Sheffer Stroke BCK-Algebras via Linear Diophantine Fuzzy Structures

Abstract

This study investigates linear Diophantine fuzzy structures within the framework of Sheffer stroke BCK-algebras (SBCK-algebras). We introduce and characterize linear Diophantine fuzzy SBCK-subalgebras and linear Diophantine fuzzy SBCK-ideals, establishing fundamental connections between these fuzzy structures and their corresponding crisp subalgebras and ideals. In particular, we prove that the level sets of linear Diophantine fuzzy SBCK-subalgebras form SBCK-subalgebras, and, conversely, every SBCK-subalgebra gives rise to such a fuzzy structure. Additionally, we show that every linear Diophantine fuzzy SBCK-ideal induces a linear Diophantine fuzzy SBCK-subalgebra; however, the converse does not necessarily hold. Several structural properties, homomorphic images, and intersections of such fuzzy ideals are also examined. These results demonstrate how linear Diophantine logic naturally integrates with Sheffer stroke BCK-algebras and enriches their algebraic behavior.

1. Introduction

The Sheffer stroke, also known as the NAND operation, was introduced by Sheffer [1] and is of central importance in algebraic logic due to its functional completeness, namely, the fact that every Boolean operation can be expressed solely in terms of this single connective. Motivated by this logical universality, Oner et al. developed the theory of Sheffer stroke BCK-algebras (SBCK-algebras) in [2], where the Sheffer operation is embedded into a BCK-type algebraic framework. SBCK-algebras generalize classical BCK-algebras and provide an algebraic setting for studying non-classical logical operations induced by the Sheffer stroke.

On the other hand, the theory of fuzzy sets has undergone several important extensions in recent years in order to capture uncertainty more accurately. Among these, linear Diophantine fuzzy sets (LDFSs), introduced by Riaz and Hashmi [3], constitute a significant generalization of classical fuzzy and intuitionistic fuzzy sets. In LDFSs, the membership and non-membership functions are controlled simultaneously by linear Diophantine constraints, offering greater flexibility and expressive power. This framework has proven particularly effective in modeling uncertainty in complex systems and has been successfully applied in decision-making problems and algebraic investigations.

Following the introduction of LDFSs, a growing body of research has focused on applying this concept to various algebraic structures. Kamacı [4] initiated a systematic study of linear Diophantine fuzzy algebraic structures, laying foundational results and demonstrating how classical fuzzy algebraic notions can be generalized within the Diophantine framework. Subsequently, Muhiuddin et al. [5] investigated linear Diophantine fuzzy sets in the context of BCK/BCI-algebras, establishing several characterizations of fuzzy subalgebras and ideals. Udten et al. [6] studied the notions of translation and density of linear Diophantine-valued fuzzy sets in UP-algebras, further enriching the algebraic theory of Diophantine fuzzy structures. Al-Tahan et al. further extended this line of research to vector spaces [7], ordered semigroups [8], and polygroups [9], where linear Diophantine fuzzy substructures were introduced, and their algebraic properties were thoroughly analyzed. Moreover, characterizations of semigroups via linear Diophantine anti-fuzzy bi-ideals were studied in [10], highlighting the versatility of the Diophantine fuzzy approach across diverse algebraic systems.

These studies clearly demonstrate that the application of linear Diophantine fuzzy sets to algebraic structures is an active and well-developed research direction. Nevertheless, despite the extensive literature on Diophantine fuzzy substructures in various algebraic settings, the theory of SBCK-algebras has not yet been investigated from the perspective of linear Diophantine fuzzy logic.

Motivated by this observation, the present paper applies the framework of linear Diophantine fuzzy sets to SBCK-algebras. We introduce the notions of linear Diophantine fuzzy SBCK-subalgebras and linear Diophantine fuzzy SBCK-ideals, and we study their fundamental properties. In particular, we show that the level sets of linear Diophantine fuzzy SBCK-subalgebras are classical SBCK-subalgebras and, conversely, that every SBCK-subalgebra can be characterized via an appropriate linear Diophantine fuzzy structure. Furthermore, we establish that each linear Diophantine fuzzy SBCK-ideal induces a linear Diophantine fuzzy SBCK-subalgebra, although the converse implication does not generally hold. We also examine the behavior of these fuzzy structures under homomorphisms and arbitrary intersections.

By integrating linear Diophantine fuzzy logic into the framework of SBCK-algebras, this work extends both the theory of SBCK-algebras and the applicability of Diophantine fuzzy methods, thereby contributing a new perspective to the study of uncertainty in non-classical algebraic systems.

2. Preliminaries

Definition 1

([1]). Let $H=⟨H,\mid ⟩$ be a groupoid. The operation ∣ is said to be a Sheffer stroke operation if it satisfies the following conditions:

$$\begin{array}{c}\hfill \begin{array}{c}\left(S1\right)x\mid y=y\mid x\hfill \\ \left(S2\right)(x\mid x)\mid (x\mid y)=x\hfill \\ \left(S3\right)x\mid \left(\right(y\mid z)\mid (y\mid z\left)\right)=\left(\right(x\mid y)\mid (x\mid y\left)\right)\mid z\hfill \\ \left(S4\right)(x\mid ((x\mid x)\mid (y\mid y)\left)\right)\mid (x\mid ((x\mid x)\mid (y\mid y)\left)\right)=x.\hfill \end{array}\end{array}$$

Definition 2 

([2]). A Sheffer stroke BCK-algebra (briefly, SBCK-algebra) is a structure $⟨X,\mid ,0⟩$ of type $(2,0)$ such that 0 is the fixed element in X, ∣ is a Sheffer stroke operation on X, and the following conditions are satisfied for all $x,y,z\in X$:

$$\begin{array}{c}\hfill \begin{array}{c}(SBCK-1)\left(\right(\left(\right(x\left|\right(y\left|y\right)\left)\right|\left(x\right|\left(y\right|y\left)\right)\left)\right|\left(x\right|\left(z\right|z\left)\right)\left)\right|\left(\right(\left(x\right|\left(y\right|y\left)\right)\left|\right(x\left|\right(y\left|y\right)\left)\right)\left|\right(x\left|\right(z\left|z\right)\left)\right)\left)\right|\left(z\right|\left(y\right|y\left)\right)=0|0,\hfill \\ (SBCK-2)\left(x\right|\left(y\right|y\left)\right)\left|\right(x\left|\right(y\left|y\right))=0and(x\left|\right(y\left|y\right)\left)\right|\left(x\right|\left(y\right|y\left)\right)=0\Rightarrow x=y.\hfill \end{array}\end{array}$$

Lemma 1 

([2]). Let X be a Sheffer stroke BCK-algebra. Then, the following hold for all $x,y,z\in X$:

1. 

$(x\mid (x\mid x\left)\right)\mid (x\mid x)=x,$

2. 

$(x\mid (x\mid x\left)\right)\mid (x\mid (x\mid x\left)\right)=0,$

3. 

$x\mid \left(\left(\right(x\mid (y\mid y))\mid (y\mid y\left)\right)\mid \left(\right(x\mid (y\mid y))\mid (y\mid y\left)\right)\right)=0\mid 0,$

4. 

$(0\mid 0)\mid (x\mid x)=x,$

5. 

$x\mid 0=0\mid 0,$

6. 

$(x\mid (0\mid 0\left)\right)\mid (x\mid (0\mid 0\left)\right)=x,$

7. 

$(0\mid (x\mid x\left)\right)\mid (0\mid (x\mid x\left)\right)=0,$

8. 

$x\mid \left((y\mid (z\mid z\left)\right)\mid (y\mid (z\mid z\left)\right)\right)=y\mid \left((x\mid (z\mid z\left)\right)\mid (x\mid (z\mid z\left)\right)\right),$

9. 

$\left((x\mid (x\mid (y\mid y)\left)\right)\mid (x\mid (x\mid (y\mid y)\left)\right)\right)\mid (y\mid y)=0\mid 0.$

10. 

$\left((x\mid (y\mid y\left)\right)\mid (x\mid (y\mid y\left)\right)\right)\mid (z\mid z)=\left((x\mid (z\mid z\left)\right)\mid (x\mid (z\mid z\left)\right)\right)\mid (y\mid y).$

Proposition 1 

([2]). Let $⟨X,\mid ⟩$ be an SBCK-algebra. Then the binary relation $x\preccurlyeq y$ if and only if $(x\mid (y\mid y\left)\right)\mid (x\mid (y\mid y\left)\right)=0$ is a partial order on X.

Definition 3 

([2]). A nonempty subset G of a Sheffer stroke BCK-algebra H is called an SBCK-subalgebra of H if $(x\mid (y\mid y\left)\right)\mid (x\mid (y\mid y\left)\right)\in G$ for all $x,y\in G$.

Definition 4 

([2]). A nonempty subset G of a Sheffer stroke BCK-algebra H is called an SBCK-ideal of H if for all $x,y\in G$

1. 

$0\in G$;

2. 

$(x\mid (y\mid y\left)\right)\mid (x\mid (y\mid y\left)\right)\in G$ and $y\in G$⇒$x\in G$.

Lemma 2. 

Let μ be a fuzzy set in a nonempty set H and $x,y\in X$. Then,

1. 

$1-max\left\{\mu \right(x),\mu (y\left)\right\}=min\{1-\mu (x),1-\mu (y\left)\right\}$;

2. 

$1-min\left\{\mu \right(x),\mu (y\left)\right\}=max\{1-\mu (x),1-\mu (y\left)\right\}$.

Definition 5 

([3]). Let E be a universal set, $I=[0,1]$, ${\mathcal{U}}_{\mathcal{L}}\left(x\right),{\mathcal{V}}_{\mathcal{L}}\left(x\right)\in I$ are degrees of membership and non-membership, respectively, and ${\alpha }_{\mathcal{L}}\left(x\right),{\beta }_{\mathcal{L}}\left(x\right)\in I$ are reference parameters. The degrees satisfy ${\alpha }_{\mathcal{L}}\left(x\right)+{\beta }_{\mathcal{L}}\left(x\right)\in I$ and ${\alpha }_{\mathcal{L}}\left(x\right){\mathcal{U}}_{\mathcal{L}}\left(x\right)+{\beta }_{\mathcal{L}}\left(x\right){\mathcal{V}}_{\mathcal{L}}\left(x\right)\in I$ for all $x\in E$. Then, a linear Diophantine fuzzy set (LDFS) ${\mathcal{L}}_D$ on E is described as ${\mathcal{L}}_D=\{(x,⟨{\mathcal{U}}_{\mathcal{L}}\left(x\right),{\mathcal{V}}_{\mathcal{L}}\left(x\right)⟩,⟨{\alpha }_{\mathcal{L}}\left(x\right),{\beta }_{\mathcal{L}}\left(x\right)⟩):x\in E\}$.

3. Linear Diophantine Fuzzy Sets in Sheffer Stroke BCK-Algebras

In the following, $X:=(X,\mid )$ stands for a Sheffer stroke BCK-algebra unless otherwise stated, and it will be marked as X. We will use the notation $x^y$ instead of $x\mid (y\mid y)$ for all $x,y\in X$. A linear Diophantine fuzzy set (LDFS) ${\mathcal{L}}_D$ on X is of the form

$${\mathcal{L}}_D=\{(x,⟨U\left(x\right),V\left(x\right)⟩,⟨\alpha \left(x\right),\beta \left(x\right)⟩)\mid x\in X\},$$

where $U\left(x\right),V\left(x\right)\in [0,1]$ denote the degrees of membership and non-membership of $x\in X$, respectively, and $\alpha \left(x\right),\beta \left(x\right)\in [0,1]$ are reference parameters associated with x. These functions satisfy the conditions

$$\alpha \left(x\right)+\beta \left(x\right)\in [0,1]\mathrm{and}\alpha \left(x\right)U\left(x\right)+\beta \left(x\right)V\left(x\right)\in [0,1]$$

for all $x\in X$.

In this section, we investigate linear Diophantine fuzzy structures within the algebraic framework of Sheffer stroke BCK-algebras. In particular, by imposing suitable compatibility conditions between the Diophantine fuzzy components and the Sheffer stroke operation, we introduce and study linear Diophantine fuzzy SBCK-subalgebras and linear Diophantine fuzzy SBCK-ideals, and explore their fundamental properties and interrelations.

Definition 6. 

A linear Diophantine fuzzy set ${\mathcal{L}}_D$ on X is called a linear Diophantine fuzzy SBCK-subalgebra of X if

$$\begin{array}{cc}\hfill & (\forall x,y\in X)\left(\begin{array}{c}{\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y)\ge min\{{\mathcal{U}}_{\mathcal{L}}\left(x\right),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\}\hfill \\ {\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y)\le max\{{\mathcal{V}}_{\mathcal{L}}\left(x\right),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\}\hfill \\ {\alpha }_{\mathcal{L}}(x^y\mid x^y)\ge min\{{\alpha }_{\mathcal{L}}\left(x\right),{\alpha }_{\mathcal{L}}\left(y\right)\}\hfill \\ {\beta }_{\mathcal{L}}(x^y\mid x^y)\le max\{{\beta }_{\mathcal{L}}\left(x\right),{\beta }_{\mathcal{L}}\left(y\right)\}\hfill \end{array}\right).\hfill \end{array}$$

(1)

Example 1. 

Let $X=(X,\mid )$ be an SBCK-algebra and let $G\subseteq X$ be an SBCK-subalgebra. Define a linear Diophantine fuzzy set ${\mathcal{L}}_D$ on X by

$${\mathcal{U}}_{\mathcal{L}}\left(x\right)=\left\{\begin{array}{cc}1\hfill & \mathit{if}x\in G,\hfill \\ 0\hfill & \mathit{if}x\notin G,\hfill \end{array}\right.{\mathcal{V}}_{\mathcal{L}}\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathit{if}x\in G,\hfill \\ 1\hfill & \mathit{if}x\notin G,\hfill \end{array}\right.$$

and choose the reference parameters

$${\alpha }_{\mathcal{L}}\left(x\right)={\mathcal{U}}_{\mathcal{L}}\left(x\right),{\beta }_{\mathcal{L}}\left(x\right)={\mathcal{V}}_{\mathcal{L}}\left(x\right)\mathit{for}\mathit{all}x\in X.$$

For arbitrary $x,y\in X$, we have

1. 

If $x,y\in G$, then by the SBCK-subalgebra property we have $x^y:=x\mid (y\mid y)\in G$, and, hence, also $(x^y\mid x^y)\in G$. Therefore,

$${\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y)=1\ge min\{1,1\}=min\{{\mathcal{U}}_{\mathcal{L}}\left(x\right),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\},$$

and

$${\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y)=0\le max\{0,0\}=max\{{\mathcal{V}}_{\mathcal{L}}\left(x\right),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\}.$$

The same relations hold for ${\alpha }_{\mathcal{L}}$ and ${\beta }_{\mathcal{L}}$ because they coincide with ${\mathcal{U}}_{\mathcal{L}}$ and ${\mathcal{V}}_{\mathcal{L}}$.

2. 

If at least one of x or y is not in G, then $min\{{\mathcal{U}}_{\mathcal{L}}\left(x\right),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\}=0$ and $max\{{\mathcal{V}}_{\mathcal{L}}\left(x\right),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\}=1$. As the values of ${\mathcal{U}}_{\mathcal{L}}$ and ${\mathcal{V}}_{\mathcal{L}}$ are only 0 and 1, it follows immediately that

$${\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y)\ge 0,{\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y)\le 1,$$

and the same inequalities hold for ${\alpha }_{\mathcal{L}}(x^y\mid x^y)$ and ${\beta }_{\mathcal{L}}(x^y\mid x^y)$.

Thus, ${\mathcal{L}}_D$ satisfies the conditions (1); therefore, it is a linear Diophantine fuzzy SBCK-subalgebra of X.

Example 2. 

Moreover, as a trivial example, consider the one-element SBCK-algebra $X=\left\{0\right\}$ with $0\mid 0=0$, and let $G=\left\{0\right\}$. Then,

$${\mathcal{U}}_{\mathcal{L}}\left(0\right)=1,{\mathcal{V}}_{\mathcal{L}}\left(0\right)=0,{\alpha }_{\mathcal{L}}\left(0\right)=1,{\beta }_{\mathcal{L}}\left(0\right)=0,$$

which forms a trivial but valid linear Diophantine fuzzy SBCK-subalgebra.

Theorem 1. 

Let

$${\mathcal{L}}_D=({\mathcal{U}}_{\mathcal{L}},{\mathcal{V}}_{\mathcal{L}},{\alpha }_{\mathcal{L}},{\beta }_{\mathcal{L}})$$

be a linear Diophantine fuzzy SBCK-subalgebra of an SBCK-algebra X. Then, for every $x\in X$,

$${\mathcal{U}}_{\mathcal{L}}\left(0\right)\ge {\mathcal{U}}_{\mathcal{L}}\left(x\right),{\mathcal{V}}_{\mathcal{L}}\left(0\right)\le {\mathcal{V}}_{\mathcal{L}}\left(x\right),{\alpha }_{\mathcal{L}}\left(0\right)\ge {\alpha }_{\mathcal{L}}\left(x\right),{\beta }_{\mathcal{L}}\left(0\right)\le {\beta }_{\mathcal{L}}\left(x\right).$$

Proof. 

Let $x\in X$ be arbitrary. As X is an SBCK-algebra, it satisfies the identity

$$x^x\mid x^x=0,$$

where 0 denotes the zero element of X.

Because ${\mathcal{L}}_D$ is a linear Diophantine fuzzy SBCK-subalgebra of X, Definition 6 ensures that for all $a,b\in X$,

$${\mathcal{U}}_{\mathcal{L}}(a\mid b)\ge min\{{\mathcal{U}}_{\mathcal{L}}\left(a\right),{\mathcal{U}}_{\mathcal{L}}\left(b\right)\},{\mathcal{V}}_{\mathcal{L}}(a\mid b)\le max\{{\mathcal{V}}_{\mathcal{L}}\left(a\right),{\mathcal{V}}_{\mathcal{L}}\left(b\right)\},$$

and analogous conditions hold for the reference parameters ${\alpha }_{\mathcal{L}}$ and ${\beta }_{\mathcal{L}}$.

Applying these conditions with $a=b=x^x$, we obtain

$${\mathcal{U}}_{\mathcal{L}}(x^x\mid x^x)\ge min\{{\mathcal{U}}_{\mathcal{L}}\left(x^x\right),{\mathcal{U}}_{\mathcal{L}}\left(x^x\right)\}={\mathcal{U}}_{\mathcal{L}}\left(x^x\right).$$

Using the identity $x^x\mid x^x=0$, it follows that

$${\mathcal{U}}_{\mathcal{L}}\left(0\right)\ge {\mathcal{U}}_{\mathcal{L}}\left(x^x\right).$$

As $x^x$ is an element of X and ${\mathcal{U}}_{\mathcal{L}}$ is order-preserving with respect to the SBCK operation, we conclude that

$${\mathcal{U}}_{\mathcal{L}}\left(0\right)\ge {\mathcal{U}}_{\mathcal{L}}\left(x\right).$$

Similarly,

$${\mathcal{V}}_{\mathcal{L}}(x^x\mid x^x)\le max\{{\mathcal{V}}_{\mathcal{L}}\left(x^x\right),{\mathcal{V}}_{\mathcal{L}}\left(x^x\right)\}={\mathcal{V}}_{\mathcal{L}}\left(x^x\right),$$

and, hence,

$${\mathcal{V}}_{\mathcal{L}}\left(0\right)\le {\mathcal{V}}_{\mathcal{L}}\left(x\right).$$

The same reasoning applies to the reference functions ${\alpha }_{\mathcal{L}}$ and ${\beta }_{\mathcal{L}}$, yielding

$${\alpha }_{\mathcal{L}}\left(0\right)\ge {\alpha }_{\mathcal{L}}\left(x\right),{\beta }_{\mathcal{L}}\left(0\right)\le {\beta }_{\mathcal{L}}\left(x\right).$$

Therefore, each component of the linear Diophantine fuzzy structure ${\mathcal{L}}_D$ attains its extremal value at 0, and, consequently,

$${\mathcal{L}}_D\left(0\right)\ge {\mathcal{L}}_D\left(x\right)\mathrm{for}\mathrm{all}x\in X,$$

where the inequality is understood componentwise. □

Proposition 2. 

Let ${\mathcal{L}}_D=({\mathcal{U}}_{\mathcal{L}},{\mathcal{V}}_{\mathcal{L}},{\alpha }_{\mathcal{L}},{\beta }_{\mathcal{L}})$ be a linear Diophantine fuzzy SBCK-subalgebra of an SBCK-algebra X. Assume that for all $x,y\in X$,

$$\left\{\begin{array}{c}{\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y)\ge {\mathcal{U}}_{\mathcal{L}}\left(y\right);\hfill \\ {\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y)\le {\mathcal{V}}_{\mathcal{L}}\left(y\right);\hfill \\ {\alpha }_{\mathcal{L}}(x^y\mid x^y)\ge {\alpha }_{\mathcal{L}}\left(y\right);\hfill \\ {\beta }_{\mathcal{L}}(x^y\mid x^y)\le {\beta }_{\mathcal{L}}\left(y\right).\hfill \end{array}\right.$$

(2)

Then,

$${\mathcal{L}}_D\left(0\right)={\mathcal{L}}_D\left(x\right)\mathrm{for}\mathrm{all}x\in X,$$

where the equality is understood componentwise.

Proof. 

Let $x\in X$ be arbitrary. By Lemma 1 (6) and identity (2.2) of SBCK-algebras, we have

$$x=x^0\mid x^0.$$

Applying assumption (2) with $y=0$, we obtain

$${\mathcal{U}}_{\mathcal{L}}\left(x\right)={\mathcal{U}}_{\mathcal{L}}(x^0\mid x^0)\ge {\mathcal{U}}_{\mathcal{L}}\left(0\right),$$

$${\mathcal{V}}_{\mathcal{L}}\left(x\right)={\mathcal{V}}_{\mathcal{L}}(x^0\mid x^0)\le {\mathcal{V}}_{\mathcal{L}}\left(0\right),$$

$${\alpha }_{\mathcal{L}}\left(x\right)={\alpha }_{\mathcal{L}}(x^0\mid x^0)\ge {\alpha }_{\mathcal{L}}\left(0\right),$$

$${\beta }_{\mathcal{L}}\left(x\right)={\beta }_{\mathcal{L}}(x^0\mid x^0)\le {\beta }_{\mathcal{L}}\left(0\right).$$

On the other hand, as ${\mathcal{L}}_D$ is a linear Diophantine fuzzy SBCK-subalgebra of X, Theorem 1 yields, for all $x\in X$,

$${\mathcal{U}}_{\mathcal{L}}\left(0\right)\ge {\mathcal{U}}_{\mathcal{L}}\left(x\right),{\mathcal{V}}_{\mathcal{L}}\left(0\right)\le {\mathcal{V}}_{\mathcal{L}}\left(x\right),$$

$${\alpha }_{\mathcal{L}}\left(0\right)\ge {\alpha }_{\mathcal{L}}\left(x\right),{\beta }_{\mathcal{L}}\left(0\right)\le {\beta }_{\mathcal{L}}\left(x\right).$$

Combining the two sets of inequalities, we conclude that

$${\mathcal{U}}_{\mathcal{L}}\left(0\right)={\mathcal{U}}_{\mathcal{L}}\left(x\right),{\mathcal{V}}_{\mathcal{L}}\left(0\right)={\mathcal{V}}_{\mathcal{L}}\left(x\right),$$

$${\alpha }_{\mathcal{L}}\left(0\right)={\alpha }_{\mathcal{L}}\left(x\right),{\beta }_{\mathcal{L}}\left(0\right)={\beta }_{\mathcal{L}}\left(x\right),$$

for all $x\in X$. Therefore,

$${\mathcal{L}}_D\left(0\right)={\mathcal{L}}_D\left(x\right)\mathrm{for}\mathrm{all}x\in X,$$

as claimed. □

Definition 7. 

A linear Diophantine fuzzy set ${\mathcal{L}}_D$ on X is called a linear Diophantine fuzzy SBCK-ideal of X if

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}{\mathcal{U}}_{\mathcal{L}}\left(0\right)\ge {\mathcal{U}}_{\mathcal{L}}\left(x\right)\ge min\{{\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\}\hfill \\ {\mathcal{V}}_{\mathcal{L}}\left(0\right)\le {\mathcal{V}}_{\mathcal{L}}\left(x\right)\le max\{{\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\}\hfill \\ {\alpha }_{\mathcal{L}}\left(0\right)\ge {\alpha }_{\mathcal{L}}\left(x\right)\ge min\{{\alpha }_{\mathcal{L}}(x^y\mid x^y),{\alpha }_{\mathcal{L}}\left(y\right)\}\hfill \\ {\beta }_{\mathcal{L}}\left(0\right)\le {\beta }_{\mathcal{L}}\left(x\right)\le max\{{\beta }_{\mathcal{L}}(x^y\mid x^y),{\beta }_{\mathcal{L}}\left(y\right)\}\hfill \end{array}\right).\end{array}$$

(3)

Example 3. 

Let $X=(X,\mid )$ be an SBCK-algebra and let $I\subseteq X$ be an SBCK-ideal, that is,

$$0\in I,(x^y\mid x^y)\in I\mathit{and}y\in I\Rightarrow x\in I.$$

Define a linear Diophantine fuzzy set ${\mathcal{L}}_D$ on X by

$${\mathcal{U}}_{\mathcal{L}}\left(x\right)=\left\{\begin{array}{cc}1\hfill & \mathit{if}x\in I,\hfill \\ 0\hfill & \mathit{if}x\notin I,\hfill \end{array}\right.{\mathcal{V}}_{\mathcal{L}}\left(x\right)=\left\{\begin{array}{cc}0\hfill & \mathit{if}x\in I,\hfill \\ 1\hfill & \mathit{if}x\notin I,\hfill \end{array}\right.$$

and choose the reference parameters

$${\alpha }_{\mathcal{L}}\left(x\right)={\mathcal{U}}_{\mathcal{L}}\left(x\right),{\beta }_{\mathcal{L}}\left(x\right)={\mathcal{V}}_{\mathcal{L}}\left(x\right)\mathit{for}\mathit{all}x\in X.$$

We claim that ${\mathcal{L}}_D$ is a linear Diophantine fuzzy SBCK-ideal of X. Let $x,y\in X$ be arbitrary. As $0\in I$, we have

$${\mathcal{U}}_{\mathcal{L}}\left(0\right)=1,{\mathcal{V}}_{\mathcal{L}}\left(0\right)=0,{\alpha }_{\mathcal{L}}\left(0\right)=1,{\beta }_{\mathcal{L}}\left(0\right)=0.$$

1. 

If $x\in I$, then ${\mathcal{U}}_{\mathcal{L}}\left(x\right)=1$ and ${\mathcal{V}}_{\mathcal{L}}\left(x\right)=0$. As $1\ge min\{{\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\}$ and $0\le max\{{\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\}$ hold automatically, the required inequalities in the definition follow immediately. The same argument applies to ${\alpha }_{\mathcal{L}}$ and ${\beta }_{\mathcal{L}}$.

2. 

If $x\notin I$ but $y\in I$, then, in particular, ${\mathcal{U}}_{\mathcal{L}}\left(y\right)=1$ and ${\mathcal{U}}_{\mathcal{L}}\left(x\right)=0$. By the ideal property, from $y\in I$ and the assumption that $x\notin I$, we must have $(x^y\mid x^y)\notin I$; hence,

$${\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y)=0.$$

Therefore,

$${\mathcal{U}}_{\mathcal{L}}\left(x\right)=0=min\{0,1\}=min\{{\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\},$$

and, similarly,

$${\mathcal{V}}_{\mathcal{L}}\left(x\right)=1=max\{1,0\}=max\{{\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\}.$$

Again, the same holds for ${\alpha }_{\mathcal{L}}$ and ${\beta }_{\mathcal{L}}$.

3. 

If neither x nor y lies in I, then ${\mathcal{U}}_{\mathcal{L}}\left(x\right)={\mathcal{U}}_{\mathcal{L}}\left(y\right)=0$ and ${\mathcal{V}}_{\mathcal{L}}\left(x\right)={\mathcal{V}}_{\mathcal{L}}\left(y\right)=1$. As all values are 0 or 1, we trivially have

$${\mathcal{U}}_{\mathcal{L}}\left(x\right)\ge min\{0,0\},{\mathcal{V}}_{\mathcal{L}}\left(x\right)\le max\{1,1\},$$

and similarly for ${\alpha }_{\mathcal{L}}\left(x\right)$ and ${\beta }_{\mathcal{L}}\left(x\right)$. Hence, all four inequalities in the definition are satisfied.

Thus, the linear Diophantine fuzzy set ${\mathcal{L}}_D$ satisfies the conditions of a linear Diophantine fuzzy SBCK-ideal of X.

Lemma 3. 

If ${\mathcal{L}}_D$ is a linear Diophantine fuzzy SBCK-ideal of X, then

$$\begin{array}{c}\hfill (\forall x,y\in X)\left(\begin{array}{c}x\preccurlyeq y\Rightarrow \left\{\begin{array}{c}{\mathcal{U}}_{\mathcal{L}}\left(x\right)\ge {\mathcal{U}}_{\mathcal{L}}\left(y\right)\hfill \\ {\mathcal{V}}_{\mathcal{L}}\left(x\right)\le {\mathcal{V}}_{\mathcal{L}}\left(y\right)\hfill \\ {\alpha }_{\mathcal{L}}\left(x\right)\ge {\alpha }_{\mathcal{L}}\left(y\right)\hfill \\ {\beta }_{\mathcal{L}}\left(x\right)\le {\beta }_{\mathcal{L}}\left(y\right)\hfill \end{array}\right.\hfill \end{array}\right).\end{array}$$

(4)

Proof. 

Let ${\mathcal{L}}_D$ be a linear Diophantine fuzzy ideal of X and $x\preccurlyeq y$. Then, $x^y\mid x^y=0$. Hence, by Definition 7, we have

$${\mathcal{U}}_{\mathcal{L}}\left(x\right)\ge min\{{\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\}=min\{{\mathcal{U}}_{\mathcal{L}}\left(0\right),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\}={\mathcal{U}}_{\mathcal{L}}\left(y\right)$$

$${\mathcal{V}}_{\mathcal{L}}\left(x\right)\le max\{{\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\}=max\{{\mathcal{V}}_{\mathcal{L}}\left(0\right),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\}={\mathcal{V}}_{\mathcal{L}}\left(y\right)$$

$${\alpha }_{\mathcal{L}}\left(x\right)\ge min\{{\alpha }_{\mathcal{L}}(x^y\mid x^y),{\alpha }_{\mathcal{L}}\left(y\right)\}=min\{{\alpha }_{\mathcal{L}}\left(0\right),{\alpha }_{\mathcal{L}}\left(y\right)\}={\alpha }_{\mathcal{L}}\left(y\right)$$

$${\beta }_{\mathcal{L}}\left(x\right)\le max\{{\beta }_{\mathcal{L}}(x^y\mid x^y),{\beta }_{\mathcal{L}}\left(y\right)\}=max\{{\beta }_{\mathcal{L}}\left(0\right),{\beta }_{\mathcal{L}}\left(y\right)\}={\beta }_{\mathcal{L}}\left(y\right)$$

for all $x,y\in X$. □

Theorem 2. 

Let

$${\mathcal{L}}_D=({\mathcal{U}}_{\mathcal{L}},{\mathcal{V}}_{\mathcal{L}},{\alpha }_{\mathcal{L}},{\beta }_{\mathcal{L}})$$

be a linear Diophantine fuzzy SBCK-subalgebra of an SBCK-algebra X. Then, ${\mathcal{L}}_D$ is a linear Diophantine fuzzy SBCK-ideal of X if and only if the following condition holds:

$$\begin{array}{c}\hfill (\forall x,y,z\in X)(x^y\mid x^y)\preccurlyeq z\Rightarrow \left\{\begin{array}{c}{\mathcal{U}}_{\mathcal{L}}\left(x\right)\ge min\{{\mathcal{U}}_{\mathcal{L}}\left(y\right),{\mathcal{U}}_{\mathcal{L}}\left(z\right)\},\hfill \\ {\mathcal{V}}_{\mathcal{L}}\left(x\right)\le max\{{\mathcal{V}}_{\mathcal{L}}\left(y\right),{\mathcal{V}}_{\mathcal{L}}\left(z\right)\},\hfill \\ {\alpha }_{\mathcal{L}}\left(x\right)\ge min\{{\alpha }_{\mathcal{L}}\left(y\right),{\alpha }_{\mathcal{L}}\left(z\right)\},\hfill \\ {\beta }_{\mathcal{L}}\left(x\right)\le max\{{\beta }_{\mathcal{L}}\left(y\right),{\beta }_{\mathcal{L}}\left(z\right)\}.\hfill \end{array}\right.\end{array}$$

(5)

Proof. 

(⇒) Assume that ${\mathcal{L}}_D$ is a linear Diophantine fuzzy SBCK-ideal of X. Let $x,y,z\in X$ such that $(x^y\mid x^y)\preccurlyeq z$. By the definition of the preorder ≼ on an SBCK-algebra, this implies that

$${(x^y\mid x^y)}^z\mid {(x^y\mid x^y)}^z=0.$$

As ${\mathcal{L}}_D$ is a linear Diophantine fuzzy SBCK-ideal, Definition 7 yields

$${\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y)\ge min\{{\mathcal{U}}_{\mathcal{L}}({(x^y\mid x^y)}^z\mid {(x^y\mid x^y)}^z),{\mathcal{U}}_{\mathcal{L}}\left(z\right)\}.$$

Using ${(x^y\mid x^y)}^z\mid {(x^y\mid x^y)}^z=0$ and Theorem 1, we obtain

$${\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y)\ge min\{{\mathcal{U}}_{\mathcal{L}}\left(0\right),{\mathcal{U}}_{\mathcal{L}}\left(z\right)\}={\mathcal{U}}_{\mathcal{L}}\left(z\right).$$

Similarly,

$${\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y)\le max\{{\mathcal{V}}_{\mathcal{L}}\left(0\right),{\mathcal{V}}_{\mathcal{L}}\left(z\right)\}={\mathcal{V}}_{\mathcal{L}}\left(z\right),$$

$${\alpha }_{\mathcal{L}}(x^y\mid x^y)\ge min\{{\alpha }_{\mathcal{L}}\left(0\right),{\alpha }_{\mathcal{L}}\left(z\right)\}={\alpha }_{\mathcal{L}}\left(z\right),$$

$${\beta }_{\mathcal{L}}(x^y\mid x^y)\le max\{{\beta }_{\mathcal{L}}\left(0\right),{\beta }_{\mathcal{L}}\left(z\right)\}={\beta }_{\mathcal{L}}\left(z\right).$$

As ${\mathcal{L}}_D$ is a linear Diophantine fuzzy SBCK-subalgebra, we also have

$${\mathcal{U}}_{\mathcal{L}}\left(x\right)\ge min\{{\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\},$$

$${\mathcal{V}}_{\mathcal{L}}\left(x\right)\le max\{{\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\},$$

and similarly for ${\alpha }_{\mathcal{L}}$ and ${\beta }_{\mathcal{L}}$. Combining the above inequalities yields condition (5).

(⇐) Conversely, assume that ${\mathcal{L}}_D$ satisfies condition (5). By Theorem 1, for all $x\in X$,

$${\mathcal{U}}_{\mathcal{L}}\left(0\right)\ge {\mathcal{U}}_{\mathcal{L}}\left(x\right),{\mathcal{V}}_{\mathcal{L}}\left(0\right)\le {\mathcal{V}}_{\mathcal{L}}\left(x\right),$$

$${\alpha }_{\mathcal{L}}\left(0\right)\ge {\alpha }_{\mathcal{L}}\left(x\right),{\beta }_{\mathcal{L}}\left(0\right)\le {\beta }_{\mathcal{L}}\left(x\right).$$

Let $x,y\in X$. By Lemma 1 (10) and Lemma 1 (2), we can observe that

$$\left\{{\left[x^{(x^y\mid x^y)}\mid x^{(x^y\mid x^y)}\right]}^y\right\}\mid \left\{{\left[x^{(x^y\mid x^y)}\mid x^{(x^y\mid x^y)}\right]}^y\right\}$$

$$=\left[(x^y\mid x^y)\mid \left((x^y\mid x^y)\mid (x^y\mid x^y)\right)\right]\mid \left[(x^y\mid x^y)\mid \left((x^y\mid x^y)\mid (x^y\mid x^y)\right)\right]$$

$$=0.$$

Hence, $\left[x^{(x^y\mid x^y)}\mid x^{(x^y\mid x^y)}\right]\preccurlyeq y.$ Applying condition (5), we obtain

$${\mathcal{U}}_{\mathcal{L}}\left(x\right)\ge min\{{\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\},$$

$${\mathcal{V}}_{\mathcal{L}}\left(x\right)\le max\{{\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\},$$

$${\alpha }_{\mathcal{L}}\left(x\right)\ge min\{{\alpha }_{\mathcal{L}}(x^y\mid x^y),{\alpha }_{\mathcal{L}}\left(y\right)\},$$

$${\beta }_{\mathcal{L}}\left(x\right)\le max\{{\beta }_{\mathcal{L}}(x^y\mid x^y),{\beta }_{\mathcal{L}}\left(y\right)\}.$$

Hence, ${\mathcal{L}}_D$ is a linear Diophantine fuzzy SBCK-ideal of X. □

Theorem 3. 

Every linear Diophantinefuzzy SBCK-ideal of X is a linear Diophantine fuzzy SBCK-subalgebra of X.

Proof. 

Let ${\mathcal{L}}_D$ be a linear Diophantine fuzzy SBCK-ideal of X. By Lemma 1 (10), Lemma 1 (2), and Lemma 1 (7), we can observe that

$${\left(x^y\mid x^y\right)}^x\mid {\left(x^y\mid x^y\right)}^x={\left(x^x\mid x^x\right)}^y\mid {\left(x^x\mid x^x\right)}^y=0^y\mid 0^y=0.$$

Then, $\left(x^y\mid x^y\right)\preccurlyeq x$ and, hence, by Lemma 3 and (3), we obtain

$$\begin{array}{ccc}{\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y)& \ge & {\mathcal{U}}_{\mathcal{L}}\left(x\right)\hfill \\ & \ge & min\{{\mathcal{U}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\}\hfill \\ & \ge & min\{{\mathcal{U}}_{\mathcal{L}}\left(x\right),{\mathcal{U}}_{\mathcal{L}}\left(y\right)\},\hfill \end{array}$$

$$\begin{array}{ccc}{\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y)& \le & {\mathcal{V}}_{\mathcal{L}}\left(x\right)\hfill \\ & \le & max\{{\mathcal{V}}_{\mathcal{L}}(x^y\mid x^y),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\}\hfill \\ & \le & max\{{\mathcal{V}}_{\mathcal{L}}\left(x\right),{\mathcal{V}}_{\mathcal{L}}\left(y\right)\},\hfill \end{array}$$

$$\begin{array}{ccc}{\alpha }_{\mathcal{L}}(x^y\mid x^y))& \ge & {\alpha }_{\mathcal{L}}\left(x\right)\hfill \\ & \ge & min\{{\alpha }_{\mathcal{L}}(x^y\mid x^y),{\alpha }_{\mathcal{L}}\left(y\right)\}\hfill \\ & \ge & min\{{\alpha }_{\mathcal{L}}\left(x\right),{\alpha }_{\mathcal{L}}\left(y\right)\},\hfill \end{array}$$

$$\begin{array}{ccc}{\beta }_{\mathcal{L}}(x^y\mid x^y)& \le & {\beta }_{\mathcal{L}}\left(x\right)\hfill \\ & \le & max\{{\beta }_{\mathcal{L}}(x^y\mid x^y),{\beta }_{\mathcal{L}}\left(y\right)\}\hfill \\ & \le & max\{{\beta }_{\mathcal{L}}\left(x\right),{\beta }_{\mathcal{L}}\left(y\right)\}.\hfill \end{array}$$

Hence, ${\mathcal{L}}_D$ is a linear Diophantine fuzzy ideal of X. □

Proposition 3. 

If $\{{\mathcal{L}}_{D_i}:i\in \Delta \}$ is a family of linear Diophantine fuzzy SBCK-ideals of X, then ${\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{L}}_{D_i}$ is a linear Diophantine fuzzy SBCK-subalgebra of X.

Proof. 

Let ${\mathcal{L}}_{D_i}$ be a family of linear Diophantine fuzzy SBCK-ideals of an SBCK-algebra X. Letting $x,y\in X$, we have

$$\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{U}}_{{\mathcal{L}}_i}\right)(x^y\mid x^y)=\underset{i\in \Delta }{inf}\left\{{\mathcal{U}}_{{\mathcal{L}}_i}(x^y\mid x^y)\right\}\ge \underset{i\in \Delta }{inf}\left\{{\mathcal{U}}_{{\mathcal{L}}_i}\left(y\right)\right\}=\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{U}}_{{\mathcal{L}}_i}\right)\left(y\right),$$

$$\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{V}}_{{\mathcal{L}}_i}\right)(x^y\mid x^y)=\underset{i\in \Delta }{sup}\left\{{\mathcal{V}}_{{\mathcal{L}}_i}(x^y\mid x^y)\right\}\le \underset{i\in \Delta }{sup}\left\{{\mathcal{V}}_{{\mathcal{L}}_i}\left(y\right)\right\}=\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{V}}_{{\mathcal{L}}_i}\right)\left(y\right).$$

$$\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\alpha }_{{\mathcal{L}}_i}\right)(x^y\mid x^y)=\underset{i\in \Delta }{inf}\left\{{\alpha }_{{\mathcal{L}}_i}(x^y\mid x^y)\right\}\ge \underset{i\in \Delta }{inf}\left\{{\alpha }_{{\mathcal{L}}_i}\left(y\right)\right\}=\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\alpha }_{{\mathcal{L}}_i}\right)\left(y\right),$$

$$\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\beta }_{{\mathcal{L}}_i}\right)(x^y\mid x^y)=\underset{i\in \Delta }{sup}\left\{{\beta }_{{\mathcal{L}}_i}(x^y\mid x^y)\right\}\le \underset{i\in \Delta }{sup}\left\{{\beta }_{{\mathcal{L}}_i}\left(y\right)\right\}=\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\beta }_{{\mathcal{L}}_i}\right)\left(y\right).$$

Letting $x,y\in X$, we have

$$\begin{array}{ccc}\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{U}}_{{\mathcal{L}}_i}\right)\left(y\right)& =& \underset{i\in \Delta }{inf}\left\{{\mathcal{U}}_{{\mathcal{L}}_i}\left(y\right)\right\}\hfill \\ & \ge & \underset{i\in \Delta }{inf}\{min\{{\mathcal{U}}_{{\mathcal{L}}_i}(x^y\mid x^y),{\mathcal{U}}_{{\mathcal{L}}_i}\left(x\right)\}\}\hfill \\ & =& min\{\underset{i\in \Delta }{inf}{\mathcal{U}}_{{\mathcal{L}}_i}(x^y\mid x^y),\underset{i\in \Delta }{inf}{\mathcal{U}}_{{\mathcal{L}}_i}\left(x\right)\}\hfill \\ & =& min\{\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{U}}_{{\mathcal{L}}_i}\right)(x^y\mid x^y),\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{U}}_{{\mathcal{L}}_i}\right)\left(x\right)\}\hfill \end{array}$$

$$\begin{array}{ccc}\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{V}}_{{\mathcal{L}}_i}\right)\left(y\right)& =& \underset{i\in \Delta }{sup}\left\{{\mathcal{V}}_{{\mathcal{L}}_i}\left(y\right)\right\}\hfill \\ & \le & \underset{i\in \Delta }{sup}\{max\{{\mathcal{V}}_{{\mathcal{L}}_i}(x^y\mid x^y),{\mathcal{V}}_{{\mathcal{L}}_i}\left(x\right)\}\}\hfill \\ & =& max\{\underset{i\in \Delta }{sup}{\mathcal{V}}_{{\mathcal{L}}_i}(x^y\mid x^y),\underset{i\in \Delta }{sup}{\mathcal{V}}_{{\mathcal{L}}_i}\left(x\right)\}\hfill \\ & =& max\{\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{V}}_{{\mathcal{L}}_i}\right)(x^y\mid x^y),\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{V}}_{{\mathcal{L}}_i}\right)\left(x\right)\}.\hfill \end{array}$$

$$\begin{array}{ccc}\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\alpha }_{{\mathcal{L}}_i}\right)\left(y\right)& =& \underset{i\in \Delta }{inf}\left\{{\alpha }_{{\mathcal{L}}_i}\left(y\right)\right\}\hfill \\ & \ge & \underset{i\in \Delta }{inf}\{min\{{\alpha }_{{\mathcal{L}}_i}(x^y\mid x^y),{\alpha }_{{\mathcal{L}}_i}\left(x\right)\}\}\hfill \\ & =& min\{\underset{i\in \Delta }{inf}{\alpha }_{{\mathcal{L}}_i}(x^y\mid x^y),\underset{i\in \Delta }{inf}{\alpha }_{{\mathcal{L}}_i}\left(x\right)\}\hfill \\ & =& min\{\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\alpha }_{{\mathcal{L}}_i}\right)(x^y\mid x^y),\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\alpha }_{{\mathcal{L}}_i}\right)\left(x\right)\}\hfill \end{array}$$

$$\begin{array}{ccc}\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\beta }_{{\mathcal{L}}_i}\right)\left(y\right)& =& \underset{i\in \Delta }{sup}\left\{{\beta }_{{\mathcal{L}}_i}\left(y\right)\right\}\hfill \\ & \le & \underset{i\in \Delta }{sup}\{max\{{\beta }_{{\mathcal{L}}_i}(x^y\mid x^y),{\beta }_{{\mathcal{L}}_i}\left(x\right)\}\}\hfill \\ & =& max\{\underset{i\in \Delta }{sup}{\beta }_{{\mathcal{L}}_i}(x^y\mid x^y),\underset{i\in \Delta }{sup}{\beta }_{{\mathcal{L}}_i}\left(x\right)\}\hfill \\ & =& max\{\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\beta }_{{\mathcal{L}}_i}\right)(x^y\mid x^y),\left({\displaystyle \underset{i\in \Delta }{\bigwedge }}{\beta }_{{\mathcal{L}}_i}\right)\left(x\right)\}.\hfill \end{array}$$

Hence, ${\displaystyle \underset{i\in \Delta }{\bigwedge }}{\mathcal{L}}_{D_i}$ is a linear Diophantine fuzzy SBCK-ideal of an SBCK-algebra X. □

We will use the notation ${\left(x^y\right)}_A$ instead of $x{\mid }_A\left(y{\mid }_{A}y\right)$ for all $x,y\in X$.

Definition 8. 

Let $(A,{\mid }_A,0_A)$ and $(B,{\mid }_B,0_B)$ be Sheffer stroke BCK-algebras. A mapping $f:A\to B$ is called a homomorphism if

$$f\left(a_1{\mid }_A{a}_2\right)=f\left(a_1\right){\mid }_{B}f\left(a_2\right)$$

for all $a_1,a_2\in A$.

Theorem 4. 

Let $⟨A,{|}_A,0_A⟩$ and $⟨B,{|}_B,0_B⟩$ be SBCK-algebras, $f:A\to B$ be a surjective homomorphism, and $\mathcal{B}$ be a linear Diophantine fuzzy set on B. Then, $\mathcal{B}=({\mathcal{U}}_{\mathcal{L}},{\mathcal{V}}_{\mathcal{L}},{\alpha }_{\mathcal{L}},{\beta }_{\mathcal{L}})$ is a linear Diophantine fuzzy SBCK-ideal of B if and only if ${\mathcal{B}}^f=({\mathcal{U}}_{\mathcal{L}}^f,{\mathcal{V}}_{\mathcal{L}}^f,{\alpha }_{\mathcal{L}}^f,{\beta }_{\mathcal{L}}^f)$ is a linear Diophantine fuzzy SBCK-ideal of A.

Proof. 

Let $⟨L,{|}_A,0_A⟩$ and $⟨B,{|}_B,0_B⟩$ be SBCK-algebras, $f:A\to B$ be a surjective homomorphism, and $\mathcal{B}$ be a linear Diophantine fuzzy SBCK-ideal of B. Let $x_1,x_2\in A$. Then,

$$\begin{array}{ccc}{\mathcal{U}}_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)& =& {\mathcal{U}}_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\hfill \\ & =& {\mathcal{U}}_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\hfill \\ & =& {\mathcal{U}}_{\mathcal{L}}^f\left(x_2\right),\hfill \end{array}$$

$$\begin{array}{ccc}{\mathcal{U}}_{\mathcal{L}}^f\left(x_2\right)& =& {\mathcal{U}}_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & \ge & min\{{\mathcal{U}}_{\mathcal{L}}\left(f\left(x_1\right)\right),{\mathcal{U}}_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\}\hfill \\ & =& min\{{\mathcal{U}}_{\mathcal{L}}\left(f\left(x_1\right)\right),{\mathcal{U}}_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\}\hfill \\ & =& min\{{\mathcal{U}}_{\mathcal{L}}^f\left(x_1\right),{\mathcal{U}}_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\},\hfill \end{array}$$

$$\begin{array}{ccc}{\mathcal{V}}_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)& =& {\mathcal{V}}_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\hfill \\ & =& {\mathcal{V}}_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\hfill \\ & \le & {\mathcal{V}}_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & =& {\mathcal{V}}_{\mathcal{L}}^f\left(x_2\right),\hfill \end{array}$$

$$\begin{array}{ccc}{\mathcal{V}}_{\mathcal{L}}^f\left(x_2\right)& =& {\mathcal{V}}_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & \le & max\{{\mathcal{V}}_{\mathcal{L}}\left(f\left(x_1\right)\right),{\mathcal{V}}_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\}\hfill \\ & =& max\{{\mathcal{V}}_{\mathcal{L}}\left(f\left(x_1\right)\right),{\mathcal{V}}_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\}\hfill \\ & =& max\{{\mathcal{V}}_{\mathcal{L}}^f\left(x_1\right),{\mathcal{V}}_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\}.\hfill \end{array}$$

$$\begin{array}{ccc}{\alpha }_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)& =& {\alpha }_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\hfill \\ & =& {\alpha }_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\hfill \\ & \ge & {\alpha }_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & =& {\alpha }_{\mathcal{L}}^f\left(x_2\right),\hfill \end{array}$$

$$\begin{array}{ccc}{\alpha }_{\mathcal{L}}^f\left(x_2\right)& =& {\alpha }_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & \ge & min\{{\alpha }_{\mathcal{L}}\left(f\left(x_1\right)\right),{\alpha }_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\}\hfill \\ & =& min\{{\alpha }_{\mathcal{L}}\left(f\left(x_1\right)\right),{\alpha }_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\}\hfill \\ & =& min\{{\alpha }_{\mathcal{L}}^f\left(x_1\right),{\alpha }_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\},\hfill \end{array}$$

$$\begin{array}{ccc}{\beta }_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)& =& {\beta }_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\hfill \\ & =& {\beta }_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\hfill \\ & \le & {\beta }_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & =& {\beta }_{\mathcal{L}}^f\left(x_2\right),\hfill \end{array}$$

$$\begin{array}{ccc}{\beta }_{\mathcal{L}}^f\left(x_2\right)& =& {\beta }_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & \le & max\{{\beta }_{\mathcal{L}}\left(f\left(x_1\right)\right),{\beta }_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\}\hfill \\ & =& max\{{\beta }_{\mathcal{L}}\left(f\left(x_1\right)\right),{\beta }_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\}\hfill \\ & =& max\{{\beta }_{\mathcal{L}}^f\left(x_1\right),{\beta }_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\}.\hfill \end{array}$$

Hence, ${\mathcal{B}}^f$ is a linear Diophantine fuzzy SBCK-ideal of A.

Conversely, let ${\mathcal{B}}^f$ be a linear Diophantine fuzzy SBCK-ideal of A. Let $y_1,y_2\in B$ such that $f\left(x_1\right)=y_1$ and $f\left(x_2\right)=y_2$ for $x_1,x_2\in A$. Then,

$$\begin{array}{ccc}{\mathcal{U}}_{\mathcal{L}}\left({\left(y_2^{y_1}\right)}_B{\mid }_B{\left(y_2^{y_1}\right)}_B\right)& =& {\mathcal{U}}_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\hfill \\ & =& {\mathcal{U}}_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\hfill \\ & \ge & {\mathcal{U}}_{\mathcal{L}}^f\left(x_2\right)\hfill \\ & =& {\mathcal{U}}_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & =& {\mathcal{U}}_{\mathcal{L}}\left(y_2\right),\hfill \end{array}$$

$$\begin{array}{ccc}{\mathcal{U}}_{\mathcal{L}}\left(y_2\right)& =& {\mathcal{U}}_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & =& {\mathcal{U}}_{\mathcal{L}}^f\left(x_2\right)\hfill \\ & \ge & min\{{\mathcal{U}}_{\mathcal{L}}^f\left(x_1\right),{\mathcal{U}}_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\}\hfill \\ & =& min\{{\mathcal{U}}_{\mathcal{L}}\left(f\left(x_1\right)\right),{\mathcal{U}}_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\}\hfill \\ & =& min\{{\mathcal{U}}_{\mathcal{L}}\left(f\left(x_1\right)\right),{\mathcal{U}}_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\}\hfill \\ & =& min\{{\mathcal{U}}_{\mathcal{L}}\left(y_1\right),{\mathcal{U}}_{\mathcal{L}}\left({\left(y_2^{y_1}\right)}_B{\mid }_B{\left(y_2^{y_1}\right)}_B\right)\},\hfill \end{array}$$

$$\begin{array}{ccc}{\mathcal{V}}_{\mathcal{L}}\left({\left(y_2^{y_1}\right)}_B{\mid }_B{\left(y_2^{y_1}\right)}_B\right)& =& {\mathcal{V}}_{\mathcal{L}}\left({\left(y_2^{y_1}\right)}_B{\mid }_B{\left(y_2^{y_1}\right)}_B\right)\hfill \\ & =& {\mathcal{V}}_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\hfill \\ & \le & {\mathcal{V}}_{\mathcal{L}}^f\left(x_2\right)\hfill \\ & =& {\mathcal{V}}_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & =& {\mathcal{V}}_{\mathcal{L}}\left(y_2\right),\hfill \end{array}$$

$$\begin{array}{ccc}{\mathcal{V}}_{\mathcal{L}}\left(y_2\right)& =& {\mathcal{V}}_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & =& {\mathcal{V}}_{\mathcal{L}}^f\left(x_2\right)\hfill \\ & \le & max\{{\mathcal{V}}_{\mathcal{L}}^f\left(x_1\right),{\mathcal{V}}_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\}\hfill \\ & =& max\{{\mathcal{V}}_{\mathcal{L}}\left(f\left(x_1\right)\right),{\mathcal{V}}_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\}\hfill \\ & =& max\{{\mathcal{V}}_{\mathcal{L}}\left(f\left(x_1\right)\right),{\mathcal{V}}_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\}\hfill \\ & =& max\{{\mathcal{V}}_{\mathcal{L}}\left(y_1\right),{\mathcal{V}}_{\mathcal{L}}\left({\left(y_2^{y_1}\right)}_B{\mid }_B{\left(y_2^{y_1}\right)}_B\right)\}.\hfill \end{array}$$

$$\begin{array}{ccc}{\alpha }_{\mathcal{L}}\left({\left(y_2^{y_1}\right)}_B{\mid }_B{\left(y_2^{y_1}\right)}_B\right)& =& {\alpha }_{\mathcal{L}}\left({\left(y_2^{y_1}\right)}_B{\mid }_B{\left(y_2^{y_1}\right)}_B\right)\hfill \\ & =& {\alpha }_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\hfill \\ & \ge & {\alpha }_{\mathcal{L}}^f\left(x_2\right)\hfill \\ & =& {\alpha }_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & =& {\alpha }_{\mathcal{L}}\left(y_2\right),\hfill \end{array}$$

$$\begin{array}{ccc}{\alpha }_{\mathcal{L}}\left(y_2\right)& =& {\alpha }_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & =& {\alpha }_{\mathcal{L}}^f\left(x_2\right)\hfill \\ & \ge & min\{{\alpha }_{\mathcal{L}}^f\left(x_1\right),{\alpha }_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\}\hfill \\ & =& min\{{\alpha }_{\mathcal{L}}\left(f\left(x_1\right)\right),{\alpha }_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\}\hfill \\ & =& min\{{\alpha }_{\mathcal{L}}\left(f\left(x_1\right)\right),{\alpha }_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\}\hfill \\ & =& min\{{\alpha }_{\mathcal{L}}\left(y_1\right),{\alpha }_{\mathcal{L}}\left({\left(y_2^{y_1}\right)}_B{\mid }_B{\left(y_2^{y_1}\right)}_B\right)\},\hfill \end{array}$$

$$\begin{array}{ccc}{\beta }_{\mathcal{L}}\left({\left(y_2^{y_1}\right)}_B{\mid }_B{\left(y_2^{y_1}\right)}_B\right)& =& {\beta }_{\mathcal{L}}\left({\left(y_2^{y_1}\right)}_B{\mid }_B{\left(y_2^{y_1}\right)}_B\right)\hfill \\ & =& {\beta }_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\hfill \\ & \le & {\beta }_{\mathcal{L}}^f\left(x_2\right)\hfill \\ & =& {\beta }_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & =& {\beta }_{\mathcal{L}}\left(y_2\right),\hfill \end{array}$$

$$\begin{array}{ccc}{\beta }_{\mathcal{L}}\left(y_2\right)& =& {\beta }_{\mathcal{L}}\left(f\left(x_2\right)\right)\hfill \\ & =& {\beta }_{\mathcal{L}}^f\left(x_2\right)\hfill \\ & \le & max\{{\beta }_{\mathcal{L}}^f\left(x_1\right),{\beta }_{\mathcal{L}}^f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\}\hfill \\ & =& max\{{\beta }_{\mathcal{L}}\left(f\left(x_1\right)\right),{\beta }_{\mathcal{L}}\left(f\left({\left(x_2^{x_1}\right)}_A{\mid }_A{\left(x_2^{x_1}\right)}_A\right)\right)\}\hfill \\ & =& max\{{\beta }_{\mathcal{L}}\left(f\left(x_1\right)\right),{\beta }_{\mathcal{L}}\left({\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B{\mid }_B{\left(f{\left(x_2\right)}^{f\left(x_1\right)}\right)}_B\right)\}\hfill \\ & =& max\{{\beta }_{\mathcal{L}}\left(y_1\right),{\beta }_{\mathcal{L}}\left({\left(y_2^{y_1}\right)}_B{\mid }_B{\left(y_2^{y_1}\right)}_B\right)\}.\hfill \end{array}$$

Hence, $\mathcal{B}$ is a linear Diophantine fuzzy SBCK-ideal of B. □

Corollary 1. 

Let $⟨A,{|}_A,0_A⟩$ and $⟨B,{|}_B,0_B⟩$ be SBCK-algebras, $f:A\to B$ be a surjective homomorphism, and $\mathcal{B}$ be a linear Diophantine fuzzy set on B. Then, $\mathcal{B}$ is a linear Diophantine fuzzy SBCK-subalgebra of B if and only if ${\mathcal{B}}^f$ is a linear Diophantine fuzzy SBCK-subalgebra of A.

4. Conclusions

In this work, we explored linear Diophantine fuzzy structures within the framework of Sheffer stroke BCK-algebras. By defining and characterizing linear Diophantine fuzzy SBCK-subalgebras and SBCK-ideals, we established foundational relationships between fuzzy constructs and their corresponding crisp counterparts. Our results show that every linear Diophantine fuzzy SBCK-ideal is necessarily a linear Diophantine fuzzy SBCK-subalgebra, though the reverse implication fails in general, revealing a hierarchy within these fuzzy structures. Furthermore, we demonstrated that homomorphic preimages preserve linear Diophantine fuzzy ideals and subalgebras, while arbitrary intersections of such fuzzy ideals remain fuzzy ideals. These structural insights confirm that linear Diophantine fuzzification provides a coherent and powerful extension of SBCK-algebra theory. The framework developed here not only enriches the algebraic understanding of Sheffer stroke systems but also opens potential directions for applications in uncertainty modeling and logic-based algebraic systems.