\left(X\right)$. Then, from Proposition 8, we can easily see that

$$\sqcap \{\mathcal{A}\in PCGP\left(X\right):{\bigsqcup }_{j\in J}{\mathcal{A}}_j⊏\mathcal{A}\}\in PCGP\left(X\right)$$

$$[\mathrm{resp}.⋒\{\mathcal{A}\in RCGP\left(X\right):{⋓}_{j\in J}{\mathcal{A}}_j⋐\mathcal{A}\}\in RCGP\left(X\right)].$$

In this case, we will denote $\sqcap \{\mathcal{A}\in PCGP\left(X\right):{\bigsqcup }_{j\in J}{\mathcal{A}}_j⊏\mathcal{A}\}\in PCGP\left(X\right)$ [resp., $\mathrm{resp}.⋒\{\mathcal{A}\in RCGP\left(X\right):{⋓}_{j\in J}{\mathcal{A}}_j⋐\mathcal{A}\}\in RCGP\left(X\right)$] as ${\bigvee }_{j\in J}^P{\mathcal{A}}_j$ [resp., ${\bigvee }_{j\in J}^R{\mathcal{A}}_j$].

It is obvious that $\left(PCGP\right(X),⊏)$ [resp., $\left(RCGP\right(X),⋐)$] is a complete lattice with the least element $\widehat{0}$ [resp., $\ddot{0}$] and the greatest element $\widehat{1}$ [resp., $\ddot{1}]$, where for each ${\left({\mathcal{A}}_j\right)}_{j\in J}\subset PCGP\left(X\right)$ [resp., $RCGP\left(X\right)$], the inf and the sup of ${\left({\mathcal{A}}_j\right)}_{j\in J}$ are $Pin{f}_{j\in J}{\mathcal{A}}_j={\bigwedge }_{j\in J}^P{\mathcal{A}}_j$ [resp., $Rin{f}_{j\in J}{\mathcal{A}}_j={\bigwedge }_{j\in J}^R{\mathcal{A}}_j$ and $Psu{p}_{j\in J}{\mathcal{A}}_j={\bigvee }_{j\in J}^P{\mathcal{A}}_j$ [resp., $Rsu{p}_{j\in J}{\mathcal{A}}_j={\bigvee }_{j\in J}^R{\mathcal{A}}_j$].

The following is an immediate consequence of Proposition 8.

Corollary 1.

For a groupoid X, let $\mathcal{A}\in CS\left(X\right)$ and let

$${\left(\mathcal{A}\right)}_P=\sqcap \{\mathcal{B}\in PCGP\left(X\right):\mathcal{A}⊏\mathcal{B}\}[\mathrm{resp}.{\left(\mathcal{A}\right)}_R=⋒\{\mathcal{B}\in RCGP\left(X\right):\mathcal{A}⋐\mathcal{B}\}].$$

Then, ${\left(\mathcal{A}\right)}_P\in PCGP\left(X\right)$[resp., ${\left(\mathcal{A}\right)}_R\in RCGP\left(X\right)$].

In this case, ${\left(\mathcal{A}\right)}_P$ [resp., ${\left(\mathcal{A}\right)}_R$] is called the P-[resp., R-]cubic subgroupoid in X generated by$\mathcal{A}$.

Proposition 9.

For a groupoid X, let $\left(A\right)$ be the subgroupoid generated by A and let ${\chi }_{{}_{{\left(\mathcal{A}\right)}_P}}=〈[{\chi }_{{}_{\left(A\right)}},{\chi }_{{}_{\left(A\right)}}],{\chi }_{{}_{\left(A\right)}}〉$[resp., ${\chi }_{{}_{{\left(\mathcal{A}\right)}_R}}=〈[{\chi }_{{}_{\left(A\right)}},{\chi }_{{}_{\left(A\right)}}],{\chi }_{{}_{\left(A^c\right)}}〉$], where $A\in 2^X$. Then, we get

$$\left({{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_P}}\right)={\chi }_{{}_{{\left(\mathcal{A}\right)}_P}}[\mathrm{resp}.\left({{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_R}}\right)={\chi }_{{}_{{\left(\mathcal{A}\right)}_R}}].$$

Proof. 

From Remark 2 (2) and Corollary 1, it is obvious that ${\chi }_{{}_{{\left(\mathcal{A}\right)}_P}}\in PCGP\left(X\right)$ [resp., ${\chi }_{{}_{{\left(\mathcal{A}\right)}_R}}\in RCGP\left(X\right)$]. Let $\mathcal{B}\in PCGP\left(X\right)$ [resp., $\mathcal{B}\in RCGP\left(X\right)$] such that $\mathcal{B}⊐{{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_P}}$ [resp., $\mathcal{B}⋑{{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_R}}$]. Then, clearly, for each $x\in A,$

$$\mathcal{B}\left(x\right)=〈[1,1],1〉[\mathrm{resp}.\mathcal{B}\left(x\right)=〈[1,1],0〉].$$

Since $\mathcal{B}\in PCGP\left(X\right)$ [resp., $\mathcal{B}\in RCGP\left(X\right)$], we have: for any for any $x,y\in A$,

$$\mathcal{B}\left(xy\right)=〈[1,1],1〉[\mathrm{resp}.\mathcal{B}\left(xy\right)=〈[1,1],0〉].$$

Thus $\mathcal{B}⊐{{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_P}}$ [resp., $\mathcal{B}⊐{{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_R}}$]. So, we get

$${\chi }_{{}_{{\left(\mathcal{A}\right)}_P}}⊏\sqcap \{\mathcal{B}\in PCGP\left(X\right):\mathcal{B}⊐{{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_P}}\}=\left({{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_P}}\right)$$

$$[\mathrm{resp}.{\chi }_{{}_{{\left(\mathcal{A}\right)}_R}}⋐⋒\{\mathcal{B}\in RCGP\left(X\right):\mathcal{B}⋑{{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_R}}\}=\left({{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_R}}\right)].$$

We can easily show that $\left({{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_P}}\right)⊏{\chi }_{{}_{{\left(\mathcal{A}\right)}_P}}$ [resp., $\left({{\chi }_{{}_{\mathcal{A}}}}_{{}_{{}_R}}\right)⋐{\chi }_{{}_{{\left(\mathcal{A}\right)}_R}}$]. Hence, the results hold. □

From the above Proposition, the subgoupoid lattice of X can be regarded as a sublattice of the cubic subgroupoid lattice of X. The following is an immediate consequence of Remark 3 and Proposition 3.3 in [35].

Proposition 10.

For a groupoid X, let ${\left({\mathcal{A}}_j\right)}_{j\in J}\subset PCLI\left(X\right)$[resp., $PCRI\left(X\right)$, $PCI\left(X\right)$, $RCLI\left(X\right)$, $RCRI\left(X\right)$ and $RCI\left(X\right)$]. Then ${\sqcap }_{j\in J}{\mathcal{A}}_j\in PCLI\left(X\right)$[resp., $PCRI\left(X\right)$ and $PCI\left(X\right)$], ${⋒}_{j\in J}{\mathcal{A}}_j\in RCLI\left(X\right)$[resp., $RCRI\left(X\right)$ and $RCI\left(X\right)$]and ${\bigsqcup }_{j\in J}{\mathcal{A}}_j\in PCLI\left(X\right)$[resp., $PCRI\left(X\right)$ and $PCI\left(X\right)$], ${⋓}_{j\in J}{\mathcal{A}}_j\in RCLI\left(X\right)$[resp., $RCRI\left(X\right)$ and $RCI\left(X\right)$].

Proposition 11.

Let $f:X\to Y$ be a groupoid homomorphism and let $\mathcal{B}\in CS\left(Y\right)$.

(1) If $\mathcal{B}\in PCGP\left(Y\right)$[resp., $RCGP\left(Y\right)$], then $f^{-1}\left(\mathcal{B}\right)\in PCGP\left(X\right)$[resp., $RCGP\left(X\right)$].

(2) If $\mathcal{B}\in PCLI\left(Y\right)$[resp., $PCRI\left(Y\right)$, $PCI\left(Y\right)$, $RCLI\left(Y\right)$, $RCRI\left(Y\right)$ and $RCI\left(Y\right)$], then $f^{-1}\left(\mathcal{B}\right)\in PCLI\left(X\right)$[resp., $PCRI\left(Y\right)$, $PCI\left(Y\right)$, $RCLI\left(Y\right)$, $RCRI\left(Y\right)$ and $RCI\left(Y\right)$.

Proof. 

(1) Suppose $\mathcal{B}\in PCGP\left(Y\right)$. Then, from Propositions 3.9 (c) in [31] and 4.1 in [35], $f^{-1}\left(\tilde{B}\right)\in IVFGP\left(X\right)$ and $f^{-1}\left(B\right)\in FGP\left(X\right)$. Thus, according to Remark 2 (1), $f^{-1}\left(\mathcal{B}\right)\in PCGP\left(X\right)$.

Now, suppose $\mathcal{B}\in RCGP\left(Y\right)$. Since $f^{-1}\left(\tilde{B}\right)\in IVFGP\left(X\right)$, it is sufficient to show that $f^{-1}\left(B\right)$ is a fuzzy anti-subgroupoid of X. Let $x,y\in X$. Then, we have

$f^{-1}\left(B\right)\left(xy\right)=B\left(f\left(xy\right)\right)=B\left(f\left(x\right)f\left(y\right)\right)$ (Since f is a groupoid homomorphism).

$\le B\left(f\right(x\left)\right)\vee B\left(f\right(y\left)\right)$ (Since B is fuzzy anti-groupoid of Y).

$=f^{-1}\left(B\right)\left(x\right)\vee f^{-1}\left(B\right)\left(y\right).$

Then, $f^{-1}\left(B\right)$ is a fuzzy anti-subgroupoid of X. Thus, according to Remark 2 (1), $f^{-1}\left(\mathcal{B}\right)\in RCGP\left(X\right)$.

(2) Suppose $\mathcal{B}\in RCLI\left(Y\right)$ and let $x,y\in X$. Since $\tilde{B}\in IVFLI\left(Y\right)$, it is sufficient to show that $f^{-1}\left(B\right)$ is a fuzzy anti-left ideal of X. Let $x,y\in X$. Then, we have

$f^{-1}\left(B\right)\left(xy\right)=B\left(f\left(xy\right)\right)=B\left(f\left(x\right)f\left(y\right)\right)$

$\le B\left(f\right(y\left)\right)$ (Since B is a fuzzy anti-left ideal of X).

$=f^{-1}\left(B\right)\left(y\right).$

Thus, $f^{-1}\left(B\right)$ is a fuzzy anti-left ideal of X. So, by Remark 3 (2), $f^{-1}\left(\mathcal{B}\right)\in RCLI\left(X\right)$. The remainder’s proofs are similar. □

Definition 13.

Let $\mathcal{A}\in CS\left(X\right)$. Then, we say that $\mathcal{A}$ has the P-sup-property [resp., R-sup-property], if for each $T\in 2^X$, there is $t_0\in T$, such that

$$\mathcal{A}\left(t_0\right)=\underset{t\in T}{\overset{P}{\bigvee }}\mathcal{A}\left(t\right)=〈\underset{t\in T}{\bigvee }\tilde{A}\left(t\right),\underset{t\in T}{\bigvee }A\left(t\right)〉$$

$$\left[\mathrm{resp}.\mathcal{A}\left(t_0\right)=\underset{t\in T}{\overset{R}{\bigvee }}\mathcal{A}\left(t\right)=〈\underset{t\in T}{\bigvee }\tilde{A}\left(t\right),\underset{t\in T}{\bigwedge }A\left(t\right)〉\right].$$

It is clear that $\mathcal{A}\in CS\left(X\right)$ has the P-sup-property if and only if $\tilde{A}$ and A have the sup-property. Furthermore, if $\mathcal{A}$ takes on only finitely many values, then it has the P-sup-property [resp., R-sup-property]. Jun et al. [19] called “$\mathcal{A}\in CS\left(X\right)$ has the R-sup-property" as $\mathcal{A}$ has the cubic property.

Proposition 12.

Let $f:X\to Y$ be a groupoid homomorphism, let $\mathcal{A}\in CS\left(X\right)$ has the P-sup-property [resp., R-sup-property].

(1) If $\mathcal{A}\in PCGP\left(X\right)$[resp., $RCGP\left(X\right)$], then $f^P\left(\mathcal{A}\right)\in PCGP\left(Y\right)$[resp., $f^R\left(\mathcal{A}\right)\in RCGP\left(Y\right)$].

(2) If $\mathcal{A}\in PCLI\left(X\right)$[resp., $PCRI\left(X\right)$, $PCI\left(X\right)$, $RCLI\left(X\right)$, $RCRI\left(X\right)$ and $RCI\left(X\right)$], then $f^P\left(\mathcal{A}\right)\in PCLI\left(Y\right)$[resp., $PCRI\left(Y\right)$, $PCI\left(Y\right)$]and $f^R\left(\mathcal{A}\right)\in RCLI\left(Y\right)$[resp., $RCRI\left(Y\right)$, $RCI\left(Y\right)$].

Proof. 

(1) Let $\mathcal{A}\in CS\left(X\right)$ have the P-sup-property and suppose $\mathcal{A}\in PCGP\left(X\right)$. Then, according to Remark 2 (1), $\tilde{A}\in IVFGP\left(X\right)$ and $A\in FGP\left(X\right)$. Thus, according to Propositions 3.11 in [31] and 4.2 in [35], $f\left(\tilde{A}\right)\in IVFGP\left(Y\right)$ and $f\left(A\right)\in FGP\left(Y\right)$. So, $f\left(\mathcal{A}\right)\in PCGP\left(Y\right)$.

Now, let $\mathcal{A}\in CS\left(X\right)$ have the R-sup-property and suppose $\mathcal{A}\in RCGP\left(X\right)$. Since $\tilde{A}\in IVFGP\left(X\right)$, it is sufficient to prove that $f^R\left(A\right)$ is a fuzzy anti-subgroupoid of Y Let $y,y^{{}^{\prime }}\in Y$. Then, we have

(i) $f^{-1}\left(y\right)\ne \varnothing$ and $f^{-1}\left(y^{{}^{\prime }}\right)\ne \varnothing$, (ii) $f^{-1}\left(y\right)\ne \varnothing$ and $f^{-1}\left(y^{{}^{\prime }}\right)=\varnothing$,

(iii) $f^{-1}\left(y\right)=\varnothing$ and $f^{-1}\left(y^{{}^{\prime }}\right)\ne \varnothing$, (iv) $f^{-1}\left(y\right)=\varnothing$ and $f^{-1}\left(y^{{}^{\prime }}\right)=\varnothing$.

We prove only the case (i) and omit the remainders. Since $\mathcal{A}$ has the R-sup-property, there are $x_0\in f^{-1}\left(y\right)$ and $x_0^{{}^{\prime }}\in f^{-1}\left(y^{{}^{\prime }}\right)$ such that

$$A\left(x_0\right)=\underset{t\in f^{-1}\left(y\right)}{\bigwedge }A\left(t\right)andA\left(x_0^{{}^{\prime }}\right)=\underset{t^{{}^{\prime }}\in f^{-1}\left(y^{{}^{\prime }}\right)}{\bigwedge }A\left(t^{{}^{\prime }}\right).$$

Then,

$\left[f^R\left(A\right)\right]\left(y{y}^{{}^{\prime }}\right)={\bigwedge }_{z\in f^{-1}\left(y{y}^{{}^{\prime }}\right)}A\left(z\right)$

$\le A\left(x_0{x}_0^{{}^{\prime }}\right)$ (Since $f\left(x_0{x}_0^{{}^{\prime }}\right)=f\left(x_0\right)f\left(x_0^{{}^{\prime }}\right)=y{y}^{{}^{\prime }}$).

$\le [A\left(x_0\right)\vee A\left(x_0^{{}^{\prime }}\right)]$ (Since A is a fuzzy anti-subgroupoid).

$={\bigwedge }_{t\in f^{-1}\left(y\right)}A\left(t\right)\vee {\bigwedge }_{t^{{}^{\prime }}\in f^{-1}\left(y^{{}^{\prime }}\right)}A\left(t^{{}^{\prime }}\right)$

$=\left[f^R\left(A\right)\right]\left(y\right)\vee \left[f^R\left(A\right)\right]\left(y^{{}^{\prime }}\right)$.

Thus, $f^R\left(A\right)$ is a fuzzy anti-subgroupoid in Y. So, $f^R\left(\mathcal{A}\right)\in RCGP\left(Y\right)$.

(2) Since the proof is similar to (1), the proofs are omitted. □

Definition 14.

Let $f:X\to Y$ be a mapping and let $\mathcal{A}\in CS\left(X\right)$. Then, $\mathcal{A}$ is said to be f-invariant, if for any $x,y\in X$, $f\left(x\right)=f\left(y\right)$ implies $\mathcal{A}\left(x\right)=\mathcal{A}\left(y\right).$

It is obvious that $\mathcal{A}$ is f-invariant if and only if $\tilde{A}$ and A are f-invariant. Moreover, we can easily see that if $\mathcal{A}$ is f-invariant, then $f^{-1}\left(f^P\left(\mathcal{A}\right)\right)=\mathcal{A}$ and $f^{-1}\left(f^R\left(\mathcal{A}\right)\right)=\mathcal{A}.$

Example 5.

Let $X=\{a,b,c\},Y=\{x,y\}$ be sets and $f:X\to Y$ be the mapping defined by $f\left(a\right)=f\left(b\right)=x$ and $f\left(c\right)=y$. Consider two IVI-octahedron sets $\mathcal{A},\mathcal{A}$ in X given by:

$$\mathcal{A}\left(a\right)=〈[0.4,0.8],0.8〉,\mathcal{A}\left(b\right)=〈[0.4,0.8],,0.8〉,\mathcal{A}\left(c\right)=〈[0.2,0.6],0.5〉,$$

$$\mathcal{B}\left(a\right)=〈[0.4,0.8],0.5〉,\mathcal{B}\left(b\right)=〈[0.3,0.7],0.7〉,\mathcal{B}\left(c\right)=〈[0.2,0.6],0.8〉.$$

Then, we can easily check that $\mathcal{A}$ is invariant but $\mathcal{B}$ is not invariant. Moreover, we can easily confirm that $f^{-1}\left(f^P\left(\mathcal{A}\right)\right)=\mathcal{A}$ and $f^{-1}\left(f^R\left(\mathcal{A}\right)\right)=\mathcal{A}.$

4. Cubic Subgroups and Cubic Normal Subgroups

Throughout this section and the next section, G and $G^{{}^{\prime }}$ are grouped with the identities e and $e^{{}^{\prime }}$, respectively, unless mentioned.

Definition 15.

Let S be a semigroup and $\mathcal{A}\in CS\left(S\right)$. Then, $\mathcal{A}$ is called a P- [resp., an R-]cubic subsemigroup of S, if it satisfies the following condition: for any $x,y\in S$,

$$\mathcal{A}\left(xy\right){\ge }_P\mathcal{A}\left(x\right){\wedge }^P\mathcal{A}\left(y\right),i.e.,\tilde{A}\left(xy\right)\ge \tilde{A}\left(x\right)\wedge \tilde{A}\left(y\right),A\left(xy\right)\ge A\left(x\right)\wedge A\left(y\right)$$

$$[\mathrm{resp}.\mathcal{A}\left(xy\right){\ge }_R\mathcal{A}\left(x\right){\wedge }^R\mathcal{A}\left(y\right),i.e.,\tilde{A}\left(xy\right)\ge \tilde{A}\left(x\right)\wedge \tilde{A}\left(y\right),A\left(xy\right)\le A\left(x\right)\vee A\left(y\right)].$$

For each $A\in I^S$, if $A\left(xy\right)\le A\left(x\right)\vee A\left(y\right)$ for any $x,y\in S$, then A will be called a fuzzy anti-subsemigroup of S.

Definition 16.

Let $\mathcal{A}\in CS\left(G\right)$. Then, $\mathcal{A}$ is called a P- [resp., R-]cubic subgroup of S, if it satisfies the following conditions: for any $x,y\in S$,

$$\mathcal{A}\left(xy\right){\ge }_P\mathcal{A}\left(x\right){\wedge }^P\mathcal{A}\left(y\right)[\mathrm{resp}.\mathcal{A}\left(xy\right){\ge }_R\mathcal{A}\left(x\right){\wedge }^R\mathcal{A}\left(y\right)],$$

(8)

$$\mathcal{A}\left(x^{-1}\right){\ge }_P\mathcal{A}\left(x\right)[\mathrm{resp}.\mathcal{A}\left(x^{-1}\right){\ge }_R\mathcal{A}\left(x\right)].$$

(9)

We will denote the set of all P- [resp., R-]cubic subgroups of G as $PCG\left(G\right)$ [resp., $RCG\left(G\right)$]. An R-cubic subgroup was called a cubic subgroup by Jun et al. [19].

Let us denote the set of all interval-valued fuzzy subgroups (See [36]) [resp., fuzzy subgroups (See [35]) and fuzzy anti-subgroups (See [37]) of G as $IVFG\left(G\right)$ [resp., $FG\left(G\right)$ and $FAG\left(G\right)$].

Remark 7.

(1) For each $\mathcal{A}\in CS\left(G\right)$,

  (a) $\mathcal{A}\in PCG\left(G\right)⟺\tilde{A}\in IVFG\left(G\right),A\in FG\left(G\right).$

  (b) $\mathcal{A}\in RCG\left(G\right)⟺\tilde{A}\in IVFG\left(G\right),A\in FAG\left(G\right).$

(2) For each $H\in 2^G$.

  (a) ${{\chi }_{{}_{\mathcal{H}}}}_{{}_{{}_P}}\in PCG\left(G\right)$ if and only if H is a subgroup.

  (b) ${{\chi }_{{}_{\mathcal{H}}}}_{{}_{{}_R}}\in RCG\left(G\right)$ if and only if H is a subgroup.

(3) Let $A\in I^G$. If $A\in FG\left(G\right)$, then $〈[A,A],A〉\in PCG\left(G\right)$ and $〈[A,A],A^c〉\in RCG\left(G\right)$.

(4) Let $A\in I^G$. If $A\in FAG\left(G\right)$, then we have

$$〈[A^c,A^c],A〉\in RCG\left(G\right)and〈[A^c,A^c],A^c〉\in PCG\left(G\right).$$

(5) Let $\tilde{A}\in IVFS\left(G\right).$ If $\tilde{A}\in IVFG\left(G\right)$, then we get

$$〈\tilde{A},A^{-}〉,〈\tilde{A},A^{+}〉\in PCG\left(G\right)and〈\tilde{A},{\left(A^{-}\right)}^c〉,〈\tilde{A},{\left(A^{+}\right)}^c〉\in RCG\left(G\right).$$

Example 6.

Consider the additive group $(\mathbb{Z},+)$. Let $\mathcal{A}$ and $\mathcal{B}$ be the cubic sets in $\mathbb{Z}$, defined as follows: for each $0\ne n\in \mathbb{Z}$,

$$\mathcal{A}\left(n\right)=\left\{\begin{array}{c}〈[0.4,0.7],0.6〉if \mathsf{n}\mathrm{is}\mathrm{odd}\hfill \\ 〈[0.6,0.8],0.7〉if \mathsf{n}\mathrm{is}\mathrm{even},\hfill \end{array}\right.$$

$$\mathcal{B}\left(n\right)=\left\{\begin{array}{c}〈[0.4,0.7],0.7〉if \mathsf{n}\mathrm{is}\mathrm{odd}\hfill \\ 〈[0.6,0.8],0.6〉if \mathsf{n}\mathrm{is}\mathrm{even},\hfill \end{array}\right.$$

$$\mathcal{A}\left(0\right)=〈[0.7,0.9],0.8〉,\mathcal{B}\left(0\right)=〈[0.7,0.9],0.4〉.$$

Then, we can easily check that $\mathcal{A}\in PCG\left(\mathbb{Z}\right)$ and $\mathcal{B}\in RCG\left(\mathbb{Z}\right).$

Proposition 13.

If ${\left({\mathcal{A}}_j\right)}_{j\in J}\subset PCG\left(G\right)$, then ${\sqcap }_{j\in J}{\mathcal{A}}_j\in PCG\left(G\right)$.

(2) If $RCG\left(G\right)$, then ${⋒}_{j\in J}{\mathcal{A}}_j\in RCG\left(G\right)$.

Proof. 

(1) The proof follows from Propositions 4.3 in [31] and 5.2 in [35].

(2) From Proposition 8 (2), it is clear that ${\bigcup }_{j\in J}{A}_j\in FAGP\left(G\right)$. Then, it is sufficient to prove that $\left({\bigcup }_{j\in J}{A}_j\right)\left(x^{-1}\right)\le \left({\bigcup }_{j\in J}{A}_j\right)\left(x\right)$ for each $x\in G$. Let $x\in G$. Then, we have

$\left({\bigcup }_{j\in J}{A}_j\right)\left(x^{-1}\right)={\bigvee }_{j\in J}{A}_j\left(x^{-1}\right)$

$\le {\bigvee }_{j\in J}{A}_j\left(x\right)$ (Since $A_j\in FAG\left(G\right)$).

$=\left({\bigcup }_{j\in J}{A}_j\right)\left(x\right).$

Thus ${\bigcup }_{j\in J}{A}_j\in FAF\left(G\right).$ So, according to Remark 7 (b) and Proposition 4.3 in [31], ${⋒}_{j\in J}{\mathcal{A}}_j\in RCG\left(G\right)$. □

Proposition 14.

(1) If $\mathcal{A}\in PCG\left(G\right)$, then $\mathcal{A}\left(x^{-1}\right)=\mathcal{A}\left(x\right)$, i.e., $\tilde{A}\left(x^{-1}\right)=\tilde{A}\left(x\right),A\left(x^{-1}\right)=A\left(x\right)$ for each $x\in G.$

(2) (See Proposition 3.6, [19]) If $\mathcal{A}\in RCG\left(G\right)$, then $\mathcal{A}\left(x^{-1}\right)=\mathcal{A}\left(x\right)$, i.e., $\tilde{A}\left(x^{-1}\right)=\tilde{A}\left(x\right),A\left(x^{-1}\right)=A\left(x\right)$ for each $x\in G.$

Proof. 

(1) The proof follows from Propositions 3.1 (i) in [36] and 5.4 in [35].

(2) See the proof of Proposition 3.6 in [19]. □

Proposition 15.

(1) If $\mathcal{A}\in PCG\left(G\right)$, then $\mathcal{A}\left(e\right){\ge }_P\mathcal{A}\left(x\right)$, i.e., $\tilde{A}\left(e\right)\ge \tilde{A}\left(x\right),A\left(e\right)\ge A\left(x\right)$ for each $x\in G.$

(2) (See Proposition 3.7, [19]) If $\mathcal{A}\in RCG\left(G\right)$, then $\mathcal{A}\left(e\right){\ge }_R\mathcal{A}\left(x\right)$, i.e., $\tilde{A}\left(e\right)\ge \tilde{A}\left(x\right),A\left(e\right)\le A\left(x\right)$ for each $x\in G.$

Proof. 

(1) The proof follows from Propositions 3.1 (ii) in [36] and 5.4 in [35].

(2) See the proof of Proposition 3.7 in [19]. □

Theorem 3.

(1) $\mathcal{A}\in PCG\left(G\right)$ if and only if $\mathcal{A}\left(x{y}^{-1}\right){\ge }_P\mathcal{A}\left(x\right){\wedge }^P\mathcal{A}\left(y\right)$, i.e., $\tilde{A}\left(x{y}^{-1}\right)\ge \tilde{A}\left(x\right)\wedge \tilde{A}\left(y\right),A\left(x{y}^{-1}\right)\ge A\left(x\right)\wedge A\left(y\right)$ for each $x\in G.$

(2) (See Theorem 3.10, [19]) $\mathcal{A}\in RCG\left(G\right)$ if and only if $\mathcal{A}\left(x{y}^{-1}\right){\ge }_R\mathcal{A}\left(x\right){\wedge }^R\mathcal{A}\left(y\right)$, i.e., $\tilde{A}\left(x{y}^{-1}\right)\ge \tilde{A}\left(x\right)\wedge \tilde{A}\left(y\right),A\left(x{y}^{-1}\right)\le A\left(x\right)\vee A\left(y\right)$ for each $x\in G.$

Proof. 

The proof follows from Propositions 3.2 in [36] and 5.6 in [35].

(2) See the proof of Theorem 3.10 in [19]. □

The following can be easily seen.

Proposition 16.

If $\mathcal{A}\in PCG\left(G\right)$[resp., $\mathcal{A}\in RCG\left(G\right)$], then $\mathcal{A}{\circ }_{{}_P}\mathcal{A}=\mathcal{A}$[resp., $\mathcal{A}{\circ }_{{}_R}\mathcal{A}=\mathcal{A}$.]

Proposition 17.

(1) Let $\mathcal{A}\in PCG\left(G\right)$. If $\mathcal{A}\left(x{y}^{-1}\right)=\mathcal{A}\left(e\right)$ for any $x,y\in G,$ then $\mathcal{A}\left(x\right)=\mathcal{A}\left(y\right)$, i.e., $\tilde{A}\left(x\right)=\tilde{A}\left(y\right),A\left(x\right)=A\left(y\right).$

(2) (See Proposition 3.7, [19]) If $\mathcal{A}\left(x{y}^{-1}\right)=\mathcal{A}\left(e\right)$ for any $x,y\in G,$ then $\mathcal{A}\left(x\right)=\mathcal{A}\left(y\right)$, i.e., $\tilde{A}\left(x\right)=\tilde{A}\left(y\right),A\left(x\right)=A\left(y\right).$

Proof. 

(1) The proof follows from Propositions 4.7 in [31] and 5.5 in [35].

(2) See the proof of Proposition 3.7 in [19]. □

Proposition 18.

For each $\mathcal{A}\in CS\left(G\right)$, consider the subset $G_{\mathcal{A}}$ of G defined as follows:

$$G_{\mathcal{A}}=\{x\in G:\mathcal{A}\left(x\right)=\mathcal{A}\left(e\right)\}=\{x\in G:\tilde{A}\left(x\right)=\tilde{A}\left(e\right),A\left(x\right)=A\left(e\right)\}.$$

If $\mathcal{A}\in PCG\left(G\right)$[resp., $\mathcal{A}\in RCG\left(G\right)$], then $G_{\mathcal{A}}$ is a subgroup of G.

Proof. 

(The proof of the first part follows from Proposition 4.6 in [31] and Corollary of 5.4 in [35]. For the proof of second part, see Theorem 3.11 in [19]. □

Theorem 4.

Let $\mathcal{A}\in CS\left(G\right)$ and let ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}\ne \varnothing$ and ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}\ne \varnothing$ for any cubic number $\tilde{\tilde{a}}$.

(1) $\mathcal{A}\in PCG\left(G\right)$ if and only if ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}$ is a subgroup of G.

(2) $\mathcal{A}\in RCG\left(G\right)$ if and only if ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$ is a subgroup of G.

Proof. 

(1) The proof is straightforward from Propositions 4.16, 4.17 in [31] and Theorems 2.1, 2.2 in [38].

(2) See the proof of Theorem 3.12 in [19]. □

Proposition 19.

(1) Every $\mathcal{A}\in PCLI\left(G\right)$[resp., $PCLI\left(G\right)$ and $PCI\left(G\right)$]is a constant.

(2) Every $\mathcal{A}\in RCLI\left(G\right)$[resp., $RCLI\left(G\right)$ and $RCI\left(G\right)$]is a constant.

Proof. 

(1) The proof is straightforward from Propositions 4.14 in [31] and 5.9 in [35].

(2) The proof is similar to (1). □

Proposition 20.

let $f:G\to G^{{}^{\prime }}$ be a group homomorphism. If $\mathcal{B}\in PCG\left(G^{{}^{\prime }}\right)$[resp., $RCG\left(G^{{}^{\prime }}\right)$], then $f^{-1}\left(\mathcal{A}\right)\in PCG\left(G\right)$[resp., $RCG\left(G\right)$].

Proof. 

From Propositions 3.4 in [36] and 5.8 in [35], the proof of the first part is easy. For the proof of the second part, see Theorem 3.13 (2) in [19]. □

Proposition 21.

let $f:G\to G^{{}^{\prime }}$ be a group homomorphism and let $\mathcal{A}\in CS\left(G\right)$.

(1) If $\mathcal{A}\in PCG\left(G\right)$ and $\mathcal{A}$ have the P-sup-property, then $f^P\left(\mathcal{A}\right)\in PCG\left(G^{{}^{\prime }}\right)$.

(2) If $\mathcal{A}\in RCG\left(G\right)$ and $\mathcal{A}$ have the R-sup-property, then $f^R\left(\mathcal{A}\right)\in PCG\left(G^{{}^{\prime }}\right)$.

Proof. 

(1) The proof follows from Propositions 4.11 (b) in [31] and 5.8 in [35]

(2) See Theorem 3.13 (1) in [19]. □

Definition 17.

Let $\mathcal{A}\in CS\left(G\right)$. Consider the following condition:

$$\mathcal{A}\left(xy\right)=\mathcal{A}\left(yx\right),i.e.,\tilde{A}\left(xy\right)=\tilde{A}\left(yx\right),A\left(xy\right)=A\left(yx\right)foranyx,y\in G.$$

(10)

Then, $\mathcal{A}$ is called:

(1) a P-cubic normal subgroup (briefly, PCNG) of G, if $\mathcal{A}\in PCG\left(G\right)$ and (10) holds,

(2) an R-cubic normal subgroup (briefly, RCNG) of G, if $\mathcal{A}\in RCG\left(G\right)$ and (10) holds.

We will denote the set of all PCNGs [resp., RCNGs] of G as $PCNG\left(G\right)$ [resp., $RCNG\left(G\right)$].

Let us denote the set of all interval-valued fuzzy normal subgroups [resp., fuzzy normal subgroups and fuzzy anti-normal subgroups] of G as $IVFNG\left(G\right)$ [resp., $FNG\left(G\right)$ and $FANG\left(G\right)$]

Remark 8.

(1) Let G be a commutative group and let $\mathcal{A}\in PCG\left(G\right)$ [resp., $RCG\left(G\right)$]. Then, $\mathcal{A}\in PCNG\left(G\right)$ [resp., $RCNG\left(G\right)$].

(2) $\mathcal{A}\in PCNG\left(G\right)$ if and only if $\tilde{A}\in IVFNG\left(G\right)$ and $A\in FNG\left(G\right)$.

(3) $\mathcal{A}\in RCNG\left(G\right)$ if and only if $\tilde{A}\in IVFNG\left(G\right)$ and $A\in FANG\left(G\right)$.

Example 7.

Consider the general linear group of degree n$GL(n;R)$. Then, clearly, $GL(n;R)$ is a noncommutative group. Let $\mathcal{A},\mathcal{B}$ be the cubic sets in $GL(n;R)$ as follows: for any $I_n\ne M\in GL(n;R)$, where $I_n$ is the unit matrix, is the unit matrix,

$$\mathcal{A}\left(M\right)=\left\{\begin{array}{c}〈[0.2,0.6],0.4〉if \mathsf{M}\mathsf{i}\mathsf{s}\mathsf{n}\mathsf{o}\mathsf{t}\mathsf{a}\mathsf{t}\mathsf{r}\mathsf{i}\mathsf{a}\mathsf{n}\mathsf{g}\mathsf{u}\mathsf{l}\mathsf{a}\mathsf{r}\mathsf{m}\mathsf{a}\mathsf{t}\mathsf{r}\mathsf{i}\mathsf{x}\hfill \\ 〈[0.3,0.5],0.8〉if \mathsf{M}\mathsf{i}\mathsf{s}\mathsf{a}\mathsf{t}\mathsf{r}\mathsf{i}\mathsf{a}\mathsf{n}\mathsf{g}\mathsf{u}\mathsf{l}\mathsf{a}\mathsf{r}\mathsf{m}\mathsf{a}\mathsf{t}\mathsf{r}\mathsf{i}\mathsf{x},\hfill \end{array}\right.$$

$$\mathcal{B}\left(M\right)=\left\{\begin{array}{c}〈[0.2,0.6],0.8〉if \mathsf{M}\mathsf{i}\mathsf{s}\mathsf{n}\mathsf{o}\mathsf{t}\mathsf{a}\mathsf{t}\mathsf{r}\mathsf{i}\mathsf{a}\mathsf{n}\mathsf{g}\mathsf{u}\mathsf{l}\mathsf{a}\mathsf{r}\mathsf{m}\mathsf{a}\mathsf{t}\mathsf{r}\mathsf{i}\mathsf{x}\hfill \\ 〈[0.3,0.5],0.4〉if \mathsf{M}\mathsf{i}\mathsf{s}\mathsf{a}\mathsf{t}\mathsf{r}\mathsf{i}\mathsf{a}\mathsf{n}\mathsf{g}\mathsf{u}\mathsf{l}\mathsf{a}\mathsf{r}\mathsf{m}\mathsf{a}\mathsf{t}\mathsf{r}\mathsf{i}\mathsf{x},\hfill \end{array}\right.$$

$$\mathcal{A}\left(I_n\right)=〈[1,1],1〉,\mathcal{B}\left(I_n\right)=〈[1,1],0〉.$$

Then, we can easily check that $\mathcal{A}\in PCNG\left(GL\right(n;R\left)\right)$ and $\mathcal{B}\in RCNG\left(GL\right(n;R\left)\right)$.

Proposition 22.

Let $\mathcal{A},\mathcal{B}\in CS\left(G\right)$.

(1) If $\mathcal{B}\in PCNG\left(G\right)$, then $\mathcal{A}{\circ }_{{}_P}\mathcal{B}=\mathcal{B}{\circ }_{{}_P}\mathcal{A}.$

(2) If $\mathcal{B}\in RCNG\left(G\right)$, then $\mathcal{A}{\circ }_{{}_R}\mathcal{B}=\mathcal{B}{\circ }_{{}_R}\mathcal{A}.$

Proof. 

(1) The proof is straightforward from Propositions 5.2 in [31] and 2.1 (i) in [30].

(2) Suppose $\mathcal{B}\in RCNG\left(G\right)$ and let $x\in G$. Then, we have

$\left(A{\circ }_{{}_R}B\right)\left(x\right)={\bigwedge }_{x=y\left(y^{-1}x\right)}[A\left(y\right)\vee B\left(y^{-1}x\right)]$ [By Remark 1 (2)]

$={\bigwedge }_{x=y\left(y^{-1}x\right)}[B\left(y^{-1}x\right)\vee A\left(y\right)]$

$={\bigwedge }_{x=\left(x{y}^{-1}\right)y)}[B\left(x{y}^{-1}\right)\vee A\left(y\right)]$ [By the hypothesis]

$=\left(B{\circ }_{{}_R}A\right)\left(x\right).$

Thus, by Remark 1 (2) and Proposition 5.2 in [31], $\mathcal{A}{\circ }_{{}_R}\mathcal{B}=\mathcal{B}{\circ }_{{}_R}\mathcal{A}.$ □

Proposition 23.

Let $\mathcal{A},\mathcal{B}\in CS\left(G\right)$.

(1) If $\mathcal{A}\in PCNG\left(G\right)$ and $\mathcal{B}\in PCG\left(G\right)$, then $\mathcal{B}{\circ }_{{}_P}\mathcal{A}\in PCG\left(G\right).$

(2) If $\mathcal{A}\in RCNG\left(G\right)$ and $\mathcal{B}\in RCG\left(G\right)$, then $\mathcal{B}{\circ }_{{}_R}\mathcal{A}\in RCG\left(G\right).$

Proof. 

(1) The proof follows from Propositions 5.3 in [31] and 2.1 (ii) in [30].

(2) Suppose $\mathcal{B}\in RCNG\left(G\right)$ and $\mathcal{B}\in RCG\left(G\right)$. Then, from Definitions 7 and 10 (ii), we can easily see that $B{\circ }_{{}_R}A\in FAGP\left(G\right).$ Thus, it is sufficient to show that $\left(B{\circ }_{{}_R}A\right)\left(x^{-1}\right)\le \left(B{\circ }_{{}_R}A\right)\left(x\right)$ for each $x\in G$. Let $x\in G$. Then, we get

$\left(B{\circ }_{{}_R}A\right)\left(x^{-1}\right)={\bigwedge }_{x^{-1}=yz}[B\left(y\right)\vee A\left(z\right)]$

$={\bigwedge }_{x=z^{-1}{y}^{-1}}[B\left({\left(y^{-1}\right)}^{-1}\right)\vee A\left({\left(z^{-1}\right)}^{-1}\right)]$

$\le {\bigwedge }_{x=z^{-1}{y}^{-1}}[B\left(y^{-1}\right)\vee A\left(z^{-1}\right)]$ (According to the second part of (10))

$=\left(A{\circ }_{{}_R}B\right)\left(x\right)$

$=\left(B{\circ }_{{}_R}A\right)\left(x\right)$. (According to Proposition 22 (2))

Thus, $B{\circ }_{{}_R}A\in FAG\left(G\right).$ So, according to Proposition 4.3 in [31] and Remark 7 (1), $\mathcal{B}{\circ }_{{}_R}\mathcal{A}\in RCG\left(G\right).$ □

Proposition 24.

Let $\mathcal{A},\mathcal{B}\in CS\left(G\right)$.

(1) If $\mathcal{A},\mathcal{B}\in PCNG\left(G\right)$, then $\mathcal{A}{\circ }_{{}_P}\mathcal{B}\in PCNG\left(G\right).$

(2) If $\mathcal{A},\mathcal{B}\in RCNG\left(G\right)$, then $\mathcal{A}{\circ }_{{}_R}\mathcal{B}\in RCNG\left(G\right).$

Proof. 

(1) The proof is clear from Corollary 5.3 in [31] and Theorem 3.11 in [39].

(2) Since $\tilde{A}\circ \tilde{B}\in IVFNG\left(G\right)$ by (Corollary 5.3 [31]), it is sufficient to show that $A{\circ }_{{}_R}B\in FANG\left(G\right).$ Let $a,b\in G$. Then, there are $x,y\in G$ such that $ab=xy.$ Since $b=a^{-1}xy$, $ba=\left(a^{-1}xa\right)\left(a^{-1}ya\right)$. Thus, we have

$\left(A{\circ }_{{}_R}B\right)\left(ab\right)={\bigwedge }_{ab=xy}[A\left(x\right)\vee B\left(y\right)]$

$={\bigwedge }_{ba=\left(a^{-1}xa\right)\left(a^{-1}ya\right)}[A\left(a^{-1}xa\right)\vee B\left(a^{-1}ya\right)]$

$=\left(A{\circ }_{{}_R}B\right)\left(ba\right).$

So $A{\circ }_{{}_R}B\in FANG\left(G\right).$ Hence, $\mathcal{A}{\circ }_{{}_R}\mathcal{B}\in RCNG\left(G\right).$ □

The following is an immediate consequence of Propositions 22, 16 and 24.

Corollary 2.

$(PCNG\left(G\right),{\circ }_{{}_P})$[resp., $(RCNG\left(G\right),{\circ }_{{}_R})$]is a semilattice (i.e., a commutative idempotent semigroup).

Proposition 25.

If $\mathcal{A}\in PCNG\left(G\right)$[resp., $\mathcal{A}\in RCNG\left(G\right)$], then $G_{\mathcal{A}}$ is a normal subgroup of G.

In this case, $G/G_{\mathcal{A}}$ is called a cubic quotient group of G with respect to $\mathcal{A}$.

Proof. 

The proof is straightforward. □

Remark 9.

It is obvious that if N is a normal subgroup of G, then ${{\chi }_{{}_{\mathcal{N}}}}_{{}_{{}_P}}\in PCNG\left(G\right)$ [resp., ${{\chi }_{{}_{\mathcal{N}}}}_{{}_{{}_R}}\in RCNG\left(G\right)$] and $G_{{{\chi }_{{}_{\mathcal{N}}}}_{{}_{{}_P}}}=N=G_{{{\chi }_{{}_{\mathcal{N}}}}_{{}_{{}_R}}}$.

Proposition 26.

Let $\mathcal{A},\mathcal{B}\in CS\left(G\right)$. If $\mathcal{A}\in PCNG\left(G\right)$[resp., $\mathcal{A}\in RCNG\left(G\right)$], then ${\pi }^{-1}\left({\pi }^P\left(\mathcal{B}\right)\right)={\chi }_{{}_{G_{\mathcal{A}}}}{\circ }_{{}_P}\mathcal{B}$[resp., ${\pi }^{-1}\left({\pi }^R\left(\mathcal{B}\right)\right)={\chi }_{{}_{G_{\mathcal{A}}}}{\circ }_{{}_R}\mathcal{B}$], where $\pi :G\to G/G_{\mathcal{A}}$ denotes the natural mapping.

Proof. 

The proof of the first part is straightforward from Propositions 5.6 in [31] and 2.3 in [30]. To prove the second part, let $x\in G$. Then, we have

$$\left[{\pi }^{-1}\left({\pi }^R\left(B\right)\right)\right]\left(x\right)=\left[{\pi }^R\left(B\right)\right]\left(\pi \left(x\right)\right)=\underset{\pi \left(x\right)=\pi \left(y\right)}{\bigwedge }B\left(y\right)=\underset{x{y}^{-1}\in G_A}{\bigwedge }B\left(y\right)$$

and

$$\left({\chi }_{{}_{G_A}}{\circ }_{R}B\right)\left(x\right)=\underset{x=zy}{\bigwedge }[{\chi }_{{}_{G_A}}\left(z\right)\vee B\left(y\right)]=\underset{z=x{y}^{-1}\in G_A}{\bigwedge }B\left(y\right).$$

Thus, ${\pi }^{-1}\left({\pi }^R\left(B\right)\right)={\chi }_{{}_{G_A}}{\circ }_{{}_R}B$. So, according to Proposition 5.6 in [31] and Definition 4, ${\pi }^{-1}\left({\pi }^R\left(\mathcal{B}\right)\right)={\chi }_{{}_{G_{\mathcal{A}}}}{\circ }_{{}_R}\mathcal{B}$. □

Theorem 5.

Let $\mathcal{A}\in CS\left(G\right)$ and let ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}\ne \varnothing$ and ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}\ne \varnothing$ for any cubic number $\tilde{\tilde{a}}$.

(1) $\mathcal{A}\in PCNG\left(G\right)$ if and only if ${\left[\mathcal{A}\right]}_{P,\tilde{\tilde{a}}}$ is a normal subgroup of G.

(2) $\mathcal{A}\in RCNG\left(G\right)$ if and only if ${\left[\mathcal{A}\right]}_{R,\tilde{\tilde{a}}}$ is a normal subgroup of G.

Proof. 

(1) From Theorem 4 (1), it is obvious that $\tilde{A}\in IVFG\left(G\right)$ if and only if ${\left[\tilde{A}\right]}_{\tilde{a}}$ is a subgroup of G and $A\in FG\left(G\right)$ if and only if ${\left[A\right]}_a$ is a subgroup of G. Moreover, $A\in FNG\left(G\right)$ if and only if ${\left[\tilde{A}\right]}_a$ is a normal subgroup of G (see Propositions 3.2 and 3.4 in [40] and [41] respectively). Then, it is sufficient to prove that $\tilde{A}\in IVFNG\left(G\right)$ if and only if ${\left[\tilde{A}\right]}_{\tilde{a}}$ is a normal subgroup of G.

Suppose $\tilde{A}\in IVFNG\left(G\right)$ and let $y\in {\left[\tilde{A}\right]}_{\tilde{a}}$, $x\in G$. Then,

$\tilde{A}\left(xy{x}^{-1}\right)=\tilde{A}\left(y\right)$ (According to the hypothesis)

$\ge \tilde{a}$. (Since $y\in {\left[\tilde{A}\right]}_{\tilde{a}}$)

Thus, $xy{x}^{-1}\in {\left[\tilde{A}\right]}_{\tilde{a}}$. So, according to Theorem 4 (1), ${\left[\tilde{A}\right]}_{\tilde{a}}$ is a normal subgroup of G.

Conversely, suppose ${\left[\tilde{A}\right]}_{\tilde{a}}$ is a normal subgroup of G. Assume that there are $x,y\in G$ such that $\tilde{A}\left(xy\right)\ne \tilde{A}\left(yx\right)$, say $\tilde{A}\left(xy\right)>\tilde{A}\left(yx\right)$. Then, there is a cubic number $\tilde{a}$, such that $\tilde{A}\left(xy\right)>\tilde{a}>\tilde{A}\left(yx\right)$. Thus, $xy\in {\left[\tilde{A}\right]}_{\tilde{a}}$ but $yx\notin {\left[\tilde{A}\right]}_{\tilde{a}}$. This contradicts the normality of ${\left[\tilde{A}\right]}_{\tilde{a}}$. So $\tilde{A}\left(xy\right)=\tilde{A}\left(yx\right)$. Hence, $\tilde{A}$ is normal.

(2) From 4 (2), it is obvious that $\tilde{A}\in IVFG\left(G\right)$ if and only if ${\left[\tilde{A}\right]}_{\tilde{a}}$ is a subgroup of G and $A\in FAG\left(G\right)$ if and only if ${\left[A\right]}_{R,a}$ is a subgroup of G. Since $\tilde{A}$ is normal if and only if ${\left[\tilde{A}\right]}_{\tilde{a}}$ is normal by (1), it is sufficient to show that the fuzzy anti-subgroup A is normal if and only if ${\left[A\right]}_{R,a}$ is normal.

Suppose $A\in FAG\left(G\right)$ is normal and let $x\in G$, $y\in {\left[A\right]}_{R,a}$. Then, we have

$$A\left(xy{x}^{-1}\right)=A\left(y\right)\le a.$$

Thus, $xy{x}^{-1}\in {\left[A\right]}_{R,a}$. So ${\left[A\right]}_{R,a}$ is normal. The proof of the converse is similar to the converse of (1). □

Proposition 27.

let $f:G\to G^{{}^{\prime }}$ be a group homomorphism. If $\mathcal{B}\in PCNG\left(G^{{}^{\prime }}\right)$[resp., $RCNG\left(G^{{}^{\prime }}\right)$], then $f^{-1}\left(\mathcal{A}\right)\in PCNG\left(G\right)$[resp., $RCNG\left(G\right)$].

Proof. 

The proof follows from Proposition 20, Definitions 4 and 17. □

Proposition 28.

let $f:G\to G^{{}^{\prime }}$ be a group homomorphism and let $\mathcal{A}\in CS\left(G\right)$.

(1) If $\mathcal{A}\in PCNG\left(G\right)$ and $\mathcal{A}$ has the P-sup-property, then $f^P\left(\mathcal{A}\right)\in PCNG\left(G^{{}^{\prime }}\right)$.

(2) If $\mathcal{A}\in RCNG\left(G\right)$ and $\mathcal{A}$ has the R-sup-property, then $f^R\left(\mathcal{A}\right)\in PCNG\left(G^{{}^{\prime }}\right)$.

Proof. 

The proof is straightforward from Proposition 21, Definitions 4 and 17. □

5. Cubic Congruences

A relation R on a groupoid X is said to be left compatible [resp., right compatible and compatible], if for any $a,b,c,d,x\in X$,

$$(a,b)\in R\mathsf{i}\mathsf{m}\mathsf{p}\mathsf{l}\mathsf{i}\mathsf{e}\mathsf{s}(xa,xb)\in R$$

$$[\mathsf{r}\mathsf{e}\mathsf{s}\mathsf{p}.(a,b)\in R\mathsf{i}\mathsf{m}\mathsf{p}\mathsf{l}\mathsf{i}\mathsf{e}\mathsf{s}(ax,bx)\in R\mathsf{a}\mathsf{n}\mathsf{d}(a,b)\in R,(c,d)\in R\mathsf{i}\mathsf{m}\mathsf{p}\mathsf{l}\mathsf{y}(ac,bd)\in R].$$

A left [resp., right] compatible equivalence relation on X is called a left [resp. right] congruence on X. A compatible equivalence relation on X is called a congruence on X. It is well known (Proposition 1.5 [25]) that a relation R on a groupoid X if and only if it is both a left and a right congruence on X.

Definition 18

(See [34]). For a groupoid X, a mapping $\mathcal{R}=〈\mathcal{R},R〉:X\times X\to \left[I\right]\times I$ is called a cubic relation on X. In fact, $\mathcal{R}$ is an interval-valued fuzzy relation on X (See [36]) and R is a fuzzy relation on X (see [2]). We denote the set of all cubic (resp., interval-valued fuzzy and fuzzy) relation on X as $CR\left(X\right)$ (resp., $IVFR\left(X\right)$ and $FR\left(X\right)$).

Definition 19.

For a groupoid X, let $\mathcal{R},\mathcal{S}\in CR\left(X\right)$.

(i) The P-composition of $\mathcal{R}$ and $\mathcal{S}$ (See [34]), denoted by $\mathcal{S}{\circ }_{{}_P}\mathcal{R}=〈\tilde{S}\circ \tilde{R},S\circ R〉$, is a cubic relation on X defined as follows: for each $(x,z)\in X\times X$,

$$\left(\mathcal{S}{\circ }_{{}_P}\mathcal{R}\right)(x,z)={\bigsqcup }_{y\in X}\left[\mathcal{R}(x,y){\wedge }^P\mathcal{S}(y,z)\right]=〈(\tilde{S}\circ \tilde{R})(x,z),(S\circ R)(x,z)〉,$$

where $(\tilde{S}\circ \tilde{R})(x,z)={\bigvee }_{y\in X}[\tilde{R}(x,y)\wedge \tilde{S}(y,z)$ and $(S\circ R)(x,z)={\bigvee }_{y\in X}[R(x,y)\wedge S(y,z).$

(ii) The R-composition of $\mathcal{R}$ and $\mathcal{S}$, denoted by $\mathcal{S}{\circ }_{{}_R}\mathcal{R}=〈\tilde{S}\circ \tilde{R},S{\circ }_{{}_R}R〉$, is a cubic relation on X defined as follows: for each $(x,z)\in X\times X$,

$$\left(\mathcal{S}{\circ }_{{}_R}\mathcal{R}\right)(x,z)={⋓}_{y\in X}\left[\mathcal{R}(x,y){\wedge }^R\mathcal{S}(y,z)\right]=〈(\tilde{S}\circ \tilde{R})(x,z),\left(S{\circ }_{{}_R}R\right)(x,z)〉,$$

where $(\tilde{S}\circ \tilde{R})(x,z)={\bigvee }_{y\in X}[\tilde{R}(x,y)\wedge \tilde{S}(y,z)$ and $\left(S{\circ }_{{}_R}R\right)(x,z)={\bigwedge }_{y\in X}[R(x,y)\vee S(y,z).$

Definition 20.

For a groupoid X, let $\mathcal{R}=〈\tilde{R},R〉\in CR\left(X\right)$. Then, $\mathcal{R}$ is said to be called a P-cubic equivalence relation on X (see [34]), if it satisfies the following conditions:

(i) P-cubic reflexive (see [34]), if $\mathcal{R}(x,x)=〈[1,1],1〉$, i.e., $\tilde{R}(x,x)=[1,1]$ and $R(x,x)=1$ for each $x\in X$,

(ii) R-cubic reflexive, if $\mathcal{R}(x,x)=〈[1,1],0〉$, i.e., $\tilde{R}(x,x)=[1,1]$ and $R(x,x)=0$ for each $x\in X$,

(iii) symmetric (see [34]), if $\mathcal{R}(x,y)=\mathcal{R}(y,x)$ for each $(x,y)\in X\times X$,

(iv) P-cubic transitive (See [34]), if $\mathcal{R}{\circ }_{{}_P}\mathcal{R}⊏\mathcal{R}$, i.e., $\tilde{R}\circ \tilde{R}\subset \tilde{R}$ (see [42]) and $R\circ R\subset R$ (see [2]),

(v) R-cubic transitive, if $\mathcal{R}{\circ }_{{}_R}\mathcal{R}⋐\mathcal{R}$, i.e., $\tilde{R}\circ \tilde{R}\subset \tilde{R}$ and $R{\circ }_{{}_R}R\supset R$,

(vi) a P-cubic equivalence relation on X, if it satisfies the conditions (i), (iii) and (iv),

(vii) an R-cubic equivalence relation on X, if it satisfies the conditions (ii), (iii) and (v).

We will denote the set of all P-cubic (resp.,, R-cubic, interval-valued fuzzy and fuzzy) equivalence relations on X as $P{C}_E\left(X\right)$ (resp.,, $R{C}_E\left(X\right)$, $IV{F}_E\left(X\right)$ and $F_E\left(X\right)$).

If a fuzzy relation R on X satisfies the second part of (ii), (iii) and the second part of (v), then R is called a fuzzy anti-equivalence relation on X, and we denote the set of all fuzzy anti-equivalence relations on X as $F{A}_E\left(X\right).$

Remark 10.

(1) Let R be a classical equivalence relation on a groupoid X. Then, clearly,

$${{\chi }_{{}_{\mathcal{R}}}}_{{}_{{}_P}}=〈[{\chi }_{{}_R},{\chi }_{{}_R}],{\chi }_{{}_R}〉\in P{C}_E\left(X\right)and{{\chi }_{{}_{\mathcal{R}}}}_{{}_{{}_R}}=〈[{\chi }_{{}_R},{\chi }_{{}_R}],{\chi }_{{}_{R^c}}〉\in R{C}_E\left(X\right).$$

(2) Let X be a groupoid and let $\tilde{R}\in IV{F}_E\left(X\right)$. Then, we can easily check that

$$〈\tilde{R},R^{-}〉,〈\tilde{R},R^{+}〉\in P{C}_E\left(X\right)and〈\tilde{R},{\left(R^{-}\right)}^c〉,〈\tilde{R},{\left(R^{+}\right)}^c〉\in R{C}_E\left(X\right).$$

(3) Let X be a groupoid and let $R\in F_E\left(X\right)$. Then, we have

$$〈[R,R],R〉\in P{C}_E\left(X\right)and〈[R,R],R^c〉\in R{C}_E\left(X\right).$$

Definition 21.

For a groupoid X, let $\mathcal{R}\in CR\left(X\right)$.

(i) $\mathcal{R}$ is said to be P-cubic left compatible, if for any $a,b,x\in G$,

$$\mathcal{R}(a,b){\le }_P\mathcal{R}(xa,xb),i.e.,\tilde{R}(a,b)\le \tilde{R}(xa,xb)andR(a,b)\le R(xa,xb),$$

(ii) $\mathcal{R}$ is said to be P-cubic right compatible, if for any $a,b,x\in G$,

$$\mathcal{R}(a,b){\le }_P\mathcal{R}(ax,bx),i.e.,\tilde{R}(a,b)\le \tilde{R}(ax,bx)andR(a,b)\le R(ax,bx),$$

(iii) $\mathcal{R}$ said to be P-cubic compatible, if for any $a,b,c,d\in G$,

$$\mathcal{R}(ac,bd){\ge }_P\mathcal{R}(a,b){\wedge }^P\mathcal{R}(c,d),i.e.,$$

$$\tilde{R}(ac,bd)\ge \tilde{R}(a,b)\wedge \tilde{R}(c,d)andR(ac,bd)\ge R(a,b)\wedge R(c,d),$$

(iv) $\mathcal{R}$ is said to be R-cubic left compatible, if for any $a,b,x\in G$,

$$\mathcal{R}(a,b){\le }_R\mathcal{R}(xa,xb),i.e.,\tilde{R}(a,b)\le \tilde{R}(xa,xb)andR(a,b)\ge R(xa,xb),$$

(v) $\mathcal{R}$ is said to be R-cubic right compatible, if for any $a,b,x\in G$,

$$\mathcal{R}(a,b){\le }_R\mathcal{R}(ax,bx),i.e.,\tilde{R}(a,b)\le \tilde{R}(ax,bx)andR(a,b)\ge R(ax,bx),$$

(vi) $\mathcal{R}$ is said to be R-cubic compatible, if for any $a,b,c,d\in G$,

$$\mathcal{R}(ac,bd){\ge }_R\mathcal{R}(a,b){\wedge }^P\mathcal{R}(c,d),i.e.,$$

$$\tilde{R}(ac,bd)\ge \tilde{R}(a,b)\wedge \tilde{R}(c,d)andR(ac,bd)\le R(a,b)\vee R(c,d).$$

If a fuzzy relation R on X satisfies the second part of (iv) (resp., (v) and (vi)), then R is called a fuzzy anti-left compatible [resp., anti-right compatible and anti-compatible].

Example 8.

Let $X=\{e,a,b\}$ be the groupoid with the following Cayley table (see

Table 6. The Cayley table of the groupoid X.

·	e	a	b
e	e	a	b
a	a	b	a
b	b	b	a

):

Consider six cubic relations ${\mathcal{R}}_i$ ($i\in \{1,2,3,4,5,6\}$) on X given by (see

Table 7. The values of the cubic relations ${\mathcal{R}}_i$.

${\mathcal{R}}_{\mathit{i}}$	e	a	b
e	${\tilde{\tilde{a}}}_{11}$	${\tilde{\tilde{a}}}_{12}$	${\tilde{\tilde{a}}}_{13}$
a	${\tilde{\tilde{a}}}_{21}$	${\tilde{\tilde{a}}}_{22}$	${\tilde{\tilde{a}}}_{23}$
b	${\tilde{\tilde{a}}}_{31}$	${\tilde{\tilde{a}}}_{32}$	${\tilde{\tilde{a}}}_{33}$

):

where ${\tilde{\tilde{a}}}_{ij}$ is a cubic number. Let us give the relationships between cubic numbers in ${\mathcal{R}}_i$.

The relationships between cubic numbers in ${\mathcal{R}}_1$:

$${\tilde{\tilde{a}}}_{21}and{\tilde{\tilde{a}}}_{31}arearbitrary,and{\tilde{\tilde{a}}}_{23}={\tilde{\tilde{a}}}_{32},{\tilde{\tilde{a}}}_{22}={\tilde{\tilde{a}}}_{33},{\tilde{\tilde{a}}}_{11}{\le }_P{\tilde{\tilde{a}}}_{22},$$

$${\tilde{\tilde{a}}}_{12}{\le }_P{\tilde{\tilde{a}}}_{23}{\wedge }^P{\tilde{\tilde{a}}}_{22},{\tilde{\tilde{a}}}_{13}{\le }_P{\tilde{\tilde{a}}}_{23}{\wedge }^P{\tilde{\tilde{a}}}_{22},{\tilde{\tilde{a}}}_{21}{\le }_P{\tilde{\tilde{a}}}_{23}{\wedge }^P{\tilde{\tilde{a}}}_{22},{\tilde{\tilde{a}}}_{31}{\le }_P{\tilde{\tilde{a}}}_{23}{\wedge }^P{\tilde{\tilde{a}}}_{22}.$$

The relationships between cubic numbers in ${\mathcal{R}}_2$:

$${\tilde{\tilde{a}}}_{11}{\le }_P{\tilde{\tilde{a}}}_{21},{\tilde{\tilde{a}}}_{12}{\le }_P{\tilde{\tilde{a}}}_{31},{\tilde{\tilde{a}}}_{13}{\le }_P{\tilde{\tilde{a}}}_{31},{\tilde{\tilde{a}}}_{21}{\le }_P{\tilde{\tilde{a}}}_{31},{\tilde{\tilde{a}}}_{32}{\le }_P{\tilde{\tilde{a}}}_{22},$$

$${\tilde{\tilde{a}}}_{33}{\le }_P{\tilde{\tilde{a}}}_{23}={\tilde{\tilde{a}}}_{22},andtheremaindersarearbitrary.$$

The relationships between cubic numbers in ${\mathcal{R}}_3$:

$${\tilde{\tilde{a}}}_{11}{\wedge }^P{\tilde{\tilde{a}}}_{12}{\le }_P{\tilde{\tilde{a}}}_{12},{\tilde{\tilde{a}}}_{11}{\wedge }^P{\tilde{\tilde{a}}}_{13}{\le }_P{\tilde{\tilde{a}}}_{13},{\tilde{\tilde{a}}}_{12}{\wedge }^P{\tilde{\tilde{a}}}_{13}{\le }_P{\tilde{\tilde{a}}}_{12},$$

$${\tilde{\tilde{a}}}_{21}{\wedge }^P{\tilde{\tilde{a}}}_{22}{\le }_P{\tilde{\tilde{a}}}_{32},{\tilde{\tilde{a}}}_{21}{\wedge }^P{\tilde{\tilde{a}}}_{23}{\le }_P{\tilde{\tilde{a}}}_{33},{\tilde{\tilde{a}}}_{22}{\wedge }^P{\tilde{\tilde{a}}}_{23}{\le }_P{\tilde{\tilde{a}}}_{32},$$

$${\tilde{\tilde{a}}}_{31}{\wedge }^P{\tilde{\tilde{a}}}_{32}{\le }_P{\tilde{\tilde{a}}}_{22},{\tilde{\tilde{a}}}_{31}{\wedge }^P{\tilde{\tilde{a}}}_{33}{\le }_P{\tilde{\tilde{a}}}_{22},{\tilde{\tilde{a}}}_{31}{\wedge }^P{\tilde{\tilde{a}}}_{33}{\le }_P{\tilde{\tilde{a}}}_{23},$$

$${\tilde{\tilde{a}}}_{32}{\wedge }^P{\tilde{\tilde{a}}}_{33}{\le }_P{\tilde{\tilde{a}}}_{22}.$$

The relationships between cubic numbers in ${\mathcal{R}}_4$:

$${\tilde{\tilde{a}}}_{21}and{\tilde{\tilde{a}}}_{31}arearbitrary,and{\tilde{\tilde{a}}}_{23}={\tilde{\tilde{a}}}_{32},{\tilde{\tilde{a}}}_{22}={\tilde{\tilde{a}}}_{33},{\tilde{\tilde{a}}}_{11}{\le }_R{\tilde{\tilde{a}}}_{22},$$

$${\tilde{\tilde{a}}}_{12}{\le }_R{\tilde{\tilde{a}}}_{23}{\wedge }^R{\tilde{\tilde{a}}}_{22},{\tilde{\tilde{a}}}_{13}{\le }_R{\tilde{\tilde{a}}}_{23}{\wedge }^R{\tilde{\tilde{a}}}_{22},{\tilde{\tilde{a}}}_{21}{\le }_R{\tilde{\tilde{a}}}_{23}{\wedge }^R{\tilde{\tilde{a}}}_{22},{\tilde{\tilde{a}}}_{31}{\le }_R{\tilde{\tilde{a}}}_{23}{\wedge }^R{\tilde{\tilde{a}}}_{22}.$$

The relationships between cubic numbers in ${\mathcal{R}}_5$:

$${\tilde{\tilde{a}}}_{11}{\le }_R{\tilde{\tilde{a}}}_{21},{\tilde{\tilde{a}}}_{12}{\le }_R{\tilde{\tilde{a}}}_{31},{\tilde{\tilde{a}}}_{13}{\le }_R{\tilde{\tilde{a}}}_{31},{\tilde{\tilde{a}}}_{21}{\le }_R{\tilde{\tilde{a}}}_{31},{\tilde{\tilde{a}}}_{32}{\le }_R{\tilde{\tilde{a}}}_{22},$$

$${\tilde{\tilde{a}}}_{33}{\le }_R{\tilde{\tilde{a}}}_{23}={\tilde{\tilde{a}}}_{22},andtheremaindersarearbitrary.$$

The relationships between cubic numbers in ${\mathcal{R}}_6$:

$${\tilde{\tilde{a}}}_{11}{\wedge }^R{\tilde{\tilde{a}}}_{12}{\le }_R{\tilde{\tilde{a}}}_{12},{\tilde{\tilde{a}}}_{11}{\wedge }^R{\tilde{\tilde{a}}}_{13}{\le }_R{\tilde{\tilde{a}}}_{13},{\tilde{\tilde{a}}}_{12}{\wedge }^R{\tilde{\tilde{a}}}_{13}{\le }_R{\tilde{\tilde{a}}}_{12},$$

$${\tilde{\tilde{a}}}_{21}{\wedge }^R{\tilde{\tilde{a}}}_{22}{\le }_R{\tilde{\tilde{a}}}_{32},{\tilde{\tilde{a}}}_{21}{\wedge }^R{\tilde{\tilde{a}}}_{23}{\le }_R{\tilde{\tilde{a}}}_{33},{\tilde{\tilde{a}}}_{22}{\wedge }^R{\tilde{\tilde{a}}}_{23}{\le }_R{\tilde{\tilde{a}}}_{32},$$

$${\tilde{\tilde{a}}}_{31}{\wedge }^R{\tilde{\tilde{a}}}_{32}{\le }_R{\tilde{\tilde{a}}}_{22},{\tilde{\tilde{a}}}_{31}{\wedge }^R{\tilde{\tilde{a}}}_{33}{\le }_R{\tilde{\tilde{a}}}_{22},{\tilde{\tilde{a}}}_{31}{\wedge }^R{\tilde{\tilde{a}}}_{33}{\le }_R{\tilde{\tilde{a}}}_{23},$$

$${\tilde{\tilde{a}}}_{32}{\wedge }^R{\tilde{\tilde{a}}}_{33}{\le }_R{\tilde{\tilde{a}}}_{22}.$$

Then, we can easily check that ${\mathcal{R}}_1$ [resp., ${\mathcal{R}}_2$ and ${\mathcal{R}}_3$] is a P-cubic left [resp., P-cubic right and P-cubic] compatible relation on X and ${\mathcal{R}}_4$ [resp., ${\mathcal{R}}_5$ and ${\mathcal{R}}_6$] is an R-cubic left [resp., R-cubic right and R-cubic] compatible relation on X.

Theorem 6.

Let R be a relation on a groupoid X. Then, the following are equivalent:

(1) R is left compatible.

(2) ${{\chi }_{{}_{\mathcal{R}}}}_{{}_{{}_P}}$ is P-cubic left compatible.

(3) ${{\chi }_{{}_{\mathcal{R}}}}_{{}_{{}_R}}$ is R-cubic left compatible.

Proof. 

(1)⟺(2): The proof follows from Lemmas 3.4 in [43] and 2.1 in [39].

(1)⟺(3): Suppose the condition (1) holds. Since $[{\chi }_{{}_R},{\chi }_{{}_R}]$ is an interval-valued fuzzy left compatible by (Lemmas 3.4 [43]), it is sufficient to prove that ${\chi }_{{}_{R^c}}$ is fuzzy left compatible. Let $a,b,x\in X$.

Case 1. Suppose $(a,b)\in R$. Then, clearly, ${\chi }_{{}_{R^c}}(a,b)=0.$ By the condition (1), $(xa,xb)\in R$. Thus, ${\chi }_{{}_{R^c}}(xa,xb)=0$. So ${\chi }_{{}_{R^c}}(xa,xb)={\chi }_{{}_{R^c}}(a,b).$

Case 2. Suppose $(a,b)\notin R$. Then, we have

$${\chi }_{{}_{R^c}}(a,b)=1\ge {\chi }_{{}_{R^c}}(xa,xb).$$

In either cases, ${\chi }_{{}_{R^c}}(a,b)\ge {\chi }_{{}_{R^c}}(xa,xb).$ Hence, ${{\chi }_{{}_{\mathcal{R}}}}_{{}_{{}_R}}$ is R-cubic left compatible.

Suppose the condition (3) holds. Let $(a,b)\in R$ and let $x\in X$. Then, according to condition (3), we have

$${{\chi }_{{}_{\mathcal{R}}}}_{{}_{{}_R}}(a,b)=〈[1,1],0〉{\le }_R{{\chi }_{{}_{\mathcal{R}}}}_{{}_{{}_R}}(xa,xb).$$

Thus, ${\chi }_{{}_{R^c}}(a,b)=0\ge {\chi }_{{}_{R^c}}(xa,xb).$ So $(xa,xb)\in R$. Hence, R is left compatible. □

Then, we follow the dual of Theorem 6.

Theorem 7.

Let R be a relation on a groupoid X. Then, the following are equivalent:

(1) R is right compatible.

(2) ${{\chi }_{{}_{\mathcal{R}}}}_{{}_{{}_P}}$ is P-cubic right compatible.

(3) ${{\chi }_{{}_{\mathcal{R}}}}_{{}_{{}_R}}$ is R-cubic right compatible.

Definition 22.

Let X be a groupoid and let $\mathcal{R}\in CR\left(X\right)$.

(i) $\mathcal{R}$ is called a P-cubic left congruence (briefly, PCLC), if $\mathcal{R}\in P{C}_E$ and $\mathcal{R}$ is P-cubic left compatible.

(ii) $\mathcal{R}$ is called a P-cubic right congruence (briefly, PCRC) on X, if $\mathcal{R}\in P{C}_E$ and $\mathcal{R}$ is P-cubic right compatible.

(iii) $\mathcal{R}$ is called a P-cubic congruence (briefly, PCC) on X, if $\mathcal{R}\in P{C}_E$ and $\mathcal{R}$ is P-cubic compatible.

(iv) $\mathcal{R}$ is called a R-cubic left congruence (briefly, RCLC) on X, if $\mathcal{R}\in R{C}_E$ and $\mathcal{R}$ is R-cubic left compatible.

(v) $\mathcal{R}$ is called a R-cubic right congruence (briefly, RCRC) on X, if $\mathcal{R}\in R{C}_E$ and $\mathcal{R}$ is R-cubic right compatible.

(vi) $\mathcal{R}$ is called a R-cubic congruence (briefly, RCC) on X, if $\mathcal{R}\in R{C}_E$ and $\mathcal{R}$ is R-cubic compatible.

A fuzzy relation R is called a fuzzy anti-left congruence (briefly, FALC) [resp., anti-right congruence (briefly, FARC) and anti-congruence (briefly, FAC)] on X, if $R\in F{A}_E\left(X\right)$ and R is fuzzy anti-left compatible [resp., anti-right compatible and anti-compatible].

We will denote the set of all PCLCs [resp., PCRCs, PCCs, RCLCs, RCRCs and RCCs] on X as $PCLC\left(X\right)$ [resp., $PCRC\left(X\right)$, $PCC\left(X\right)$, $RCLC\left(X\right)$, $RCRC\left(X\right)$ and $RCC\left(X\right)$].

Example 9.

Let $X=\{e,a,b\}$ be the groupoid given in Example 8. Consider two cubic relations, $\mathcal{R}$ and $\mathcal{S}$, on X, defined as follows (see

Table 8. The values of the cubic relation $\mathcal{R}$.

$\mathcal{R}$	e	a	b
e	$〈[1,1],1〉$	$〈0.4,0.6],0.6〉$	$〈[0.4,0.6],0.6〉$
a	$〈[0.4,0.6],0.6〉$	$〈[1,1],1〉$	$〈0.2,0.7],0.7〉$
b	$〈[0.4,0.6],0.6〉$	$〈0.2,0.7],0.7〉$	$〈[1,1],1〉$

and

Table 9. The values of the cubic relation $\mathcal{S}$.

$\mathcal{S}$	e	a	b
e	$〈[1,1],0〉$	$〈0.4,0.6],0.6〉$	$〈[0.4,0.6],0.6〉$
a	$〈[0.4,0.6],0.6〉$	$〈[1,1],0〉$	$〈0.2,0.7],0.4〉$
b	$〈[0.4,0.6],0.6〉$	$〈0.2,0.7],0.4〉$	$〈[1,1],0〉$

):

Then, we can easily check that $\mathcal{R}\in PCC\left(X\right)$ and $\mathcal{S}\in RCC\left(X\right)$.

Let us denote the set of all fuzzy left congruences [resp., right congruences and congruences] and all interval-valued fuzzy left congruences [resp., right congruences and congruences] on X as $FLC\left(X\right)$ [resp., $FRC\left(X\right)$ and $FC\left(X\right)$] and $IVFLC\left(X\right)$ [resp., $IVFRC\left(X\right)$ and $IVFC\left(X\right)$].

Remark 11.

(1) For a groupoid X, let $R\in FLC\left(X\right)$ [resp., $FRC\left(X\right)$, $FC\left(X\right)$, $FALC\left(X\right)$, $FARC\left(X\right)$ and $FAC\left(X\right)$]. Then, $〈[R,R],R〉\in PCLC\left(X\right)$ [resp., $PCRC\left(X\right)$, $PCC\left(X\right)$, $RCLC\left(X\right)$, $RCRC\left(X\right)$ and $RCC\left(X\right)$].

(2) For a groupoid X, let $\mathcal{R}\in IVFLC\left(X\right)$ [resp., $IVFRC\left(X\right)$ and $IVFC\left(X\right)$]. Then, $〈\mathcal{R},R^{-}〉,$$〈\mathcal{R},R^{+}〉$$\in PCLC\left(X\right)$ [resp., $PCRC\left(X\right)$ and $PCC\left(X\right)$]. Moreover, $〈\mathcal{R},{\left(R^{-}\right)}^c〉,〈\mathcal{R},{\left(R^{+}\right)}^c〉\in RCLC\left(X\right)$ [resp., $RCRC\left(X\right)$ and $RCC\left(X\right)$].

Theorem 8.

For a groupoid X, let $\mathcal{R}\in P{C}_E\left(X\right)$ or $\mathcal{R}\in R{C}_E\left(X\right)$. Then,

(1) $\mathcal{R}\in PCC\left(X\right)$ if and only if $\mathcal{R}\in PCLC\left(X\right)$ and $\mathcal{R}\in PCRC\left(X\right)$.

(2) $\mathcal{R}\in RCC\left(X\right)$ if and only if $\mathcal{R}\in RCLC\left(X\right)$ and $\mathcal{R}\in RCRC\left(X\right)$.

Proof. 

(1) The proof is straightforward from Proposition 3.8 in [43] and Theorem 2.3 in [39].

(2) From Proposition 3.8 in [43], it is obvious that $\tilde{R}\in IVFC\left(X\right)$ if and only if $\tilde{R}\in IVFLC\left(X\right)$ and $\tilde{R}\in IVFRC\left(X\right)$. Then, it is sufficient to show that $R\in FAC\left(X\right)$ if and only if $R\in FALC\left(X\right)$ and $R\in FARC\left(X\right)$.

Suppose $R\in FAC\left(X\right)$ and let $a,b,x\in X$. Then, we have

$R(xa,xb)\le R(x,x)\vee R(a,b)$ (Since R is fuzzy anti-compatible).

$=0\vee R(a,b)$ (Since R is fuzzy anti-reflexive)

$=R(a,b)$.

Thus, R is fuzzy anti-left compatible. It is clear that $R\in F{A}_E\left(X\right)$. So $R\in FALC\left(X\right)$. Similarly, we can see that $R\in FARC\left(X\right)$.

Conversely, suppose $R\in FALC\left(X\right)$ and $R\in FARC\left(X\right)$. Let $a,!b,c,d\in X$. Then, we get

$R(ac,bd)\le {\bigwedge }_{x\in X}[R(ac,x)\vee R(x,bd)]$ [Since $R\subset R{\circ }_{R}R$]

$\le R(ac,bc)\vee R(bc,bd)$ (Since $bc\in X$)

$\le R(a,b)\vee R(c,d)$. (According to the hypothesis)

Thus R is fuzzy anti-compatible. So, $R\in FAC\left(X\right)$. □

From Remark 10 (1), Theorems 6, 7 and 8, we obtain the following.

Theorem 9.

Let R be a relation on a groupoid X. Then, the following are equivalent:

(1) R is a congruence.

(2) ${{\chi }_{{}_{\mathcal{R}}}}_{{}_{{}_P}}\in PCC\left(X\right)$.

(3) ${{\chi }_{{}_{\mathcal{R}}}}_{{}_{{}_R}}\in RCC\left(X\right)$.

Lemma 1.

For a groupoid X, let $\mathcal{R},\mathcal{S}\in CR\left(X\right)$.

(1) If $\mathcal{R}$ and $\mathcal{S}$ are P-cubic reflexive, then $\mathcal{S}{\circ }_{{}_P}\mathcal{R}$ is P-cubic reflexive.

(2) If $\mathcal{R}$ and $\mathcal{S}$ are R-cubic reflexive, then $\mathcal{S}{\circ }_{{}_R}\mathcal{R}$ is R-cubic reflexive.

Proof. 

(1) Let $x\in X$. Then, we have

$(\tilde{S}\circ \tilde{R})(x,x)={\bigvee }_{y\in X}[\tilde{R}(x,y)\wedge \tilde{R}(y,x)]$

$\ge \tilde{R}(x,x)\wedge \tilde{R}(x,x)$

$=[1,1]$. (Since $\tilde{R}$ and $\tilde{S}$ are reflexive).

It is obvious that $(\tilde{S}\circ \tilde{R})(x,x)\le [1,1]$. Thus, $(\tilde{S}\circ \tilde{R})(x,x)=[1,1].$ So $\tilde{S}\circ \tilde{R}$ is interval-valued fuzzy reflexive. Moreover, $S\circ R$ is fuzzy reflexive from Proposition 4.7 in [44]. Hence, $\mathcal{S}{\circ }_{{}_P}\mathcal{R}$ is P-cubic reflexive.

(2) Let $x\in X$. Then, we have

$\left(S{\circ }_{{}_R}R\right)(x,x)={\bigwedge }_{y\in X}[R(x,y)\vee S(y,x)]$

$\le R(x,x)\vee S(x,x)$

$=0.$ (Since R and S are fuzzy anti-reflexive).

It is obvious that $\left(S{\circ }_{{}_R}R\right)(x,x)\ge 0.$ Thus, $\left(S{\circ }_{R}R\right)(x,x)=0$. So, $S{\circ }_{{}_R}R$ is fuzzy anti-reflexive. From (1), $\tilde{S}\circ \tilde{R}$ is interval-valued fuzzy reflexive. Hence, $\mathcal{S}{\circ }_{{}_R}\mathcal{R}$ is R-cubic reflexive. □

Lemma 2.

For a groupoid X, let $\mathcal{R},\mathcal{S}\in CR\left(X\right)$.

(1) If $\mathcal{R}$ and $\mathcal{S}$ are P-cubic compatible, then $\mathcal{S}{\circ }_{{}_P}\mathcal{R}$ is P-cubic compatible.

(2) If $\mathcal{R}$ and $\mathcal{S}$ are R-cubic compatible, then $\mathcal{S}{\circ }_{{}_R}\mathcal{R}$ is R-cubic compatible.

Proof. 

(1) The proof is clear from Lemmas 3.13 in [43] and 2.7 in [39].

(2) Let $a,b,x\in X$. Then, we have

$\left(S{\circ }_{{}_R}R\right)(xa,xb)={\bigwedge }_{z\in X}[R(xa,z)\vee S(z,xb)]$

$\le R(xa,xz)\vee S(xz,xb)$

$\le R(a,z)\vee S(z,b)$

[Since R and S are fuzzy anti-left compatible]

$\le {\bigwedge }_{z\in X}[R(a,z)\vee S(z,b)]$

$=\left(S{\circ }_{R}R\right)(a,b).$

Thus, $S{\circ }_{{}_R}R$ is fuzzy anti-left compatible. Similarly, we can see that $S{\circ }_{{}_R}R$ is fuzzy anti-right compatible. So $S{\circ }_{R}R$ is fuzzy anti-compatible. From Lemma 3.13 in [43], $\tilde{S}\circ \tilde{R}$ is interval-valued fuzzy compatible. Hence, $\mathcal{S}{\circ }_{{}_R}\mathcal{R}$ is R-cubic compatible. □

The following is an immediate consequence of (Theorem 3.14 [43]) and (Theorem 2.8 [39]).

Theorem 10.

Let $\mathcal{R}$ and $\mathcal{S}$ be P-cubic congruences on a groupoid X. Then, the following are equivalent:

(1) $\mathcal{S}{\circ }_{{}_P}\mathcal{R}\in PCC\left(X\right)$.

(2) $\mathcal{S}{\circ }_{{}_P}\mathcal{R}\in P{C}_E\left(X\right)$.

(3) $\mathcal{S}{\circ }_{{}_P}\mathcal{R}$ is cubic symmetric.

(4) $\mathcal{S}{\circ }_{{}_P}\mathcal{R}=\mathcal{R}{\circ }_P\mathcal{S}.$

Lemma 3.

Let R and S be fuzzy anti-congruences on a groupoid X. Then, the following are equivalent:

(1) $S{\circ }_{{}_R}R\in FAC\left(X\right)$.

(2) $S{\circ }_{{}_R}R\in F{A}_E\left(X\right)$.

(3) $S{\circ }_{{}_R}R$ is fuzzy symmetric.

(4) $S{\circ }_{{}_R}R=R{\circ }_{R}S.$

Proof. 

(1)⟺(2): The proof is straightforward.

(2)⟺(3): The proof is straightforward.

(3)⟺(4): Suppose the condition (3) holds and let $(x,z)\in X\times X$. Then, we have

$\left(R{\circ }_{{}_R}S\right)(x,z)={\bigwedge }_{y\in X}[S(x,y)\vee R(y,z)]$

$={\bigwedge }_{y\in X}[S(y,x)\vee R(z,y)]$

(Since R and S are fuzzy symmetric)

$={\bigwedge }_{y\in X}[R(z,y)\vee S(y,x)]$

$=\left(S{\circ }_{{}_R}R\right)(y,z)$

$=\left(S{\circ }_{{}_R}R\right)(x,y)$. (According to the hypothesis)

Thus, $R{\circ }_{{}_R}S=S{\circ }_{R}R.$

(4)⟺(1): Suppose the condition (4) holds. Then, we have

$\left(S{\circ }_{{}_R}R\right){\circ }_{{}_R}\left(S{\circ }_{{}_R}R\right)=S{\circ }_{{}_R}\left(R{\circ }_{{}_R}S\right){\circ }_{{}_R}R$

$=S{\circ }_{{}_R}\left(S{\circ }_{{}_R}R\right){\circ }_{{}_R}R$

$=\left(S{\circ }_{{}_R}S\right){\circ }_R\left(R{\circ }_{{}_R}R\right)$

$\supset S{\circ }_{{}_R}R$.

(Since R and S are fuzzy anti-transitive).

Thus, $S{\circ }_{{}_R}R$ is fuzzy anti-transitive. It is obvious that $S{\circ }_{{}_R}R$ is fuzzy anti-reflexive due to Lemma 1 (2). Moreover, $S{\circ }_{{}_R}R$ is fuzzy symmetric from the proof’s procedure of Theorem 2.8 in [39]. So $S{\circ }_{{}_R}R\in F{A}_E\left(X\right)$. It follows from Lemma 2 (2) that $S{\circ }_{{}_R}R$ is fuzzy anti-compatible. Hence, (1) holds. □

The following is an immediate consequence of Theorem 3.14 in [43] and Lemma 3.

Theorem 11.

Let $\mathcal{R}$ and $\mathcal{S}$ be R-cubic congruences on a groupoid X. Then, the following are equivalent:

(1) $\mathcal{S}{\circ }_{{}_R}\mathcal{R}\in RCC\left(X\right)$.

(2) $\mathcal{S}{\circ }_{{}_R}\mathcal{R}\in R{C}_E\left(X\right)$.

(3) $\mathcal{S}{\circ }_{{}_R}\mathcal{R}$ is cubic symmetric.

(4) $\mathcal{S}{\circ }_{{}_R}\mathcal{R}=\mathcal{R}{\circ }_R\mathcal{S}.$

Proposition 29.

If $\mathcal{R}\in PCC\left(G\right)$[resp., $\mathcal{R}\in RCC\left(G\right)$], then the following holds:

$$\mathcal{R}\left(xay\right)=\mathcal{R}(xa,xb)=\mathcal{R}(ay,by)=\mathcal{R}(a,b)\mathsf{f}\mathsf{o}\mathsf{r}\mathsf{a}\mathsf{l}\mathsf{l},a,b,x,y\in G.$$

Proof. 

The proof of the first part is straightforward from (Lemma 3.16 [43]) and Lemma 3.1 [39]), and the second part can be easily proved. □

Proposition 30.

If $\mathcal{R}\in PCC\left(G\right)$[resp., $\mathcal{R}\in RCC\left(G\right)$], then the following holds:

$$\mathcal{R}(x^{-1},y^{-1})=\mathcal{R}(x,y)\mathsf{f}\mathsf{o}\mathsf{r}\mathsf{a}\mathsf{n}\mathsf{y}x,y\in G.$$

Proof. 

The proof of the first part is straightforward from (Lemma 3.16 [43]) and Lemma 3.1 [39]) and the second part can be easily proved. □

Proposition 31.

Let $\mathcal{A}\in PCNG\left(G\right)$[resp., $\mathcal{A}\in RCNG\left(G\right)$]and let ${\mathcal{R}}_{\mathcal{A}}=〈{\tilde{R}}_{\tilde{A}},R_A〉$ be the cubic relation on G defined as follows: for each $(x,y)\in G\times G$,

$${\mathcal{R}}_{\mathcal{A}}(x,y)=\mathcal{A}\left(x{y}^{-1}\right)=〈\tilde{A}\left(x{y}^{-1}\right),A\left(x{y}^{-1}\right)〉.$$

Then, ${\mathcal{R}}_{\mathcal{A}}\in PCC\left(G\right)$[resp., ${\mathcal{R}}_{\mathcal{A}}\in RCC\left(G\right)$].

In this case, we will call ${\mathcal{R}}_{\mathcal{A}}$ as a P-cubic congruence [resp., an R-cubic congruence] on G induced by $\mathcal{A}$.

Proof. 

The proof of the first part is straightforward from (Lemma 3.16 [43]) and Lemma 3.1 [39]), and the second part can be easily proved. □

Proposition 32.

Let $\mathcal{A},\mathcal{B}\in PCG\left(G\right)$[resp., $RCG\left(G\right)$]. Then

$${\mathcal{R}}_{\mathcal{B}}{\circ }_{{}_P}{\mathcal{R}}_{\mathcal{A}}={\mathcal{R}}_{\mathcal{A}{\circ }_{{}_P}\mathcal{B}}[\mathrm{resp}.,{\mathcal{R}}_{\mathcal{B}}{\circ }_{{}_R}{\mathcal{R}}_{\mathcal{A}}={\mathcal{R}}_{\mathcal{A}{\circ }_{{}_R}\mathcal{B}}].$$

Proof. 

To show the first part, let $(a,b)\in G\times G$ and let $z\in G$. Let $a{z}^{-1}=x$ and $z{b}^{-1}=y$. Then, clearly, $xy=a{b}^{-1}.$ Thus, we get

$({\tilde{R}}_{\tilde{B}}\circ {\tilde{R}}_{\tilde{A}})(a,b)={\bigvee }_{z\in G}[{\tilde{R}}_{\tilde{A}}(a,z)\wedge {\tilde{R}}_{\tilde{B}}(z,b)]$

$={\bigvee }_{z\in G}[\tilde{A}\left(a{z}^{-1}\right)\wedge \tilde{B}\left(z{b}^{-1}\right)]$

$={\bigvee }_{a{b}^{-1}=xy}[\tilde{A}\left(x\right)\wedge \tilde{B}\left(y\right)]$

$=(\tilde{A}\circ \tilde{B})\left(a{b}^{-1}\right)$

$=({\tilde{R}}_{\tilde{A}}\circ {\tilde{R}}_{\tilde{B}})(a,b).$

So ${\mathcal{R}}_{\mathcal{B}}{\circ }_{{}_P}{\mathcal{R}}_{\mathcal{A}}={\mathcal{R}}_{\mathcal{A}{\circ }_{{}_P}\mathcal{B}}.$

Now, let us prove the second part. Since ${\tilde{R}}_{\tilde{B}}\circ {\tilde{R}}_{\tilde{A}}={\tilde{R}}_{\tilde{A}\circ \tilde{B}}$ from the above, it is sufficient to show that $R_B{\circ }_{{}_R}{R}_A=R_{A{\circ }_{{}_R}B}$. Then, we have

$\left(R_B{\circ }_{{}_R}{R}_A\right)(a,b)={\bigwedge }_{z\in G}[R_A(a,z)\vee R_B(z,b)]$

$={\bigwedge }_{z\in G}[A\left(a{z}^{-1}\right)\vee B\left(z{b}^{-1}\right)]$

$={\bigwedge }_{a{b}^{-1}=xy}[A\left(x\right)\vee B\left(y\right)]$

$=\left(A{\circ }_{{}_R}B\right)\left(a{b}^{-1}\right)$

$=\left(R_A{\circ }_{{}_R}{R}_B\right)(a,b).$

Thus, $R_B{\circ }_{{}_R}{R}_A=R_A{\circ }_{{}_R}{R}_B.$ So, the second part holds. □

Proposition 33.

$(PCC\left(G\right),{\circ }_{{}_P})$[resp., $(RCC\left(G\right),{\circ }_{{}_R})$]is a semilattice.

Proof. 

Let $\tilde{R},\tilde{S}\in IVFC\left(G\right)$ and let $(x,z)\in G\times G$. Then, we have

$(\tilde{S}\circ \tilde{R})(x,z)={\bigvee }_{y\in G}[\tilde{R}(x,y)\wedge \tilde{S}(y,z)]$

$={\bigvee }_{y\in G}[\tilde{R}(x{y}^{-1},e)\wedge \tilde{S}(e,y^{-1}z)]$

$={\bigvee }_{y\in G}[\tilde{S}(e,y^{-1}z)\wedge \tilde{R}(x{y}^{-1},e)]$

$={\bigvee }_{x{y}^{-1}z}[\tilde{S}(x,x{y}^{-1}z)\wedge \tilde{R}(x{y}^{-1}z,z)]$

$=(\tilde{R}\circ \tilde{S})(x,z).$

Thus, $\tilde{S}\circ \tilde{R}=\tilde{R}\circ \tilde{S}.$ From Theorem 10, it is obvious that $\tilde{S}\circ \tilde{R}\in IVFC\left(G\right).$ Thus $\left(IVFC\right(G),\circ )$ is a semigroup. We can easily see that $\tilde{R}\circ \tilde{R}=\tilde{R}.$ So $\left(IVFC\right(G),\circ )$ is a semilattice. It is well-known (Theorem 3.7 [39]) that $\left(FC\right(G),\circ )$ is a semilattice. Hence, $(PCC\left(G\right),{\circ }_{{}_P})$ is a semilattice.

To prove the second part, it is sufficient to show that $(FAC\left(G\right),{\circ }_{{}_R})$ is a semilattice. We can easily see that $S{\circ }_{{}_R}R=R{\circ }_{{}_R}S$ and $R{\circ }_{{}_R}R=R$ for any $R,S\in FAC\left(G\right)$. Moreover, from Theorem 10, $S{\circ }_{{}_R}R\in FAC\left(G\right)$ for any $R,S\in FAC\left(G\right)$. Then, $(FAC\left(G\right),{\circ }_{{}_R})$ is a semilattice. □

Proposition 34.

Let $\mathcal{R}\in PCC\left(G\right)$[resp., $\mathcal{R}\in RCC\left(G\right)$]and let ${\mathcal{A}}_{\mathcal{R}}=〈{\tilde{A}}_{\tilde{R}},A_R〉$ be the cubic set in G defined as follows: for each $x\in G$,

$${\mathcal{A}}_{\mathcal{R}}\left(x\right)=\mathcal{R}(x,e)=〈\tilde{R}(x,e),R(x,e)〉.$$

Then, ${\mathcal{A}}_{\mathcal{R}}\in PCNG\left(G\right)$[resp., ${\mathcal{A}}_{\mathcal{R}}\in RCNG\left(G\right)$].

In this case, we will call ${\mathcal{A}}_{\mathcal{R}}$ a P-cubic normal subgroup [resp., an R-cubic normal subgroup] of G induced by $\mathcal{R}$.

Proof. 

To prove the first part, let $x,y\in G$. Then, we have

${\tilde{A}}_{\tilde{R}}\left(x{y}^{-1}\right)=\tilde{R}(x{y}^{-1},e)$

$=\tilde{R}(x,y)$ (Since $\tilde{R}$ is compatible)

$\ge {\bigvee }_{z\in G}[\tilde{R}(x,z)\wedge \tilde{R}(z,y)]$ (Since $\tilde{R}$ is transitive)

$\ge \tilde{R}(x,e)\wedge \tilde{R}(e,y)$

$=\tilde{R}(x,e)\wedge \tilde{R}(y,e)$ (Since $\tilde{R}$ is symmetric)

$={\tilde{A}}_{\tilde{R}}\left(x\right)\wedge {\tilde{A}}_{\tilde{R}}\left(y\right).$

Thus ${\tilde{A}}_{\tilde{R}}\in IVFG\left(G\right)$. On the other hand, we get

${\tilde{A}}_{\tilde{R}}\left(xy\right)=\tilde{R}(xy,e)$

$=\tilde{R}(y\left(xy\right)y^{-1},ye{y}^{-1})$ (Since $\tilde{R}$ is compatible)

$=\tilde{R}(yx,e)$

$={\tilde{A}}_{\tilde{R}}\left(yx\right).$

So, according to Theorem 3 (1), ${\tilde{A}}_{\tilde{R}}\in IVFNG\left(G\right)$. It is well known (Theorem 3.4 [39]) that $A_R\in FNG\left(G\right)$. Hence, ${\mathcal{A}}_{\mathcal{R}}\in PCNG\left(G\right)$

Now let us show the second part. Since ${\tilde{A}}_{\tilde{R}}\in IVFNG\left(G\right)$, it is sufficient to prove that $A_R\in FANG\left(G\right)$. Let $x,y\in G.$ Then, we have

$A_R\left(x{y}^{-1}\right)=R(x{y}^{-1},e)$

$=R(x,y)$

$\le {\bigwedge }_{z\in G}[R(x,z)\vee R(z,y)]$ (Since R is fuzzy anti-transitive)

$\le R(x,e)\vee R(e,y)$

$=R(x,e)\vee R(y,e)$ (Since R is fuzzy symmetric)

$=A_R\left(x\right)\vee A_R\left(y\right).$

Thus, according to Theorem 3 (2), $A_R\in FANG\left(G\right).$ So ${\mathcal{A}}_{\mathcal{R}}\in RCNG\left(G\right)$. □

Now, we give the extension of (1) already mentioned in the introduction via cubic sets.

Proposition 35.

There is a bijection between $PCN\left(G\right)$ and $PCC\left(G\right)$[resp., $RCN\left(G\right)$ and $RCC\left(G\right)$].

Proof. 

We define $\Psi :PCN\left(G\right)\to PCC\left(G\right)$ [resp., $\Phi :RCN\left(G\right)\to RCC\left(G\right)$] as follows: for each $\mathcal{A}\in PCN\left(G\right)$ [resp., $RCN\left(G\right)$],

$$\Psi \left(\mathcal{A}\right)={\mathcal{R}}_{\mathcal{A}}[\mathrm{resp}.\Phi \left(\mathcal{A}\right)={\mathcal{R}}_{\mathcal{A}}].$$

Then, we can easily see that $\Psi$ [resp., $\Phi$] is well defined. Furthermore, it can be seen that $\Psi$ [resp., $\Phi$] is bijective. □

6. Conclusions

We saw thar P-cubic subgroupoids (resp., ideals, subgroups and congruences) have almost similar properties to fuzzy subgroupoids (resp., ideals, subgroups and comgruences) and interval-valued fuzzy groupoids (resp., ideals, subgroups and congruences). Although R-cubic subgroupoids (resp., ideals, subgroups and congruences) have a different structure than P-cubic subgroupoids (resp., ideals, subgroups and congruences), we could confirm it has properties similar to those of the corresponding fuzzy and interval-valued fuzzy concepts. In the future, we expect that one will study various algebraic structures based on cubic sets. In particular, we want to find the characteristics of several types of semigroups by cubic ideals and cubic bi-ideals. Moreover, we will study the graph problem based on a cubic set and present a smooth transportation network system as an example.

7. Discussions

In classical semigropus, special semigroups—for example, a semigroup which is regular (resp., complete regular, intra-regular, a semilattice of simple semigroups, a semilattice of groups)—are known for their characterization by ideal and bi-ideal. Can such characterizations be extended by cubic-ideal and cubic bi-ideal? We think this study is necessary.