A New Perspective on L-Fuzzy Ideals in MV-Algebras

Abstract

In this paper, we investigate some properties of L-fuzzy ideals in MV-algebras from a new perspective by taking the completely distributive lattice L as the lattice-valued context. Firstly, we introduce the notion of the L-fuzzy ideal degree in an MV-algebra, which can illustrate the extent to which an L-fuzzy subset qualifies as an L-fuzzy ideal. Secondly, we further use four kinds of cut sets to characterize the L-fuzzy ideal degree. Furthermore, we show that the L-fuzzy ideal degree is precisely an L-fuzzy convex structure in an MV-algebra and discuss its related properties. Finally, we demonstrate that the collection of all L-fuzzy ideals in an MV-algebra forms an L-convex structure and present its L-convex hull formula.

1. Introduction

The profound symmetry between classical and fuzzy sets establishes a conceptual bridge, connecting the discrete binary world of sharp boundaries to a continuous realm characterized by gradual shades of membership. This duality fundamentally extends the absolute logic of “in or out” into a nuanced continuum, enabling the formal mathematical representation of vagueness and partial truth. With the development of fuzzy logic, scholars have proposed various logical algebras such as Residuated lattice [1,2], BL-algebra [3] and MV-algebra [4]. Among these logical algebras, MV-algebras originated with Chang’s work, serving as the algebraic semantics for the many-valued Łukasiewicz propositional logic, and have attracted extensive attention from scholars and been systematically studied from multiple perspectives such as topological properties [5,6], state theory [7,8], convex structure [9], and ideal theory [10,11].

The concept of ideals has been systematically extended to the fuzzy setting, marking a significant advancement in the study of non-classical logical algebras. In 1994, Hoo [12] initiated this line of research by applying fuzzy set theory to MV-algebras and introducing the notion of fuzzy ideals within this framework. Subsequently, in a further contribution [13], Hoo deepened the investigation by examining various types of fuzzy ideals and demonstrating the fundamental equivalence between fuzzy implicative ideals and fuzzy Boolean ideals in MV-algebras. Building upon this foundation, Forouzesh [14] formulated the concept of fuzzy p-ideals in MV-algebras and established its equivalence with both fuzzy Boolean ideals and fuzzy implication ideals, thereby reinforcing the interconnections among these generalized ideal types. For additional research developments concerning fuzzy ideals in MV-algebras, interested readers may consult references [15,16]. Concurrently with these algebraic developments, the evolution of fuzzy set theory has prompted a substantial generalization of truth-value structures, expanding from the conventional unit interval $[0,1]$ to more abstract and broadly defined lattice-based frameworks, particularly including completely distributive lattices. This inspires us to wonder if there exists a more general case than L-fuzzy ideals, which prompts us to extend the notion of L-fuzzy ideals to a more general framework, so we study L-fuzzy ideals in a degree of approach that can be used to characterize L-fuzzy ideals from a new perspective.

Convexity [17,18] is a fundamental concept in modern mathematical research. It originated from the study of simple geometric problems in low-dimensional Euclidean spaces. Furthermore, convexity is also presented in numerous mathematical areas [9,19,20,21]. The extension of convex structures into fuzzy settings represents an important development in the fusion of algebraic and fuzzy mathematics. In 1994, Rosa [22] pioneered this direction by generalizing classical convex structures to the fuzzy framework, thereby introducing the fundamental concept of fuzzy convex structures. This line of research was further advanced in 2009 when Maruyama [23] expanded the theoretical foundation by adopting a completely distributive lattice as the lattice-value context, subsequently defining the more general L-convex structures. A significant conceptual contribution came from Shi and Xiu [24], who formulated the notion of M-fuzzifying convex structures—an innovative framework that quantifies the degree to which a given subset satisfies the properties of a convex set. Building on this, they subsequently developed the comprehensive framework of $(L,M)$-fuzzy convex structures [25]. When the lattices L and M coincide, an $(L,L)$-fuzzy convexity is conveniently abbreviated as an L-fuzzy convex structure, a case that has since been extensively investigated from multiple perspectives [26,27,28,29]. It is well known that MV-algebras are closely related to topology. Convex structures, as structures similar to topological structures, also have a certain connection with MV-algebras. In our paper, we aim to construct L-fuzzy convex structures and L-convex structures in MV-algebras.

In a parallel development inspired by the quantitative approach of degree theory, Shi and Xin [30] introduced the definition of L-fuzzy subgroup degree, which provides a precise measure of how closely an arbitrary L-fuzzy subset approximates the structure of an L-fuzzy subgroup. The degree-based methodology has proven highly fruitful, serving as a foundation for subsequent extensions to various other algebraic structures [31,32,33]. Motivated by these converging research streams, our paper aims to introduce the concept of L-fuzzy ideal degree in MV-algebras; it can describe the degree to which an L-fuzzy subset becomes an L-fuzzy ideal. Thus the L-fuzzy ideal degree can be regarded as a more general case of L-fuzzy ideals. We aim to characterize L-fuzzy ideals and study some properties of L-fuzzy ideals from the view of L-fuzzy ideal degree. In addition, we can construct an L-fuzzy convex structure based on the L-fuzzy ideal degree in an MV-algebra. Through the investigation of the L-fuzzy ideal degree, it can establish its fundamental connections with L-fuzzy ideals and L-fuzzy convex structures. It is worth noting that the study of L-fuzzy ideal degree can enrich the theory of L-fuzzy ideals in MV-algebras and the theory of lattice-valued convex structures, thereby creating new bridges between fuzzy algebraic structures and fuzzy convex structures.

The organization of this paper is outlined below. In Section 2, we collect some key notations and fundamental definitions with respect to MV-algebras and L-fuzzy convex structures. In Section 3, we propose the notion of L-fuzzy ideal degree in MV-algebras based on the implication operator on a completely distributive lattice L. Furthermore, we characterize the L-fuzzy ideal degree by using four types of cut sets of L-fuzzy subsets, which can establish the connection between L-fuzzy ideal degree and L-fuzzy ideal. In Section 4, the L-fuzzy convex structure is established from the L-fuzzy ideal degree in an MV-algebra. We also focus on studying some properties of the L-fuzzy convex structure, proving that the set of all L-fuzzy ideals forms an L-convex structure and giving its concrete L-convex hull formula.

2. Preliminaries

This section is dedicated to providing the essential mathematical preliminaries and foundational results that will serve as the basis for the subsequent developments in this paper.

2.1. MV-Algebras

Definition 1 

([4]). An MV-algebra $(A,⊞,\ast ,0)$ is a structure which fulfills the following conditions: for each $s,z\in A$,

(1)

$(A,⊞,0)$ is a commutative monoid;

(2)

${\left(s^{\ast }\right)}^{\ast }=s$;

(3)

$s⊞0^{\ast }=0^{\ast }$;

(4)

${(s^{\ast }⊞z)}^{\ast }⊞z={(z^{\ast }⊞s)}^{\ast }⊞s$.

The MV-algebra $(A,⊞,\ast ,0)$ is denoted by A. Put $1=0^{\ast }$. The relation ≤ can be induced by

$$s\le z⟺s^{\ast }⊞z=1,$$

then ≤ is a partial order on A. In this paper, A will denote an MV-algebra, unless stated otherwise.

$$s\vee z:={(s^{\ast }⊞z)}^{\ast }⊞z, s\wedge z:={(s^{\ast }\vee z^{\ast })}^{\ast }.$$

Definition 2 

([34]). Let I be an empty subset of A. We call I an ideal if it fulfills the following assertions:

(I1)

$s\le z$ and $z\in I$⟹$s\in I$;

(I2)

$s,z\in I$⟹$s⊞z\in I$.

Definition 3 

([34]). If $g:A_1⟶A_2$ is a mapping between two MV-algebras, then

(1)

g is an order-preserving mapping provided $s\le z$ implies $g\left(s\right)\le g\left(z\right)$ for all $s,z\in A_1$;

(2)

g is a monotonic mapping provided $g\left(s\right)\le g\left(z\right)$ implies $s\le z$ for all $s,z\in A_1$;

(3)

g is an ⊞-morphism provided that for all $s,z\in A_1$,

$$g\left(s\right)⊞g\left(z\right)=g(s⊞z).$$

2.2. L-Fuzzy Convex Structures

This section provides a systematic overview of fundamental concepts and established results drawn from lattice theory and the theory of lattice-valued convex structures, all of which form the necessary theoretical foundation for the developments presented in this paper.

A complete lattice [35] is a partially ordered set L such that every subset $S\subseteq L$ has both a least upper bound (denoted as $\bigvee S$) and a greatest lower bound (denoted as $\bigwedge S$) in L. Suppose L is a complete lattice with the top element $1_L$ and bottom element $0_L$. If an element $d\in L$ fulfills for all $e,c\in L$, $e\wedge c\le d$ implies $e\le d$ or $c\le d$, then d is called a prime element. If an element $d\in L$ fulfills for all $e,c\in L$, $d\le e\vee c$ implies $d\le e$ or $d\le c$, then d is called a co-prime element. The collection of non-top prime elements is written as as $P\left(L\right)$ and the collection of non-bottom co-prime elements is written as $J\left(L\right)$.

Define a binary relation ≺ in a complete lattice L using the following definition: for $d,e\in L$, we say $d\prec e$ if for any subset $H\subseteq L$, $e\le \bigvee H$ implies that there exists some $c\in H$ satisfying $d\le c$. The collection $\left\{d\in L|d\prec e\right\}$ is denoted by $\beta \left(e\right)$ [35]. For $d,e\in L$, we say $e{\prec }^{op}d$ if for any subset $H\subseteq L$, $\bigwedge H\le e$ implies there exists some $c\in H$ satisfying $c\le a$. The collection $\left\{d\in L|e{\prec }^{op}d\right\}$ is denoted by $\alpha \left(e\right)$. It holds that $\alpha$ is an $\bigwedge -\bigcup$ mapping, i.e., for every family ${\left\{d_i\right\}}_{i\in \Omega }\subseteq L$,

$$\alpha \left(\underset{i\in \Omega }{\bigwedge }{d}_i\right)=\bigcup _{i\in \Omega }\alpha \left(d_i\right),$$

the mapping $\beta$ keeps union-preserving, i.e., for any family ${\left\{d_i\right\}}_{i\in \Omega }\subseteq L$,

$$\beta \left(\underset{i\in \Omega }{\bigvee }{d}_i\right)=\bigcup _{i\in \Omega }\beta \left(d_i\right).$$

We say that a complete lattice L is a completely distributive lattice [35] if for every $e\in L$, the following equation holds:

$$e=\bigvee \beta \left(e\right)=\bigwedge \alpha \left(e\right).$$

For each e in a completely distributive lattice L, define ${\beta }^{\ast }\left(e\right)=\beta \left(e\right)\cap J\left(L\right)$ and ${\alpha }^{\ast }\left(e\right)=\alpha \left(e\right)\cap P\left(L\right)$. Moreover, the following equation holds: for every element $e\in L$,

$$e=\bigvee \beta \left(e\right)=\bigvee {\beta }^{\ast }\left(e\right)=\bigwedge \alpha \left(e\right)=\bigwedge {\alpha }^{\ast }\left(e\right).$$

The mapping from a set X to L is called an L-fuzzy subset of X, and the collection of all L-fuzzy subsets of X is written as $L^X$. Also, $1_X$ is the top element in $L^X$, $0_X$ is the the bottom element in $L^X$. Let $g:X⟶Y$ be a mapping between two nonempty sets. Define the image of $\varphi$, $g_L^{\to }:L^X⟶L^Y$ and the preimage of $\eta$, $g_L^{\leftarrow }:L^Y⟶L^X$ according to the following: for any $\varphi \in L^X,\eta \in L^Y$, $s\in X$ and $v\in Y$,

$$g_L^{\to }\left(\varphi \right)\left(v\right)=\underset{g\left(s\right)=v}{\bigvee }\varphi \left(s\right) \mathrm{and} g_L^{\leftarrow }\left(\eta \right)\left(s\right)=\eta \left(g\left(s\right)\right).$$

In the subsequent sections, L will denote a completely distributive lattice, unless stated otherwise. The right adjoint (implication operation →) for the meet operation ∧ on L can be induced by the following: for all $e,c\in L$,

$$c\to e=\bigvee \left\{d\in L|c\wedge d\le e\right\}.$$

We present some basic properties for the implication operation as follows.

Lemma 1 

([36]). The operation → fulfills the following properties: for any $d,e,v\in L$ and ${\left\{v_i\right\}}_{i\in J}\subseteq L$,

(1)

$1_L\to v=v$;

(2)

$d\le v\to e$⟺$d\wedge v\le e$;

(3)

$d\to v=1_L$⟺$d\le v$;

(4)

$d\to \left({\displaystyle \underset{i\in J}{\bigwedge }}{v}_i\right)={\displaystyle \underset{i\in J}{\bigwedge }}\left(d\to v_i\right)$, it implies $d\to e\le d\to v$ if $e\le v$;

(5)

$\left({\displaystyle \underset{i\in J}{\bigvee }}{v}_i\right)\to e={\displaystyle \underset{i\in J}{\bigwedge }}\left(v_i\to e\right)$, it implies $v\to e\le d\to e$ whenever $d\le v$;

(6)

$(d\to e)\wedge (e\to v)\le d\to v$.

Definition 4 

([37]). Let X be a nonempty set, $\varphi \in L^X$ and $c\in L$. Define

$$\begin{array}{cccc}\hfill {\varphi }_{\left[c\right]}& =\left\{s\in X\left|\varphi \right(s)\ge c\right\};\hfill & \hfill {\varphi }^{\left(c\right)}& =\left\{s\in X\left|\varphi \right(s)\nleqq c\right\};\hfill \\ \hfill {\varphi }_{\left(c\right)}& =\left\{s\in X|c\in \beta \left(\varphi \left(s\right)\right)\right\};\hfill & \hfill {\varphi }^{\left[c\right]}& =\left\{s\in X|c\notin \alpha \left(\varphi \left(s\right)\right)\right\}.\hfill \end{array}$$

The definition of $(L,M)$-fuzzy convex structures was proposed by Shi and Xiu in [25]. Setting $M=L$ yields the following definition.

Definition 5 

([25]). If the mapping $\mathcal{C}:L^X⟶L$ fulfills the following assertions

(1)

$\mathcal{C}\left(0_X\right)=\mathcal{C}\left(1_X\right)=1_L$;

(2)

if ${\left\{{\varphi }_i\right\}}_{i\in J}\subseteq L^X$, then $\mathcal{C}\left({\displaystyle \underset{i\in J}{\bigwedge }}{\varphi }_i\right)\ge {\displaystyle \underset{i\in J}{\bigwedge }}\mathcal{C}\left({\varphi }_i\right)$;

(3)

if ${\left\{{\varphi }_i\right\}}_{i\in J}\subseteq L^X$ is nonempty and directed, then $\mathcal{C}\left({{\displaystyle \underset{i\in J}{\bigvee }}}^{\uparrow }{\varphi }_i\right)\ge {\displaystyle \underset{i\in J}{\bigwedge }}\mathcal{C}\left({\varphi }_i\right)$,

then $\mathcal{C}$ is said to be an L-fuzzy convex structure and the pair $(X,\mathcal{C})$ is said to be an L-fuzzy convex space.

Definition 6  

([25]). Consider the mapping $g:(X,{\mathcal{C}}_X)⟶(Y,{\mathcal{C}}_Y)$ between two L-fuzzy convex spaces.

(1)

If ${\mathcal{C}}_X\left(g_L^{\leftarrow }\left(\eta \right)\right)\ge {\mathcal{C}}_Y\left(\eta \right)$ for every $\eta \in L^Y$, then we say g is L-fuzzy convexity-preserving;

(2)

If ${\mathcal{C}}_Y\left(g_L^{\to }\left(\varphi \right)\right)\ge {\mathcal{C}}_X\left(\varphi \right)$ for every $\varphi \in L^X$, then we say g is L-fuzzy convex-to-convex.

Definition 7  

([23]). Let $\mathcal{C}$ be a family of L-subsets of X. If $\mathcal{C}$ fulfills the following assertions

(1)

$0_X,1_X\in \mathcal{C}$;

(2)

if $\{{\varphi }_i\mid i\in J\}\subseteq \mathcal{C}$, then ${\displaystyle \underset{i\in J}{\bigwedge }}{\varphi }_i\in \mathcal{C}$;

(3)

if a nonempty family $\{{\varphi }_i\mid i\in J\}\subseteq \mathcal{C}$ is directed, then ${{\displaystyle \underset{i\in J}{\bigvee }}}^{\uparrow }{\varphi }_i\in \mathcal{C}$,

then $\mathcal{C}$ is said to be an L-convex structure and the pair $(X,\mathcal{C})$ is said to be an L-convex space. For every $\varphi \in L^X$, $c{o}_{\mathcal{C}}\left(\varphi \right)=\bigwedge \{\eta \in L^X\mid \varphi \le \eta \in \mathcal{C}\}$ is said to be the L-convex hull in $\mathcal{C}$.

3. L-Fuzzy Ideal Degree in MV-Algebras

This section presents a new perspective to investigate L-fuzzy ideals in an MV-algebra, which is called L-fuzzy ideal degree. It can illustrate the extent to which an L-fuzzy subset qualifies as an L-fuzzy ideal.

The definition of L-fuzzy ideal in residuated lattices was introduced by Kengne [38]. Since MV-algebras are special cases of residuated lattices, we obtain the following definition of L-fuzzy ideal in MV-algebras.

Definition 8 

([38]). Let $\varphi \in L^A$. Then we say ϕ is an L-fuzzy ideal provided that

(LI1)

for every $s,z\in A$, $s\le z$⟹$\varphi \left(z\right)\le \varphi \left(s\right)$;

(LI2)

for every $s,z\in A$, $\varphi \left(s\right)\wedge \varphi \left(z\right)\le \varphi (s⊞z)$.

We next propose the concept of L-fuzzy ideal degree.

Definition 9. 

Define the mapping $\mathfrak{I}:L^A⟶L$ as follows: $\forall \varphi \in L^A,$

$$\mathfrak{I}\left(\varphi \right)=\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigwedge }\left(\varphi \left(s\right)\to \varphi \left(z\right)\right)\wedge \left(\varphi \left(v\right)\wedge \varphi \left(u\right)\to \varphi (v⊞u)\right).$$

Then we say that $\mathfrak{I}$ is the L-fuzzy ideal degree in A.

Remark 1. 

Let $\varphi \in L^A$. In fact, $\mathfrak{I}\left(\varphi \right)$ can characterize the extent to which ϕ is an L-fuzzy ideal. Particularly, if $\mathfrak{I}\left(\varphi \right)=1_L$, then for all $s,z,v,u\in A$ with $z\le s$,

$$\left(\varphi \left(s\right)\to \varphi \left(z\right)\right)\wedge \left(\varphi \left(u\right)\wedge \varphi \left(v\right)\to \varphi (v⊞u)\right)=1_L.$$

This implies

$$\varphi \left(s\right)\le \varphi \left(z\right) \mathrm{and} \varphi \left(u\right)\wedge \varphi \left(v\right)\le \varphi (v⊞u),$$

By Definition 8, we get ϕ is an L-fuzzy ideal. Conversely, assuming ϕ is an L-fuzzy ideal, we can easily get that $\mathfrak{I}\left(\varphi \right)=1_L$. Hence, ϕ is an L-fuzzy ideal if and only if $\mathfrak{I}\left(\varphi \right)=1_L$.

We next present examples for the L-fuzzy ideal degree.

Example 1. 

Consider an MV-algebra $A=\{0_A,s,z,u,v,1_A\}$. The Hasse diagram and Cayley tables associated with A are shown below.Let $L=\{0_L,d,e,1_L\}$ be a completely distributive lattice with Hasse diagram as follows. It is not difficult to see that the implication operation on L is induced as follows:

$$h\to m=\left\{\begin{array}{cc}{1}_L,\hfill & \mathit{if} h\le m;\hfill \\ m,\hfill & \mathit{if} h\nleqq m,h,m\in \{d,e\} \mathit{or} h=1_L;\hfill \\ e,\hfill & \mathit{if} h=d,m=0_L;\hfill \\ d,\hfill & \mathit{if} h=e,m=0_L.\hfill \end{array}\right.$$

(1) Define $\varphi \in L^A$ by

$$\varphi \left(0_A\right)=1_L,\varphi \left(s\right)=\varphi \left(u\right)=\varphi \left(v\right)=e,\varphi \left(z\right)=d,\varphi \left(1_A\right)=0_L.$$

By Definition 9, it can be proved that $\mathfrak{I}\left(\varphi \right)=d$. Since $z\le v$, we obtain

$$\varphi \left(v\right)=e\nleqq d=\varphi \left(z\right).$$

Thus ϕ is not an L-fuzzy ideal.

(2) Define $\varphi \in L^A$ as follows:

$$\varphi \left(0_A\right)=\varphi \left(s\right)=\varphi \left(z\right)=\varphi \left(u\right)=\varphi \left(v\right)=\varphi \left(1_A\right)=e.$$

By Definition 9, we get $\mathfrak{I}\left(\varphi \right)=1_L$. Thus ϕ is an L-fuzzy ideal.

Example 2.  

Consider the MV-algebra $A=\{0_A,s,z,u,v,1_A\}$ defined in Example 1 and $L=[0,1]$.

(1) Define $\varphi \in L^A$ as follows:

$$\varphi \left(0_A\right)=0.9,\varphi \left(z\right)=\varphi \left(u\right)=\varphi \left(v\right)=0.7,\varphi \left(s\right)=0.6,\varphi \left(1_A\right)=0.5.$$

It can be proved that $\mathfrak{I}\left(\varphi \right)=0.5$. Since $s\le u$, we obtain

$$\varphi \left(u\right)=0.7\nleqq 0.6=\varphi \left(s\right).$$

Thus ϕ is not an L-fuzzy ideal.

(2) Define $\varphi \in L^A$ as follows:

$$\varphi \left(0_A\right)=0.9,\varphi \left(z\right)=0.7,\varphi \left(s\right)=\varphi \left(u\right)=\varphi \left(v\right)=\varphi \left(1_A\right)=0.5.$$

It can be proved that $\mathfrak{I}\left(\varphi \right)=1$. By Remark 1, we get ϕ is an L-fuzzy ideal.

To give some characterizations for the L-fuzzy ideal degree, we first give the lemma as follows.

Lemma 2.  

Let $\varphi \in L^A$. Then for any $d\in L$, $d\le \mathfrak{I}\left(\varphi \right)$ if and only if for all $s,z,v,u\in A$ with $z\le s$,

$$d\wedge \varphi \left(s\right)\le \varphi \left(z\right) \mathit{and} \varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u).$$

Proof. 

Take $d\in L$ such that $d\le \mathfrak{I}\left(\varphi \right)$. Then

$$d\le \underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigwedge }\left(\varphi \left(s\right)\to \varphi \left(z\right)\right)\wedge \left(\varphi \left(u\right)\wedge \varphi \left(v\right)\to \varphi (v⊞u)\right)$$

⟺ for any $s,z,v,u\in A$ with $z\le s$,

$$d\le \left(\varphi \left(s\right)\to \varphi \left(z\right)\right)\wedge \left(\varphi \left(u\right)\wedge \varphi \left(v\right)\to \varphi (v⊞u)\right),$$

⟺ for any $s,z,v,u\in A$ with $z\le s$,

$$d\le \varphi \left(s\right)\to \varphi \left(z\right) \mathrm{and} d\le \varphi \left(u\right)\wedge \varphi \left(v\right)\to \varphi (v⊞u)$$

⟺ for any $s,z,v,u\in A$ with $z\le s$,

$$d\wedge \varphi \left(s\right)\le \varphi \left(z\right) \mathrm{and} \varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (u⊞v),$$

as desired. □

We next study the characterization for the L-fuzzy ideal degree.

Theorem 1.  

Let $\varphi \in L^A$. Then

$$\mathfrak{I}\left(\varphi \right)=\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}.$$

Proof. 

Let $t\in L$ fulfill

$$t\le \mathfrak{I}\left(\varphi \right)=\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigwedge }\left(\varphi \left(s\right)\to \varphi \left(z\right)\right)\wedge \left(\varphi \left(u\right)\wedge \varphi \left(v\right)\to \varphi (v⊞u)\right).$$

By Lemma 2, we know that for all $s,z,v,u\in A$ with $z\le s$,

$$t\wedge \varphi \left(s\right)\le \varphi \left(z\right) \mathrm{and} t\wedge \varphi \left(u\right)\wedge \varphi \left(v\right)\le \varphi (v⊞u).$$

This results that

$$t\le \underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}.$$

Thus we obtain

$$\mathfrak{I}\left(\varphi \right)\le \underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}.$$

Let $t\in L$ fulfill

$$t\prec \underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}.$$

Then there exists $d\in L$ satisfying for all $s,z,v,u\in A$ with $z\le s$,

$$d\wedge \varphi \left(s\right)\le \varphi \left(z\right) \mathrm{and} \varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi \left(v\right),$$

such that $t\le d$. Then it follows that for all $s,z,v,u\in A$ with $z\le s$,

$$t\le d\le \varphi \left(s\right)\to \varphi \left(z\right) \mathrm{and} t\le d\le \left(\varphi \left(u\right)\wedge \varphi \left(v\right)\right)\to \varphi (v⊞u).$$

Hence

$$t\le \underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigwedge }\left(\varphi \left(s\right)\to \varphi \left(z\right)\right)\wedge \left(\varphi \left(u\right)\wedge \varphi \left(v\right)\to \varphi (v⊞u)\right)=\mathfrak{I}\left(\varphi \right).$$

The arbitrariness of t implies

$$\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}\le \mathfrak{I}\left(\varphi \right)$$

as desired. □

In [30,33], some characterizations for L-fuzzy filter degree and L-fuzzy subgroup degree are given via cut sets of L-fuzzy sets. We follow the proof idea in [30,33] and treat the empty set as an ideal; we explore some characterizations for the L-fuzzy ideal degree in MV-algebras in the following.

Theorem 2.  

Let $\varphi \in L^A$. Then

(1)

$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\le d,{\varphi }_{\left[e\right]}is an ideal of A\right\}$;

(2)

$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\in J\left(L\right),e\le d,{\varphi }_{\left[e\right]}is an ideal of A\right\}.$

Proof. 

(1) Suppose for each $e\le d$, ${\varphi }_{\left[e\right]}$ is an ideal of A. Take $s,z\in A$ with $z\le s$. Let $t=d\wedge \varphi \left(s\right)$. Then we obtain $t\le d$ and $t\le \varphi \left(s\right)$, so $s\in {\varphi }_{\left[t\right]}$. By the assumption, ${\varphi }_{\left[t\right]}$ is an ideal. Since $z\le s$ and $s\in {\varphi }_{\left[t\right]}$, we obtain $z\in {\varphi }_{\left[t\right]},$ which means

$$d\wedge \varphi \left(s\right)=t\le \varphi \left(z\right).$$

Let $v,u\in A$ and $c=d\wedge \varphi \left(v\right)\wedge \varphi \left(u\right)$. Then we obtain $c\le d$ and $c\le \varphi \left(v\right),\varphi \left(u\right)$. By the assumption, ${\varphi }_{\left[c\right]}$ is an ideal and $v,u\in {\varphi }_{\left[c\right]}$. Thus $v⊞u\in {\varphi }_{\left[c\right]}$, so

$$d\wedge \varphi \left(v\right)\wedge \varphi \left(u\right)=c\le \varphi (v⊞u).$$

This gives that

$$\begin{array}{cc}\hfill \mathfrak{I}\left(\varphi \right)& =\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}\hfill \\ \hfill & \ge \bigvee \left\{d\in L|\forall e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}.\hfill \end{array}$$

Conversely, suppose for all $s,z,v,u\in A$ with $z\le s$, $d\wedge \varphi \left(s\right)\le \varphi \left(z\right)$ and $\varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u)$. For any $e\le d$, we prove ${\varphi }_{\left[e\right]}$ is an ideal of A. If $s\in {\varphi }_{\left[e\right]}$ and $z\le s$, then $e\le \varphi \left(s\right)$. This suggests that

$$e\le d\wedge \varphi \left(s\right)\le \varphi \left(z\right).$$

So we obtain $z\in {\varphi }_{\left[e\right]}$. If $u,v\in {\varphi }_{\left[e\right]}$, then

$$e\le \varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u).$$

This gives that $v⊞u\in {\varphi }_{\left[e\right]}.$ Hence, by Definition 2, ${\varphi }_{\left[e\right]}$ is an ideal. Thus we get the result

$$\begin{array}{cc}\hfill \mathfrak{I}\left(\varphi \right)& =\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}\hfill \\ \hfill & \le \bigvee \left\{d\in L|\forall e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\},\hfill \end{array}$$

as desired.

(2) By (1), one can see that that

$$\begin{array}{cc}\hfill \mathfrak{I}\left(\varphi \right)& =\bigvee \left\{d\in L|\forall e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}\hfill \\ & \le \bigvee \left\{d\in L|\forall e\in J\left(L\right),e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}.\hfill \end{array}$$

Conversely, we state that

$$\left\{d\in L|\forall e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}\supseteq \left\{d\in L|\forall e\in J\left(L\right),e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}.$$

Let $d\in \left\{d\in L|\forall e\in J\left(L\right),e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}$. Then for any $e\in J\left(L\right),e\le d$, ${\varphi }_{\left[e\right]}$ is an ideal. For any $c\in L,c\le d$, since $c=\bigvee \beta \left(c\right)\cap J\left(L\right)$, we obtain

$${\varphi }_{\left[c\right]}={\varphi }_{[\bigvee \beta (c)\cap J(L\left)\right]}=\bigcap \left\{{\varphi }_{\left[b_i\right]}\right|b_i\in \beta \left(c\right)\cap J\left(L\right)\}.$$

It follows from $b_i\in J\left(L\right)$ and $b_i\le d$ that ${\varphi }_{\left[b_i\right]}$ is an ideal for any $b_i\in \beta \left(c\right)\cap J\left(L\right)$. Thus $\bigcap \left\{{\varphi }_{\left[b_i\right]}\right|b_i\in \beta \left(c\right)\cap J\left(L\right)\}$ is an ideal, implying that ${\varphi }_{\left[c\right]}$ is an ideal. Hence

$$\left\{d\in L|\forall e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}\supseteq \left\{d\in L|\forall e\in J\left(L\right),e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}.$$

This gives that

$$\begin{array}{cc}\hfill \mathfrak{I}\left(\varphi \right)& =\bigvee \left\{d\in L|\forall e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}\hfill \\ & \ge \bigvee \left\{d\in L|\forall e\in J\left(L\right),e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\},\hfill \end{array}$$

as desired. □

Theorem 3.  

Let $\varphi \in L^A$. Then

(1)

$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathit{is} \mathit{an} \mathit{ideal} \mathit{of} A\right\}$;

(2)

$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\in P\left(L\right),e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathit{is} \mathit{an} \mathit{ideal} \mathit{of} A\right\}$.

Proof. 

(1) Suppose that for any $s,z,v,u\in A$ with $z\le s$, take $d\in L$ with $d\wedge \varphi \left(s\right)\le \varphi \left(z\right)$ and $d\wedge \varphi \left(u\right)\wedge \varphi (v⊞u)\le \varphi \left(v\right)$. Let $e\notin \alpha \left(d\right)$. We prove that ${\varphi }^{\left[e\right]}$ is an ideal.

If $z\le s$ and $s\in {\varphi }^{\left[e\right]}$, then $e\notin \alpha \left(\varphi \right(s\left)\right)$. From $d\wedge \varphi \left(s\right)\le \varphi \left(z\right)$, we have

$$\alpha \left(\varphi \right(z\left)\right)\subseteq \alpha (d\wedge \varphi (s\left)\right)=\alpha \left(\varphi \right(s\left)\right)\cup \alpha \left(d\right).$$

From $e\notin \alpha \left(\varphi \right(s\left)\right)\cup \alpha \left(d\right)$, we get $e\notin \alpha \left(\varphi \right(z\left)\right).$ Thus $z\in {\varphi }^{\left[e\right]}$. If $u,v\in {\varphi }^{\left[e\right]}$, then

$$e\notin \alpha \left(\varphi \right(u\left)\right)\cup \alpha \left(\varphi \right(v\left)\right)\cup \alpha \left(d\right)=\alpha \left(\varphi \right(u)\wedge d\wedge \varphi (v\left)\right).$$

Since

$$\varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u),$$

we obtain

$$\alpha \left(\varphi \right(v⊞u\left)\right)\subseteq \alpha \left(\varphi \right(u)\wedge d\wedge \varphi (v\left)\right).$$

Thus $e\notin \alpha \left(\varphi \right(v⊞u\left)\right).$ This gives that $v⊞u\in {\varphi }^{\left[e\right]}$. So ${\varphi }^{\left[e\right]}$ is an ideal. It results that

$$\begin{array}{cc}\hfill \mathfrak{I}\left(\varphi \right)& =\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}\hfill \\ \hfill & \le \bigvee \left\{d\in L|\forall e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}.\hfill \end{array}$$

Conversely, let $d,e\in L$ such that $e\notin \alpha \left(d\right)$. Suppose that ${\varphi }^{\left[e\right]}$ is an ideal. Then for all $s,z,v,u\in A$ with $z\le s$, we next show that

$$d\wedge \varphi \left(s\right)\le \varphi \left(z\right) \mathrm{and} \varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u).$$

If $e\notin \alpha (d\wedge \varphi (s\left)\right)$, then it follows from

$$\alpha (d\wedge \varphi (s\left)\right)=\alpha \left(d\right)\cup \alpha \left(\varphi \right(s\left)\right)$$

that

$$e\notin \alpha \left(d\right) \mathrm{and} e\notin \alpha \left(\varphi \right(s\left)\right).$$

So we obtain $s\in {\varphi }^{\left[e\right]}$. This gives ${\varphi }^{\left[e\right]}$ is an ideal of A, which implies $z\in {\varphi }^{\left[e\right]}$. Thus $e\notin \alpha \left(\varphi \right(z\left)\right).$ Since e is arbitrary, we obtain $\alpha \left(\varphi \right(z\left)\right)\subseteq \alpha (d\wedge \varphi (s\left)\right).$ Hence

$$d\wedge \varphi \left(s\right)=\bigwedge \alpha (d\wedge \varphi (s\left)\right)\le \bigwedge \alpha \left(\varphi \right(z\left)\right)=\varphi \left(z\right).$$

Similarly, for any $v,u\in A$, we can get

$$\varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u).$$

This results that

$$\begin{array}{cc}\hfill \mathfrak{I}\left(\varphi \right)& =\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}\hfill \\ \hfill & \ge \bigvee \left\{d\in L|\forall e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}.\hfill \end{array}$$

(2) Since

$$\begin{array}{cc}\hfill & \left\{d\in L|\forall e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathit{of} A\right\}\hfill \\ \hfill & \subseteq \left\{d\in L|\forall e\in P\left(L\right),e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathit{of} A\right\},\hfill \end{array}$$

we obtain

$$\begin{array}{cc}\hfill & \bigvee \left\{d\in L|\forall e\in P\left(L\right),e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathit{of} A\right\}\hfill \\ \hfill & \ge \bigvee \left\{d\in L|\forall e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathit{of} A\right\}.\hfill \end{array}$$

By (1) we first obtain

$$\mathfrak{I}\left(\varphi \right)\le \bigvee \left\{d\in L|\forall e\in P\left(L\right),e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathit{of} A\right\}.$$

Conversely, suppose $d\in L$ and ${\varphi }^{\left[e\right]}$ is an ideal for all $e\in P\left(L\right)$ satisfying $e\notin \alpha \left(d\right)$. In the following, we prove $d\wedge \varphi \left(s\right)\le \varphi \left(z\right)$ and $\varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u)$ for all $s,z,u,v\in A$ satisfying $s\le z$. Let $e\in P\left(L\right)$ and $e\notin {\alpha }^{\ast }(d\wedge \varphi \left(s\right))$. Then

$$e\notin \alpha (d\wedge \varphi (s\left)\right)=\alpha \left(d\right)\cup \alpha \left(\varphi \right(s\left)\right).$$

Thus $e\notin \varphi \left(s\right)$ and $e\notin \alpha \left(d\right)$, which implies $s\in {\varphi }^{\left[e\right]}$. Since ${\varphi }^{\left[e\right]}$ is an ideal, we obtain $z\in {\varphi }^{\left[e\right]}$, i.e., $e\notin \alpha \left(\varphi \right(z\left)\right)$. So $e\notin {\alpha }^{\ast }\left(\varphi \left(z\right)\right)$; this gives ${\alpha }^{\ast }(d\wedge \varphi \left(s\right)\supseteq {\alpha }^{\ast }\left(\varphi \left(z\right)\right)$. Thus

$$d\wedge \varphi \left(s\right)=\bigwedge {\alpha }^{\ast }(d\wedge \varphi \left(s\right))\le \bigwedge {\alpha }^{\ast }\left(\varphi \left(z\right)\right)=\varphi \left(z\right).$$

We can similarly prove $d\wedge \varphi \left(s\right)\wedge \varphi \left(z\right)\le \varphi (v⊞u)$. This shows

$$\mathfrak{I}\left(\varphi \right)\ge \bigvee \left\{d\in L|\forall e\in P\left(L\right),e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} of A\right\}.$$

Hence

$$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\in P\left(L\right),e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} of A\right\}.$$

as desired. □

Theorem 4.  

Let $\varphi \in L^A$. Then

$$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\in P\left(L\right),d\nleqq e,{\varphi }^{\left(e\right)}\mathit{is} \mathit{an} \mathit{ideal} A\right\}.$$

Proof. 

Let $d\in L$ fulfill $d\wedge \varphi \left(s\right)\le \varphi \left(z\right)$ and $\varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u)$ for any $s,z,v,u\in A$ with $z\le s$. Suppose that $e\in P\left(L\right)$ such that $d\nleqq e$. We next show that ${\varphi }^{\left(e\right)}$ is an ideal of A. Let $s\in {\varphi }^{\left(e\right)}$ and $z\le s$. If $z\notin {\varphi }^{\left(e\right)}$, then $\varphi \left(z\right)\le e$. It follows from $d\wedge \varphi \left(s\right)\le \varphi \left(z\right)$ that $d\wedge \varphi \left(s\right)\le e.$ By the fact that $e\in P\left(L\right)$ and $s\in {\varphi }^{\left(e\right)}$, i.e., $\varphi \left(s\right)\nleqq e$, we get $d\le e$, a contradiction. Thus we can get $z\in {\varphi }^{\left(e\right)}.$ Similarly, for all $v,u\in A$, we obtain

$$u,v\in {\varphi }^{\left(e\right)} \mathrm{implies} v⊞u\in {\varphi }^{\left(e\right)}.$$

This results that

$$\begin{array}{cc}\hfill \mathfrak{I}\left(\varphi \right)& =\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}\hfill \\ \hfill & \le \bigvee \left\{d\in L|\forall e\in P\left(L\right),\varphi \nleqq e,{\varphi }^{\left(e\right)}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}.\hfill \end{array}$$

Conversely, let $d\in L$ and $e\in P\left(L\right)$ fulfill $d\nleqq e$. Suppose that ${\varphi }^{\left(e\right)}$ is an ideal of A. Next, we show that for all $s,z,v,u\in A$ with $z\le s$, it holds

$$d\wedge \varphi \left(s\right)\le \varphi \left(z\right) \mathrm{and} \varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u).$$

For any $s,z\in A$ with $z\le s$, let $e\in P\left(L\right)$ such that $d\wedge \varphi \left(s\right)\nleqq e$. Then we obtain

$$d\nleqq e \mathrm{and} \varphi \left(s\right)\nleqq e.$$

It results that $s\in {\varphi }^{\left(e\right)}$. By the assumption, ${\varphi }^{\left(e\right)}$ is an ideal. This gives that $z\in {\varphi }^{\left(e\right)}$, i.e., $\varphi \left(z\right)\nleqq e.$ Since e is arbitrary, we obtain

$$d\wedge \varphi \left(s\right)\le \varphi \left(z\right).$$

Similarly, for any $v,u\in A$, we can get

$$\varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u).$$

It results that

$$\begin{array}{cc}\hfill \mathfrak{I}\left(\varphi \right)& =\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}\hfill \\ \hfill & \ge \bigvee \left\{d\in L|\forall e\in P\left(L\right),d\nleqq e,{\varphi }^{\left(e\right)}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\},\hfill \end{array}$$

as desired. □

Theorem 5.  

Let $\varphi \in L^A$. If $\beta (d\wedge e)=\beta \left(d\right)\cap \beta \left(e\right)$ for all $d,e\in L$. Then

$$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\in \beta \left(d\right),{\varphi }_{\left(e\right)}\mathit{is} \mathit{an} \mathit{ideal} \mathit{of} A\right\}.$$

Proof. 

Suppose that $d\in L$ such that $d\wedge \varphi \left(s\right)\le \varphi \left(z\right)$ and $\varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u)$ for all $s,z,v,u\in A$ with $z\le s$. For any $e\in \beta \left(d\right)$, we next prove that ${\varphi }_{\left(e\right)}$ is an ideal of A.

If $s\in {\varphi }_{\left(e\right)}$ and $z\le s$, then

$$e\in \beta \left(\varphi \right(s\left)\right)\cap \beta \left(d\right)=\beta \left(\varphi \right(s)\wedge d)\subseteq \beta \left(\varphi \right(z\left)\right).$$

So we obtain $z\in {\varphi }_{\left(e\right)}$. If $u,v\in {\varphi }_{\left(e\right)}$, then

$$e\in \beta \left(\varphi \right(u\left)\right)\cap \beta \left(\varphi \right(v\left)\right)\cap \beta \left(d\right)=\beta \left(\varphi \right(u)\wedge \varphi (v)\wedge d)\subseteq \beta \left(\varphi \right(v⊞u\left)\right).$$

Thus $u⊞v\in {\varphi }_{\left(e\right)}.$ This gives that

$$\begin{array}{cc}\hfill \mathfrak{I}\left(\varphi \right)& =\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}\hfill \\ \hfill & \le \bigvee \left\{d\in L|\forall e\in \beta \left(d\right),{\varphi }_{\left(e\right)}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}.\hfill \end{array}$$

Conversely, let $d\in L$. Suppose for every $e\in L$ satisfying $e\in \beta \left(d\right)$, ${\varphi }_{\left(e\right)}$ is an ideal A, we next show

$$d\wedge \varphi \left(s\right)\le \varphi \left(z\right) \mathrm{and} \varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)\le \varphi (v⊞u).$$

Let $e\in \beta (d\wedge \varphi (s\left)\right)$. Since

$$\beta (d\wedge \varphi (s\left)\right)=\beta \left(\varphi \right(s\left)\right)\cap \beta \left(d\right),$$

we obtain

$$e\in \beta \left(d\right) \mathrm{and} e\in \beta \left(\varphi \right(s\left)\right).$$

This gives $s\in {\varphi }_{\left(e\right)}$. So we can get that ${\varphi }_{\left(e\right)}$ is an ideal of A. It follows from $z\le s$ that $z\in {\varphi }_{\left(e\right)}$. So we obtain $e\in \beta \left(\varphi \right(z\left)\right).$ From the arbitrariness of e, it holds that

$$\beta (d\wedge \varphi (s\left)\right)\subseteq \beta \left(\varphi \right(z\left)\right).$$

Hence $d\wedge \varphi \left(s\right)\le \varphi \left(z\right)$.

Let $e\in \beta \left(\varphi \right(u)\wedge d\wedge \varphi (v\left)\right)$. By

$$\beta \left(\varphi \right(u)\wedge d\wedge \varphi (v\left)\right)=\beta \left(d\right)\cap \beta \left(\varphi \right(u\left)\right)\cap \beta \left(\varphi \right(v\left)\right),$$

we get

$$e\in \beta \left(d\right),e\in \beta \left(\varphi \right(u\left)\right) \mathrm{and} e\in \beta \left(\varphi \right(v\left)\right).$$

This gives that $u,v\in {\varphi }_{\left(e\right)}.$ By the assumption, we can get ${\varphi }_{\left(e\right)}$ is an ideal of A. Thus $v⊞u\in {\varphi }_{\left(e\right)},$ implying that $e\in \beta \left(\varphi \right(v⊞u\left)\right)$. From the arbitrariness of e, it holds that

$$\beta \left(\varphi \right(u)\wedge d\wedge \varphi (v\left)\right)\subseteq \beta \left(\varphi \right(v⊞u\left)\right).$$

This gives

$$\varphi \left(u\right)\wedge d\wedge \varphi \left(v\right)=\bigvee \beta \left(\varphi \right(u)\wedge d\wedge \varphi (v\left)\right)\le \bigvee \beta \left(\varphi \right(v⊞u\left)\right)=\varphi (v⊞u).$$

Hence,

$$\begin{array}{cc}\hfill \mathfrak{I}\left(\varphi \right)& =\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigvee }\left\{d\in L|d\wedge \varphi (s)\le \varphi (z),\varphi (u)\wedge d\wedge \varphi (v)\le \varphi (v⊞u)\right\}\hfill \\ \hfill & \ge \bigvee \left\{d\in L|\forall e\in \beta \left(d\right),{\varphi }_{\left(e\right)}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\},\hfill \end{array}$$

as desired. □

By the fact $\varphi \in L^A$ is an L-fuzzy ideal if and only if $\mathfrak{I}\left(\varphi \right)=1_L$, we provide some characterizations for L-fuzzy ideals by using their cut sets.

Proposition 1.  

Let $\varphi \in L^A$. Then the statements listed below are equivalent:

(1)

ϕ is an L-fuzzy ideal;

(2)

Each $d\in L$, ${\varphi }_{\left[d\right]}$ is an ideal;

(3)

Each $d\in J\left(L\right)$, ${\varphi }_{\left[d\right]}$ is an ideal;

(4)

Each $d\in L$, ${\varphi }^{\left[d\right]}$ is an ideal;

(5)

Each $d\in P\left(L\right)$, ${\varphi }^{\left[d\right]}$ is an ideal;

(6)

Each $d\in P\left(L\right)$, ${\varphi }^{\left(d\right)}$ is an ideal.

Proof. 

(1) ⇒ (2) Suppose $\varphi$ is an L-fuzzy ideal. Let $s,z\in A$, $d\in L$ such that $z\le s$ and $s\in {\varphi }_{\left[d\right]}$. Since $\varphi$ is an L-fuzzy ideal, we obtain

$$\varphi \left(z\right)\ge \varphi \left(s\right)\ge d.$$

Thus $z\in {\varphi }_{\left[d\right]}.$

Let $v,u\in {\varphi }_{\left[d\right]}$. Then $\varphi \left(u\right)\ge d,\varphi \left(v\right)\ge d$. Since $\varphi$ is an L-fuzzy ideal, by (LI2), we obtain

$$\varphi (v⊞u)\ge \varphi \left(u\right)\wedge \varphi \left(v\right)\ge d,$$

i.e., $v\in {\varphi }_{\left[d\right]}$. Hence ${\varphi }_{\left[d\right]}$ is an ideal.

(2) ⇒ (3) is obvious.

(3) ⇒ (1) By Theorem 2 (2), we get

$$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\in J\left(L\right),e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}.$$

By the fact that for any $d\in J\left(L\right)$, ${\varphi }_{\left[d\right]}$ is an ideal, we obtain

$$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\in J\left(L\right),e\le d,{\varphi }_{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} \mathrm{of} A\right\}=1_L.$$

It follows from Remark 1 that $\varphi$ is an L-fuzzy ideal.

(1) ⇒ (4) Suppose that $\varphi$ is an L-fuzzy ideal. For every $d\in L$, let $s\le z,s\in {\varphi }^{\left[d\right]}$, i.e., $d\notin \alpha \left(\varphi \right(s\left)\right)$. Since $\varphi$ is an L-fuzzy ideal, we get

$$d\notin \alpha \left(\varphi \right(z\left)\right).$$

Hence $z\in {\varphi }^{\left[d\right]}.$

Let $d\in L$ and $v,u\in {\varphi }^{\left[d\right]}$. Then $d\notin \alpha \left(\varphi \right(u\left)\right)$ and $d\notin \alpha \left(\varphi \right(v\left)\right)$. It implies

$$d\notin \alpha \left(\varphi \right(u\left)\right)\cup \alpha \left(\varphi \right(v\left)\right).$$

Since $\alpha$ is an $\bigwedge -\bigcup$ mapping, we obtain

$$d\notin \alpha \left(\varphi \right(u)\wedge \varphi (v\left)\right).$$

From (I2), we get $d\notin \alpha \left(\varphi \right(v⊞u\left)\right),$ i.e., $v⊞u\in {\varphi }^{\left[d\right]}$. Therefore ${\varphi }^{\left[d\right]}$ is an ideal.

(4) ⇒ (5) is obvious.

(5) ⇒ (1) Theorem 3 (2) implies that

$$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\in P\left(L\right),e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} A\right\}.$$

Since ${\varphi }^{\left[d\right]}$ is an ideal of A for all $d\in P\left(L\right)$, it holds that

$$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\in P\left(L\right),e\notin \alpha \left(d\right),{\varphi }^{\left[e\right]}\mathrm{is} \mathrm{an} \mathrm{ideal} A\right\}=1_L.$$

Again by Remark 1, we can get that $\varphi$ is an L-fuzzy ideal.

(1) ⇒ (6) Let $\varphi$ be an L-fuzzy ideal. For every $d\in P\left(L\right)$, let $z\le ssatisfyings\in {\varphi }^{\left(d\right)}$, i.e., $\varphi \left(s\right)\nleqq d$. We thus obtain $\varphi \left(s\right)\le \varphi \left(z\right).$ This implies $\varphi \left(z\right)\nleqq d$, i.e., $z\in {\varphi }^{\left(d\right)}$.

Let $v,u\in A,d\in P\left(L\right)$ and $u,v\in {\varphi }^{\left(d\right)}$, i.e., $\varphi \left(u\right)\nleqq d,\varphi \left(v\right)\nleqq d$, then

$$\varphi \left(u\right)\wedge \varphi \left(v\right)\nleqq d.$$

Since $\varphi$ is an L-fuzzy ideal, it holds that

$$\varphi \left(u\right)\wedge \varphi \left(v\right)\le \varphi (v⊞u).$$

This implies $\varphi (v⊞u)\nleqq d$, i.e., $v\in {\varphi }^{\left(d\right)}$. Therefore ${\varphi }^{\left(d\right)}$ is an ideal.

$\left(6\right)\Rightarrow \left(1\right)$ By Theorem 4, we know that

$$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\in P\left(L\right),d\nleqq e,{\varphi }^{\left(e\right)}\mathrm{is} \mathrm{an} \mathrm{ideal} A\right\}.$$

Since ${\varphi }^{\left(d\right)}$ is an ideal for every $d\in P\left(L\right)$, it holds that

$$\mathfrak{I}\left(\varphi \right)=\bigvee \left\{d\in L|\forall e\in P\left(L\right),d\nleqq e,{\varphi }^{\left(e\right)}\mathrm{is} \mathrm{an} \mathrm{ideal} A\right\}=1_L.$$

Again by Remark 1, we can get that $\varphi$ is an L-fuzzy ideal. □

Proposition 2.  

Let $\varphi \in L^A$. If $\beta (d\wedge e)=\beta \left(d\right)\cap \beta \left(e\right)$ for all $d,e\in L$, then we obtain the following equivalent statements:

(1)

ϕ is an L-fuzzy ideal of A;

(2)

${\varphi }_{\left(d\right)}$ is an ideal for any $d\in J\left(L\right)$;

(3)

${\varphi }_{\left(d\right)}$ is an ideal for any $d\in L$.

Proof. 

The proof is analogous to that of Theorem 1, and we omit it. □

4. L-Fuzzy Convex Structure in MV-Algebras

This section establishes that the degree measure of L-fuzzy ideals within an MV-algebra precisely corresponds to an L-fuzzy convex structure, and systematically examines its fundamental properties. Furthermore, we demonstrate that the family of all L-fuzzy ideals in an MV-algebra constitutes a well-defined L-convex structure, and derive its explicit L-convex hull formulation.

Next, we prove that $\mathfrak{I}$ is an L-fuzzy convex structure in an MV-algebra A.

Theorem 6.  

Let $\mathfrak{I}$ be the L-fuzzy ideal degree in A. Then $\mathfrak{I}$ is an L-fuzzy convex structure on A.

Proof. 

It is easily seen that $\mathfrak{I}\left(0_A\right)=\mathfrak{I}\left(1_A\right)=1_L$.

Take any subfamily ${\left\{{\varphi }_i\right\}}_{i\in J}\subseteq L^A$. Then

$$\begin{array}{cc}\hfill \mathfrak{I}\left(\underset{i\in J}{\bigwedge }{\varphi }_i\right)& =\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigwedge }\left(\underset{i\in J}{\bigwedge }{\varphi }_i\left(s\right)\to \underset{i\in J}{\bigwedge }{\varphi }_i\left(z\right)\right)\wedge \left(\underset{i\in J}{\bigwedge }{\varphi }_i\left(u\right)\wedge \underset{i\in J}{\bigwedge }{\varphi }_i\left(v\right)\to \underset{i\in J}{\bigwedge }{\varphi }_i(v⊞u)\right)\hfill \\ \hfill & =\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigwedge }\underset{i\in J}{\bigwedge }\left(\underset{j\in J}{\bigwedge }{\varphi }_j\left(s\right)\to {\varphi }_i\left(z\right)\right)\wedge \underset{i\in J}{\bigwedge }\left(\underset{j\in J}{\bigwedge }{\varphi }_j\left(u\right)\wedge \underset{j\in J}{\bigwedge }{\varphi }_j\left(v\right)\to {\varphi }_i(v⊞u)\right)\hfill \\ \hfill & =\underset{\begin{array}{c}s,z,v,u\in A\\ z\le s\end{array}}{\bigwedge }\underset{i\in J}{\bigwedge }\left(\left(\underset{j\in J}{\bigwedge }{\varphi }_j\left(s\right)\to {\varphi }_i\left(z\right)\right)\wedge \left(\underset{j\in J}{\bigwedge }{\varphi }_j\left(u\right)\wedge \underset{j\in J}{\bigwedge }{\varphi }_j\left(v\right)\to {\varphi }_i(v⊞u)\right)\right)\hfill \\ \hfill & \ge \underset{i\in J}{\bigwedge }\underset{\begin{array}{c}s,z,v,wu\in A\\ z\le s\end{array}}{\bigwedge }\left({\varphi }_i\left(s\right)\to {\varphi }_i\left(z\right)\right)\wedge \left({\varphi }_i\left(u\right)\wedge {\varphi }_i\left(v\right)\to {\varphi }_i(v⊞u)\right)\hfill \\ \hfill & =\underset{i\in J}{\bigwedge }\mathfrak{I}\left({\varphi }_i\right).\hfill \end{array}$$

For any directed subfamily ${\left\{{\varphi }_i\right\}}_{i\in J}\subseteq L^A$, we next show

$$\mathfrak{I}\left(\underset{i\in J}{\bigvee }{\varphi }_i\right)\ge \underset{i\in J}{\bigwedge }\mathfrak{I}\left({\varphi }_i\right).$$

Take $d\in L$ with $d\le {\displaystyle \underset{i\in J}{\bigwedge }}\mathfrak{I}\left({\varphi }_i\right)$. Then $d\le \mathfrak{I}\left({\varphi }_i\right)$ for all $i\in J$. By Lemma 2, we know for any $s,z,v,u\in A$ with $z\le s$,

$$d\wedge {\varphi }_i\left(s\right)\le {\varphi }_i\left(z\right) \mathrm{and} d\wedge {\varphi }_i\left(u\right)\wedge {\varphi }_i\left(v\right)\le {\varphi }_i(v⊞u).$$

Next we show for all $s,z,v,u\in A$ with $z\le s$.

$$d\wedge \left(\underset{i\in J}{\bigvee }{\varphi }_i\left(s\right)\right)\le \underset{i\in J}{\bigvee }{\varphi }_i\left(z\right) \mathrm{and} d\wedge \left(\underset{i\in J}{\bigvee }{\varphi }_i\left(u\right)\right)\wedge \left(\underset{i\in J}{\bigvee }{\varphi }_i\left(v\right)\right)\le \underset{i\in J}{\bigvee }{\varphi }_i(v⊞u).$$

Take any $c\prec d\wedge \left(\underset{i\in J}{\bigvee }{\varphi }_i\left(u\right)\right)\wedge \left(\underset{i\in J}{\bigvee }{\varphi }_i\left(v\right)\right)$. So there exists an $i\in J$ and $j\in J$ such that

$$c\le {\varphi }_i\left(u\right),c\le {\varphi }_j\left(v\right) \mathrm{and} c\le d.$$

Since ${\left\{{\varphi }_i\right\}}_{i\in J}$ is directed, there exists $k\in J$ satisfying ${\varphi }_i\le {\varphi }_k$ and ${\varphi }_j\le {\varphi }_k$. This gives that

$${\varphi }_i\left(u\right)\le {\varphi }_k\left(u\right) \mathrm{and} {\varphi }_j\left(v\right)\le {\varphi }_k\left(v\right),$$

which implies that for all $v,u\in A$,

$$d\le d\wedge {\varphi }_k\left(u\right)\wedge {\varphi }_k\left(v\right)\le {\varphi }_k(v⊞u)\le \underset{i\in J}{\bigvee }{\varphi }_i(v⊞u).$$

Hence, we obtain for any $v,u\in A$,

$$d\wedge \left(\underset{i\in J}{\bigvee }{\varphi }_i\left(u\right)\right)\wedge \left(\underset{i\in J}{\bigvee }{\varphi }_i\left(v\right)\right)\le \underset{i\in J}{\bigvee }{\varphi }_i(v⊞u).$$

Similarly, we get for any $s,z\in A$ with $z\le s$,

$$d\wedge \left(\underset{i\in J}{\bigvee }{\varphi }_i\left(s\right)\right)\le \underset{i\in J}{\bigvee }{\varphi }_i\left(z\right).$$

By Lemma 2, we get

$$d\le \mathfrak{I}\left(\underset{i\in J}{\bigvee }{\varphi }_i\right).$$

From the arbitrariness of d, we obtain $\mathfrak{I}\left({\displaystyle \underset{i\in J}{\bigvee }}{\varphi }_i\right)\ge {\displaystyle \underset{i\in J}{\bigwedge }}\mathfrak{I}\left({\varphi }_i\right)$.

Hence, by Definition 5, we obtain that $\mathfrak{I}$ is an L-fuzzy convex structure on A. □

It is known that L-fuzzy convexity-preserving mappings and L-fuzzy convex-to-convex mappings are the links connecting two L-fuzzy convex structures. In [33], the authors discussed the connection between morphisms in effect algebras and L-fuzzy convex-to-convex mapping. Following their proof method, we relax the condition of morphisms and require a mapping between two MV-algebras to be an ⊞-morphism; we next explore the relationship between the ⊞-morphisms and L-fuzzy convexity-preserving mappings, L-fuzzy convex-to-convex mapping.

Theorem 7.  

Let $g:A_1⟶A_2$ be an order-preserving ⊞-morphism between two MV-algebras. If ${\mathfrak{I}}_1$, ${\mathfrak{I}}_2$ are the L-fuzzy ideal degrees on $A_1$ and $A_2$, respectively, then $g:(A_1,{\mathfrak{I}}_1)⟶(A_2,{\mathfrak{I}}_2)$ is L-fuzzy convexity-preserving.

Proof. 

Take $\eta \in L^{A_2}$. Then

$$\begin{array}{cc}\hfill & {\mathfrak{I}}_1\left(g_L^{\leftarrow }\left(\eta \right)\right)\hfill \\ \hfill & =\underset{\begin{array}{c}s,z,v,u\in A_1\\ z\le s\end{array}}{\bigwedge }\left(g_L^{\leftarrow }\left(\eta \right)\left(s\right)\to g_L^{\leftarrow }\left(\eta \right)\left(z\right)\right)\wedge \left(g_L^{\leftarrow }\left(\eta \right)\left(u\right)\wedge g_L^{\leftarrow }\left(\eta \right)\left(v\right)\to g_L^{\leftarrow }\left(\eta \right)(v⊞u)\right)\hfill \\ \hfill & =\underset{\begin{array}{c}s,z,v,\in A_1\\ z\le s\end{array}}{\bigwedge }\left(\eta \left(g\right(s\left)\right)\to \eta \left(g\right(z\left)\right)\right)\wedge \left(\eta \left(g\right(u\left)\right)\wedge \eta \left(g\right(v\left)\right)\to \eta \left(g\right(v⊞u)\right)\hfill \\ \hfill & \stackrel{(★)}{=}\underset{\begin{array}{c}s,z,v,u\in A_1\\ z\le s\end{array}}{\bigwedge }\left(\eta \left(g\right(s\left)\right)\to \eta \left(g\right(z\left)\right)\right)\wedge \left(\eta \left(g\right(u\left)\right)\wedge \eta \left(g\right(v\left)\right)\to \eta \left(g\right(u\left)\right)⊞\eta \left(g\right(v\left)\right)\right)\hfill \\ \hfill & \ge \underset{\begin{array}{c}{x}_1,y_1,z_1,w_1\in A_2\\ y_1\le x_1,\end{array}}{\bigwedge }\left(\eta \left(x_1\right)\to \eta \left(y_1\right)\right)\wedge \left(\eta \left(w_1\right)\wedge \eta \left(w_1\right)\to \eta (w_1⊞z_1)\right)(g\mathrm{is}\mathrm{order-preserving})\hfill \\ \hfill & ={\mathfrak{I}}_2\left(\eta \right).\hfill \end{array}$$

The equation (★) holds from the following fact: since g is an ⊞-morphism, we obtain $\eta \left(g\right(v⊞u)=\eta (g\left(u\right))⊞\eta (g\left(v\right))$.

Therefore, by Definition 7 (1), g is L-fuzzy convexity-preserving. □

Theorem 8.  

Let $g:A_1⟶A_2$ be a monotonic ⊞-morphism between two MV-algebras. If ${\mathfrak{I}}_1$, ${\mathfrak{I}}_2$ are the L-fuzzy ideal degrees on $A_1$ and $A_2$, respectively, then $g:(A_1,{\mathfrak{I}}_1)⟶(A_2,{\mathfrak{I}}_2)$ is an L-fuzzy convex-to-convex mapping.

Proof. 

Take $\varphi \in L^{A_1}$. Then

$$\begin{array}{cc}\hfill & {\mathfrak{I}}_2\left(g_L^{\to }\left(\varphi \right)\right)\hfill \\ \hfill & =\underset{\begin{array}{c}s,z,v,u\in A_2\\ z\le s\end{array}}{\bigwedge }\left(g_L^{\to }\left(\varphi \right)\left(s\right)\to g_L^{\to }\left(\varphi \right)\left(z\right)\right)\wedge \left(g_L^{\to }\left(\varphi \right)\left(u\right)\wedge g_L^{\to }\left(\varphi \right)\left(v\right)\to g_L^{\to }\left(\varphi \right)(v⊞u)\right)\hfill \\ \hfill & =\underset{\begin{array}{c}s,z,v,u\in A_2\\ z\le s\end{array}}{\bigwedge }\left(\underset{g\left(o\right)=s}{\bigvee }\varphi \left(o\right)\to \underset{g\left(m\right)=z}{\bigvee }\varphi \left(m\right)\right)\wedge \left(\underset{g\left(h\right)=u}{\bigvee }\varphi \left(h\right)\wedge \underset{g\left(r\right)=v}{\bigvee }\varphi \left(r\right)\to \underset{g\left(k\right)=v⊞u}{\bigvee }\varphi \left(k\right)\right)\hfill \\ \hfill & \stackrel{(★)}{\ge }\underset{\begin{array}{c}s,z,v,u\in A_2\\ z\le s\end{array}}{\bigwedge }\left(\underset{g\left(o\right)=s}{\bigvee }\varphi \left(o\right)\to \underset{g\left(m\right)=z}{\bigvee }\varphi \left(m\right)\right)\wedge \left(\underset{g\left(h\right)=u}{\bigvee }\underset{g\left(r\right)=v}{\bigvee }(\varphi \left(h\right)\wedge \varphi \left(r\right))\to \underset{g\left(k\right)=v⊞u}{\bigvee }\varphi \left(k\right)\right)\hfill \\ \hfill & =\underset{\begin{array}{c}s,z,v,u\in A_2\\ z\le s\end{array}}{\bigwedge }\underset{g\left(r\right)=v}{\bigwedge }\underset{g\left(h\right)=u}{\bigwedge }\underset{g\left(o\right)=s}{\bigwedge }\left(\varphi \left(o\right)\to \underset{g\left(m\right)=z}{\bigvee }\varphi \left(m\right)\right)\wedge \left(\varphi \left(h\right)\wedge \varphi \left(r\right)\to \underset{g\left(k\right)=v⊞u}{\bigvee }\varphi \left(k\right)\right)\hfill \\ \hfill & \ge \underset{\begin{array}{c}s,z,v,u\in A_2\\ z\le s\end{array}}{\bigwedge }\underset{g\left(r\right)=v}{\bigwedge }\underset{g\left(h\right)=u}{\bigwedge }\underset{g\left(o\right)=s}{\bigwedge }\underset{g\left(m\right)=z}{\bigvee }\underset{g\left(k\right)=v⊞u}{\bigvee }\left(\varphi \left(o\right)\to \varphi \left(m\right)\right)\wedge \left(\varphi \left(h\right)\wedge \varphi \left(r\right)\to \varphi \left(k\right)\right)\hfill \\ \hfill & \stackrel{(△)}{\ge }\underset{\begin{array}{c}{x}_1,y_1,z_1,w_1\in A_1\\ y_1\le x_1,\end{array}}{\bigwedge }\left(\varphi \left(x_1\right)\to \varphi \left(y_1\right)\right)\wedge \left(\varphi \left(w_1\right)\wedge \varphi \left(z_1\right)\to \varphi (w_1⊞z_1)\right)\hfill \\ \hfill & ={\mathfrak{I}}_1\left(\varphi \right).\hfill \end{array}$$

The inequality (★) holds from the following fact: we claim that

$$\underset{g\left(h\right)=u}{\bigvee }\varphi \left(h\right)\wedge \underset{g\left(r\right)=v}{\bigvee }\varphi \left(r\right)\le \underset{g\left(h\right)=u}{\bigvee }\underset{g\left(r\right)=v}{\bigvee }(\varphi \left(h\right)\wedge \varphi \left(r\right)).$$

Take $a\prec {\displaystyle \underset{g\left(h\right)=u}{\bigvee }}\varphi \left(h\right)\wedge {\displaystyle \underset{g\left(r\right)=v}{\bigvee }}\varphi \left(r\right)$. Since

$$\underset{g\left(h\right)=u}{\bigvee }\varphi \left(h\right)\wedge \underset{g\left(r\right)=v}{\bigvee }\varphi \left(r\right)\le \underset{g\left(h\right)=u}{\bigvee }\varphi \left(h\right),\underset{g\left(r\right)=v}{\bigvee }\varphi \left(r\right),$$

we have $a\prec {\displaystyle \underset{g\left(h\right)=u}{\bigvee }}\varphi \left(h\right)$ and $a\prec {\displaystyle \underset{g\left(r\right)=v}{\bigvee }}\varphi \left(r\right)$. Then there exist $h_0\in A_1$ and $r_0\in A_1$ with $g\left(h_0\right)=u$ and $g\left(r_0\right)=v$ such that $a\le \varphi \left(h_0\right)$ and $a\le \varphi \left(r_0\right)$, thus $a\le \varphi \left(h_0\right)\wedge \varphi \left(r_0\right)\le {\displaystyle \underset{g\left(h\right)=u}{\bigvee }}{\displaystyle \underset{g\left(r\right)=v}{\bigvee }}(\varphi \left(h\right)\wedge \varphi \left(r\right)).$ Hence

$$\underset{g\left(h\right)=u}{\bigvee }\varphi \left(h\right)\wedge \underset{g\left(r\right)=v}{\bigvee }\varphi \left(r\right)\le {\displaystyle \underset{g\left(h\right)=u}{\bigvee }}{\displaystyle \underset{g\left(r\right)=v}{\bigvee }}(\varphi \left(h\right)\wedge \varphi \left(r\right)).$$

Therefore, by Lemma 1 (5) we get

$$\underset{g\left(h\right)=u}{\bigvee }\varphi \left(h\right)\wedge \underset{g\left(r\right)=v}{\bigvee }\varphi \left(r\right)\to \underset{g\left(k\right)=v⊞u}{\bigvee }\varphi \left(k\right)\ge \underset{g\left(h\right)=u}{\bigvee }\underset{g\left(r\right)=v}{\bigvee }(\varphi \left(h\right)\wedge \varphi \left(r\right))\to \underset{g\left(k\right)=v⊞u}{\bigvee }\varphi \left(k\right).$$

The inequality (△) holds from the following fact: since g is a monotonic ⊞-morphism, $z\le s$, by Definition 3, we obtain $g\left(m\right)=z\le s=g\left(o\right)$ implies $m\le o$. Furthermore, since g is an ⊞-morphism, we have $g(r⊞h)=g\left(r\right)⊞g\left(h\right)=v⊞u$. Since $g\left(k\right)=v⊞u$, we get $g(r⊞h)=g\left(k\right)$. By the fact that $g\left(w_1\right)⊞g\left(z_1\right)=g(w_1⊞z_1)$ in △, thus the inequality (△) holds.

Therefore, by Definition 7 (2), g is L-fuzzy convex-to-convex. □

From the perspective of L-fuzzy ideal degree, some properties of L-fuzzy ideals are investigated as follows.

Proposition 3.  

Let $g:A_1⟶A_2$ be an order-preserving ⊞-morphism between two MV-algebras. If η is an L-fuzzy ideal of $A_2$, then $g_L^{\leftarrow }\left(\eta \right)$ is an L-fuzzy ideal of $A_1$.

Proof. 

Assume ${\mathfrak{I}}_1$ and ${\mathfrak{I}}_2$ are the L-fuzzy ideal degrees of $A_1$ and $A_2$, respectively. Since $\eta$ is an L-fuzzy ideal of $A_2$, we have ${\mathfrak{I}}_2\left(\eta \right)=1_L$. From Theorem 7, we get g is L-fuzzy convexity-preserving, implying that ${\mathfrak{I}}_2\left(\eta \right)\le {\mathfrak{I}}_1\left(g_L^{\leftarrow }\left(\eta \right)\right).$ Hence

$$1_L={\mathfrak{I}}_2\left(\eta \right)\le {\mathfrak{I}}_1\left(g_L^{\leftarrow }\left(\eta \right)\right).$$

Thus ${\mathfrak{I}}_1(g_L^{\leftarrow }\left(\eta \right))=1_L$. Therefore, $g_L^{\leftarrow }\left(\eta \right)$ is an L-fuzzy ideal of $A_1$. □

Proposition 4.  

Let $g:A_1⟶A_2$ be a monotonic ⊞-morphism between two MV-algebras. If ϕ is an L-fuzzy ideal of $A_1$, then $g_L^{\to }\left(\varphi \right)$ is an L-fuzzy ideal of $A_2$.

Proof. 

Assume ${\mathfrak{I}}_1$ and ${\mathfrak{I}}_2$ are the L-fuzzy ideal degrees of $A_1$ and $A_2$, respectively. By Theorem 8, we get g is L-fuzzy convex-to-convex, implying that ${\mathfrak{I}}_1\left(\varphi \right)\le {\mathfrak{I}}_2\left(g_L^{\to }\left(\varphi \right)\right)$. Since $\varphi$ is an L-fuzzy ideal of $A_1$, we have ${\mathfrak{I}}_1\left(\varphi \right)=1_L.$ Hence ${\mathfrak{I}}_2\left(g_L^{\to }\left(\varphi \right)\right)=1_L$. Therefore $g_L^{\to }\left(\varphi \right)$ is an L-fuzzy ideal of $A_2$. □

Example 3.  

Let $L=[0,1]$. Consider the MV-algebra $A=\{0_A,s,z,u,v,1_A\}$ defined in Example 1 and the MV-algebra $B=\{0_B,a,b,1_B\}$ whose Hasse diagram and Cayley tables are defined as follows:

Define the mapping $g:A⟶B$ by

$$g\left(0_A\right)=0_B,g\left(s\right)=g\left(u\right)=a,g\left(z\right)=b,g\left(v\right)=g\left(1_A\right)=1_B.$$

It can be checked that g is an order-preserving ⊞-morphism. Define $\eta \in L^B$ by

$$\eta \left(0_B\right)=0.9,\eta \left(b\right)=0.7,\eta \left(a\right)=\eta \left(1_B\right)=0.5.$$

It can be checked that η is an L-fuzzy ideal of the MV-algebra B. By Theorem 3, we get $g_L^{\leftarrow }\left(\eta \right)$ is an L-fuzzy ideal of the MV-algebra A, where

$$g_L^{\leftarrow }\left(\eta \right)\left(0_A\right)=0.9,g_L^{\leftarrow }\left(\eta \right)\left(z\right)=0.7,g_L^{\leftarrow }\left(\eta \right)\left(s\right)=g_L^{\leftarrow }\left(\eta \right)\left(u\right)=g_L^{\leftarrow }\left(\eta \right)\left(v\right)=g_L^{\leftarrow }\left(\eta \right)\left(1_A\right)=0.5.$$

In fact, $g_L^{\leftarrow }\left(\eta \right)$ is precisely the L-fuzzy ideal ϕ defined in Example 2, thus $g_L^{\leftarrow }\left(\eta \right)$ is an L-fuzzy ideal of A.

Corollary 1.  

Let ${\left\{A_i\right\}}_{i\in J}$ be a collection of MV-algebras and ${\displaystyle \prod _{i\in J}}{\varphi }_i$ be the direct product of ${\left\{{\varphi }_i\right\}}_{i\in J}$, where ${\varphi }_i\in L^{A_i}$. If ${\lambda }_i:{\displaystyle \prod _{i\in J}}{A}_i⟶A_i$ is the projection, then ${\lambda }_i$: $({\displaystyle \prod _{i\in J}}{A}_i,{\mathfrak{I}}_{{\displaystyle \prod _{i\in J}}{A}_i})⟶(A_i,{\mathfrak{I}}_{A_i})$ is an L-fuzzy convexity-preserving mapping for every $i\in J$.

Proof. 

Obviously ${\lambda }_i:{\displaystyle \prod _{i\in J}}{A}_i⟶A_i$ is an order-preserving ⊞-morphism between two MV-algebras, so by Theorem 7, we get ${\lambda }_i$ is an L-fuzzy convexity preserving mapping for all $i\in J$. □

Proposition 5.  

Let ${\left\{A_i\right\}}_{i\in J}$ be a collection of MV-algebras and ${\displaystyle \prod _{i\in J}}{\varphi }_i$ be the direct product of ${\left\{{\varphi }_i\right\}}_{i\in J}$, where ${\varphi }_i\in L^{A_i}$. Then ${\mathfrak{I}}_{{\displaystyle \prod _{i\in J}}{A}_i}\left({\displaystyle \prod _{i\in J}}{\varphi }_i\right)\ge {\displaystyle \underset{i\in J}{\bigwedge }}{\mathfrak{I}}_{A_i}\left({\varphi }_i\right)$.

Proof. 

Let ${\lambda }_i:{\displaystyle \prod _{i\in J}}{A}_i⟶A_i$ be the projection. It is clear that ${\displaystyle \prod _{i\in J}}{\varphi }_i={\displaystyle \underset{i\in J}{\bigwedge }}{\lambda }_i^{\leftarrow }\left({\varphi }_i\right)$. Since ${\mathfrak{I}}_{{\displaystyle \prod _{i\in J}}{A}_i}$ is an L-fuzzy ideal degree, we obtain

$${\mathfrak{I}}_{{\displaystyle \prod _{i\in J}}{A}_i}\left({\displaystyle \prod _{i\in J}}{\varphi }_i\right)={\mathfrak{I}}_{{\displaystyle \prod _{i\in J}}{A}_i}\left({\displaystyle \underset{i\in J}{\bigwedge }}{\lambda }_i^{\leftarrow }\left({\varphi }_i\right)\right)\ge {\displaystyle \underset{i\in J}{\bigwedge }}{\mathfrak{I}}_{{\displaystyle \prod _{i\in J}}{A}_i}\left({\lambda }_i^{\leftarrow }\left({\varphi }_i\right)\right)\ge {\displaystyle \underset{i\in J}{\bigwedge }}{\mathfrak{I}}_{A_i}\left({\varphi }_i\right).$$

Since ${\lambda }_i$ is L-fuzzy convexity preserving by Corollary 1, the last inequality holds. □

From [25], we get ${\mathfrak{I}}_{\left[d\right]}$ is an L-convex structure for any $d\in L$. The L-convex hull of the L-convex structure ${\mathfrak{I}}_{\left[1_L\right]}$ is defined as follows: for $\varphi \in L^A$,

$$co\left(\varphi \right)=\bigwedge \{\eta \in L^A\mid \varphi \le \eta \in {\mathfrak{I}}_{\left[1_L\right]}\}.$$

We denote all L-fuzzy ideals in an MV-algebra by $\mathcal{LI}$. By Remark 1 we know, $\mathcal{LI}={\mathfrak{I}}_{\left[1_L\right]}$. Thus $co\left(\varphi \right)$ is precisely the least L-fuzzy ideal containing $\varphi$. In the following, we will give the concrete L-convex hull formula.

Theorem 9.  

Let $\varphi \in L^A$ and $s\in A$. Define

$$c{o}_{\mathcal{LI}}\left(\varphi \right)\left(s\right)=\bigvee \{\varphi \left(s_1\right)\wedge \varphi \left(s_2\right)\wedge ···\wedge \varphi \left(s_n\right)\mid \exists s_1,s_2,···,s_n\in As.t.s\le s_1⊞s_2⊞···⊞s_n\}.$$

Then $c{o}_{\mathcal{LI}}$ is the L-convex hull of the L-convex structure $\mathcal{LI}$.

Proof. 

We first show that

$$\begin{array}{cc}\hfill & c{o}_{\mathcal{LI}}\left(\varphi \right)\hfill \\ \hfill & =\bigvee \{\varphi \left(s_1\right)\wedge \varphi \left(s_2\right)\wedge ···\wedge \varphi \left(s_n\right)\mid \exists s_1,s_2,···,s_n\in As.t.s\le s_1⊞s_2⊞s_3⊞···⊞s_n\}\hfill \\ & =\bigwedge \{\eta \in L^A\mid \varphi \le \eta \in \mathcal{LI}\}.\hfill \end{array}$$

Firstly, we prove that $c{o}_{\mathcal{LI}}\left(\varphi \right)\in \mathcal{LI}$, that is, $c{o}_{\mathcal{LI}}\left(\varphi \right)$ is an L-fuzzy ideal of A.

Let $z\le s$. For any $s_1,s_2,···,s_n\in A$ such that $s\le s_1⊞s_2⊞s_3⊞···⊞s_n$, it must hold that $z\le s_1⊞s_2⊞s_3⊞···⊞s_n$. Thus

$$\begin{array}{cc}\hfill & c{o}_{\mathcal{LI}}\left(\varphi \right)\left(s\right)\hfill \\ \hfill & =\bigvee \{\varphi \left(s_1\right)\wedge \varphi \left(s_2\right)\wedge ···\wedge \varphi \left(s_n\right)\mid \exists s_1,s_2,···,s_n\in As.t.s\le s_1⊞s_2⊞s_3⊞···⊞s_n\}\hfill \\ \hfill & \le \bigvee \{\varphi \left(z_1\right)\wedge \varphi \left(z_2\right)\wedge ···\wedge \varphi \left(z_n\right)\mid \exists z_1,z_2,···,z_n\in As.t.z\le z_1⊞z_2⊞z_3⊞···⊞z_n\}\hfill \\ \hfill & =c{o}_{\mathcal{LI}}\left(\varphi \right)\left(z\right).\hfill \end{array}$$

Let $s,z\in A$. Take $d\in L$ such that $d\prec c{o}_{\mathcal{LI}}\left(\varphi \right)\left(s\right)\wedge c{o}_{\mathcal{LI}}\left(\varphi \right)\left(z\right)$. Then we obtain $d\prec c{o}_{\mathcal{LI}}\left(\varphi \right)\left(s\right)$ and $d\prec c{o}_{\mathcal{LI}}\left(\varphi \right)\left(z\right)$. It follows that there exist $s_1,s_2,···,s_n\in A$ with $s\le s_1⊞s_2⊞s_3⊞···⊞s_n$ such that

$$d\le \varphi \left(s_1\right)\wedge \varphi \left(s_2\right)\wedge \varphi \left(s_3\right)\wedge ···\wedge \varphi \left(s_n\right),$$

and there exist $z_1,z_2,···,z_n\in A$ with $z\le z_1⊞z_2⊞z_3⊞···⊞z_n$ such that

$$d\le \varphi \left(z_1\right)\wedge \varphi \left(z_2\right)\wedge \varphi \left(z_3\right)\wedge ···\wedge \varphi \left(z_n\right).$$

Hence we obtain

$$s⊞z\le s_1⊞s_2⊞s_3⊞···⊞s_n⊞z_1⊞z_2⊞z_3⊞···⊞z_n,$$

and

$$d\le \varphi \left(s_1\right)\wedge \varphi \left(s_2\right)\wedge \varphi \left(s_3\right)\wedge ···\wedge \varphi \left(s_n\right)\wedge \varphi \left(z_1\right)\wedge \varphi \left(z_2\right)\wedge \varphi \left(z_3\right)\wedge ···\wedge \varphi \left(z_n\right).$$

This gives that

$$\begin{array}{cc}\hfill & c{o}_{\mathcal{LI}}\left(\varphi \right)(s⊞z)\hfill \\ \hfill & =\bigvee \{\varphi \left(s_1\right)\wedge \varphi \left(s_2\right)\wedge ···\wedge \varphi \left(s_n\right)\mid \exists s_1,s_2,···,s_n\in As.t.s⊞z\le s_1⊞s_2⊞···⊞s_n\}\hfill \\ \hfill & \ge \varphi \left(s_1\right)\wedge \varphi \left(s_2\right)\wedge \varphi \left(s_3\right)\wedge ···\wedge \varphi \left(s_n\right)\wedge \varphi \left(z_1\right)\wedge \varphi \left(z_2\right)\wedge \varphi \left(z_3\right)\wedge ···\wedge \varphi \left(z_n\right)\hfill \\ \hfill & \ge d.\hfill \end{array}$$

Thus

$$c{o}_{\mathcal{LI}}\left(\varphi \right)(s⊞z)\ge c{o}_{\mathcal{LI}}\left(\varphi \right)\left(s\right)\wedge c{o}_{\mathcal{LI}}\left(\varphi \right)\left(z\right).$$

Therefore, $c{o}_{\mathcal{LI}}\in \mathcal{LI}$.

Next, we show that for any $\eta \in \mathcal{LI}$ with $\varphi \le \eta$, it holds that $c{o}_{\mathcal{LI}}\left(\varphi \right)\le \eta$. Since for any $s\in A$,

$$\begin{array}{cc}\hfill & c{o}_{\mathcal{LI}}\left(\varphi \right)\left(s\right)\hfill \\ \hfill & =\bigvee \{\varphi \left(s_1\right)\wedge \varphi \left(s_2\right)\wedge ···\wedge \varphi \left(s_n\right)\mid \exists s_1,s_2,···,s_n\in As.t.s\le s_1⊞s_2⊞s_3⊞···⊞s_n\}\hfill \\ \hfill & \le \bigvee \{\eta \left(s_1\right)\wedge \eta \left(s_2\right)\wedge ···\wedge \eta \left(s_n\right)\mid \exists s_1,s_2,···,s_n\in As.t.s\le s_1⊞s_2⊞s_3⊞···⊞s_n\}\hfill \\ \hfill & =c{o}_{\mathcal{LI}}\left(\eta \right)\left(s\right)=\eta \left(s\right).\hfill \end{array}$$

we obtain $c{o}_{\mathcal{LI}}\left(\varphi \right)\le \eta$.

Hence $c{o}_{\mathcal{LI}}\left(\varphi \right)=\bigwedge \{\eta \in L^A\mid \varphi \le \eta \in \mathcal{LI}\}.$ Therefore, $c{o}_{\mathcal{LI}}$ is the L-convex hull of the L-convex structure $\mathcal{LI}$. □

5. Conclusions

This paper first put forward the definition of L-fuzzy ideal degree in an MV-algebra, which serves as a quantitative measure to illustrate the extent to which a given L-fuzzy subset satisfies the conditions of being an L-fuzzy ideal. Next, we provided various characterizations for the L-fuzzy ideal degree and analyzed some of its relevant properties through the use of cut sets, thereby establishing a clear and operable theoretical foundation. In the final part, we showed that the L-fuzzy ideal degree is precisely an L-fuzzy convex structure, and we further explored the properties of this L-fuzzy convex structure. This paper not only enriches the theoretical framework of L-fuzzy ideals but also opens up new perspectives for the study of MV-algebras.

It is worth noting that we can further apply the degree approach to other types of logical algebras, such as BL-algebras, EQ-algebras and residuated lattices. By applying similar approaches, it is possible to develop systematic theories of “ideal degrees” or “filter degrees” in these structures, thereby offering new tools for analyzing uncertainty and gradation in non-classical logics.