# t-Intuitionistic Fuzzy Structures on PMS-Ideals of a PMS-Algebra

Authors: Beza Lamesgin Derseh; Berhanu Assaye Alaba; Yohannes Gedamu Wondifraw
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

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## Abstract

In this article, we apply the concept of at-intuitionistic fuzzy set to PMS-ideals in PMS-algebras. The notion of the t-intuitionistic fuzzy PMS-ideal of PMS-algebra is introduced, and several related properties are studied. The relationships between a t-intuitionistic fuzzy PMS-ideal and a t-intuitionistic fuzzy PMS-subalgebra of a PMS-algebra, as well as the relationships between an intuitionistic fuzzy PMS-ideal and a t-intuitionistic fuzzy PMS-ideal are discussed in detail. A condition for an intuitionistic fuzzy set to be a t-intuitionistic fuzzy PMS-ideal is provided. The t-intuitionistic fuzzy PMS-ideals of PMS-algebra are described using their α,β level cuts. The homomorphism of a t-intuitionistic fuzzy PMS-ideal of a PMS-algebra is studied, and its homomorphic image and inverse image are explored. The Cartesian product of any two t-intuitionistic fuzzy PMS-ideals is discussed, and some related results are derived. The Cartesian product of the t-intuitionistic fuzzy PMS-ideals is also characterized using its α,β level cuts. The strongest t-intuitionistic fuzzy PMS-relation in a PMS-algebra is defined. Finally, the relationships between the strongest t-intuitionistic fuzzy PMS-relation and t-intuitionistic fuzzy PMS-ideal are studied.


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## Body

## 1. Introduction

In 1965, Zadeh[1] introduced the concept of a fuzzy set for dealing with uncertainty and vagueness in real-world problems. Since then, several researchers have applied it to a wide range of algebraic structures, such as BCI-algebras, BCK-algebras, BG-algebras, KU-algebras, etc. Akram and Dar [2] introduced the notions of T-fuzzy subalgebras and T-fuzzy H-ideals in BCI-algebras using a t-norm T and investigated some of their properties. Akram and Zhan [3] introduced the notion of sensible fuzzy ideals of BCK-algebras with respect to a t-conorm and investigated some of their properties. Senapati et al. [4] introduced the notion of T-fuzzy subalgebras and T-fuzzy closed ideals of BG-algebras and investigated their related results. The notion of T-fuzzy KU-ideals of KU-algebras are introduced using t-norm T and their related results are investigated by Senapati [5]. He further investigated images and pre-images of KU-ideals under homomorphism and the Cartesian product and T-product of T-fuzzy KU-ideals of KU-algebras. After the introduction of fuzzy sets by Zadeh, many mathematicians have worked to extend this fundamental concept in a variety of ways. In 1975, Zadeh [6] developed a type-2 fuzzy set as an extension of the fuzzy set with a membership grade of fuzzy set in the unit interval [0, 1] rather than a point in [0, 1]. Torra and Narukawa [7, 8] developed the concept of a hesitant fuzzy set as one of the extensions of the fuzzy set to express hesitant information more thoroughly than other extensions of the fuzzy set, as it permits several possible values for the membership degree of an element.Atanassov [9, 10] introduced the idea of an intuitionistic fuzzy set as a generalization of the fuzzy set. An intuitionistic fuzzy set is more effective than a fuzzy set in dealing with ambiguity and uncertainty since it assigns a membership and nonmembership degree to each element of a set. Since its appearance, mathematicians have applied this fundamental concept to a number of algebraic structures. Jun et al. [11] introduced the notion of an intuitionistic fuzzy quasi-associative ideal of a BCI-algebra and investigated some related properties. Many fundamental characteristics of intuitionistic fuzzy subgroups were also explored by Sharma [12, 13]. Panigrahi and Nanda [14] studied the idea of intuitionistic fuzzy relations over intuitionistic fuzzy subsets and found several interesting properties of intuitionistic fuzzy relations in intuitionistic fuzzy subsets. Peng [15] introduced the notion of intuitionistic fuzzy B-algebras in B-algebra and studied some properties of the homomorphic image and inverse image of intuitionistic fuzzy B-algebras. Jana et al. [16] introduced the concept of intuitionistic fuzzy set to G-subalgebras of G-algebras and investigated several properties. Sharma [17, 18] developed the concept of the t-intuitionistic fuzzy set as an extension of the intuitionistic fuzzy set to deal with uncertainty and vagueness within the context of some reference points in the unit interval [0,1] and then introduced the concepts of t-intuitionistic fuzzy subgroups and t-intuitionistic fuzzy subrings. Shuaib et al. [19] introduced the notion of η-intuitionistic fuzzy subgroup over η-intuitionistic fuzzy subset and studied some algebraic aspects of η-fuzzy subgroups. Barbhuiya [20] introduced the notion of t-intuitionistic fuzzy subalgebra and t-intuitionistic fuzzy normal subalgebra of BG-algebra and studied their properties. He also investigated the homomorphic image and inverse image of both t-intuitionistic fuzzy subalgebra and t-intuitionistic fuzzy normal subalgebra of a BG-algebra.Iseki and Tanaka [21] introduced a class of abstract algebras called BCK-algebras. Iseki [22] introduced another class of abstract algebra called BCI-algebra as a generalization of BCK-algebra. In 2016, Sithar Selvam and Nagalakshmi [23] introduced a new algebraic structure, known as PMS-algebra, as a generalization of BCK\BCI\TM\PS-algebras and investigated various related results. In the same year, Sithar Selvam and Nagalakshmi [24] also fuzzified a PMS-ideal in a PMS-algebra and investigated its basic properties. The study of intuitionistic fuzzification of PMS-subalgebras and PMS-ideals of PMS-algebras was done by Derseh et al. [25]. The notion of t-intuitionistic fuzzy subalgebra has been studied in several algebraic structures (see [17–20, 26, 27]. However, to the best of our knowledge, no studies on t-intuitionistic fuzzy ideals of any algebraic structure, including PMS-algebra, are available. This motivated us to develop t-intuitionistic fuzzy PMS-ideals in PMS-algebra.In this manuscript, we use the concept of at-intuitionistic fuzzy set to PMS-ideals in PMS-algebras. We introduce the notion of a t-intuitionistic fuzzy PMS-ideal of PMS-algebra and study its properties. We consider the relationships between a t-intuitionistic fuzzy PMS-ideal and a t-intuitionistic fuzzy PMS-subalgebra of a PMS-algebra as well as the relationships between an intuitionistic fuzzy PMS-ideal and a t-intuitionistic fuzzy PMS-ideal of a PMS-algebra. We establish a condition for an intuitionistic fuzzy set in a PMS-algebra to be a t-intuitionistic fuzzy PMS-ideal of a PMS-algebra. We describe the t-intuitionistic fuzzy PMS-ideals of PMS-algebra using their α,β level cuts. We consider a t-intuitionistic fuzzy PMS-ideal in a PMS-algebra under homomorphism and explore its homomorphic image and inverse image in a PMS-algebra. Furthermore, we study the Cartesian product of any two t-intuitionistic fuzzy PMS-ideals of PMS-algebra and find some interesting results. We also characterize the Cartesian product of the t-intuitionistic fuzzy PMS-ideals using their α,β level cuts. We finally define the strongest t-intuitionistic fuzzy PMS-relation in a PMS-algebra and study the relationship between the strongest t-intuitionistic fuzzy PMS-relation and a t-intuitionistic fuzzy PMS-ideal.


## 2. Preliminaries

In this section, we consider some basic definitions, results, and some important concepts in PMS-algebras that are needed for our work.Definition 1.
(see [23]). A PMS-algebra is a nonempty set X with a constant 0 and a binary operation ∗ of type (2, 0) satisfying the following axioms:(i)
0∗x=x(ii)
y∗x∗z∗x=z∗y, for all x,y,z∈X
We can define a binary relation≤ in X by x≤y if and only if x∗y=0.Definition 2.
(see [23]). A nonempty subset S of a PMS-algebra is called a PMS-subalgebra of X if x∗y∈S, for all x,y∈S.Definition 3.
(see [23]). A nonempty subset I of a PMS-algebra X,∗,0 is said to be a PMS-ideal of X if it satisfies the following conditions:(i)
0∈ I(ii)
z∗y,z∗x∈ I ⇒y∗x∈ I, for all x,y∈ I.Proposition 1 (see [23]).
LetX,∗,0 be a PMS-algebra. Then the following properties hold for all x,y,z∈X,(i)
x∗x=0(ii)
y∗x∗x=y(iii)
x∗y∗x=y∗0(iv)
y∗x∗z=z∗x∗y(v)
x∗y∗0=y∗x=0∗y∗0∗xDefinition 4.
(see [1]). A fuzzy subset A in a nonempty set X is defined as A=x,μAx|x∈X, where the mapping μA:X⟶0,1 defines the degree of membershipDefinition 5.
(see [9, 10]). An intuitionistic fuzzy set A in a nonempty set X is an object having the form A=x,μAx,νAx|x∈X, where the functions μA:X⟶0,1 and νA:X⟶0,1 define the degree of membership and the degree of nonmembership of each element x∈X to the set A, respectively, with the condition 0≤μAx+νAx≤1, ∀x∈X.Definition 6.
(see [9, 10]). Let A and B be two intuitionistic fuzzy subsets of the set X, where A=x,μAx,νAx|x∈X and B=x,μBx,νBx|x∈X, then(i)
A∩B=x,minμAx,μBx,maxνAx,νBx|x∈X(ii)
A∪B=x,maxμAx,μBx,minνAx,νBx|x∈X(iii)
A¯=x,νAx,μAx|x∈X(iv)
□A=x,μAx,1−μAx|x∈X(v)
A=x,1−νAx,νAx|x∈XDefinition 7.
(see [25]). An intuitionistic fuzzy subset A=μA,νA of a PMS-algebra X is called an intuitionistic fuzzy PMS-subalgebra of X if μAx∗y≥minμAx,νAy and νAx∗y≤maxνAx,νAy, for all x,y∈X.Definition 8.
(see [28]). An intuitionistic fuzzy set A=μA,νA in X is called an intuitionistic fuzzy PMS-ideal of X if it satisfies the following conditions for all x,y,z∈X.(i)
μA0≥μAx and νA0≤νAx,(ii)
μAy∗x≥minμAz∗y,μAz∗x,(iii)
νAy∗x≤maxνAz∗y,νAz∗xDefinition 9.
(see [29]). Let A be a fuzzy set in a nonempty set X with membership function μA:X⟶0,1 and let t∈0,1. Then the fuzzy set At in X is called the t-fuzzy subset of X whose membership function is μAt (w.r.t fuzzy set A) and is defined by μAtx=minμAx,t, for all x∈X.Definition 10.
(see [18]). Let A=μA,νA be an intuitionistic fuzzy set in a nonempty set X and t∈0,1. Then the t-intuitionistic fuzzy set (t-IFS) At in a nonempty set X is an object having the form At=x,μAtx,νAtx|x∈X, where the function μAt:X⟶0,1 and νAt:X⟶0,1 denote the degree of membership and degree of nonmembership, respectively, such that μAtx=minμAx,t and νAtx=maxνAx,1−t satisfying the condition 0≤μAtx+νAtx≤1, for all x∈X.
Note: For the sake of simplicity, we shall use the symbolAt=μAt,νAt, for t-IFS At=x,μAtx,νAtx|x∈X.Remark 1.
LetAt=x,μAtx,νAtx|x∈X be a t-IFSs of the set X. Then
□At=x,μAtx,1−μAtx|x∈X=x,μAtx,μ¯Atx|x∈X and(1)♢At=x,1−νAtx,νAtx|x∈X=x,ν¯Atx,νAtx|x∈X.Remark 2.
(see [18, 27]). Let At=μAt,νAt and Bt=μBt,νBt be any two t-intuitionistic fuzzy subsets of any nonempty set X, then A∩Bt=At∩Bt and A∪Bt=At∪Bt.Definition 11.
(see [23]). Let X and Y be PMS-algebras. The mapping f:X⟶Y is called a homomorphism of PMS-algebras if fx∗y=fx∗fy, for all x,y∈X. A homomorphism f:X⟶Y is called an epimorphism of PMS-algebras if fX=Y.
Note: Iff is a homomorphism of PMS-algebras, then f0=0.Remark 3.
(see [18]). Let f:X⟶Y be a mapping and A such that B are any two t-IFSs of X and Y, respectively, then fAt=fAt and f−1Bt=f−1Bt, for all t∈0,1.Definition 12.
(see [26]). Let At=μAt,νAt and At=μAt,vBt be two t-intuitionistic fuzzy subsets of X and Y, respectively. Then their Cartesian product of At and Bt denoted by At×Bt is defined as At×Bt=x,y,μAt×Bt,νAt×Bt|x∈Xandy∈Y, where μAt×Btx∗y=minμAtx,μBty and νAt×Btx∗y=maxνAtx,νBty, for all x∈X and y∈Y.Remark 4.
LetX and Y be any two PMS-algebras, for every x,y,u,v∈X×Y, we define ′∗′ on X×Y by x,y∗u,v=x∗u,y∗v. Clearly, X×Y,∗,0,0 is a PMS-algebra.Definition 13.
(see [26]). Let At be t-IFS of X w.r.t IFS A. Then the α,β-cut of At is a crisp subset Cα,βAt of X and is given by Cα,βAt=x∈X:μAtx≥α,νAtx≤β, where α,β∈0,1 with α+β≤1.


## 3.t-Intuitionistic Fuzzy PMS-Ideals of a PMS-Algebra

In this section, we study the notion of at-intuitionistic fuzzy PMS-ideal in a PMS-algebra and investigate several interesting results. In what follows, let X and Y denote PMS-algebra unless otherwise specified.Definition 14.
Lett∈0,1. A t-IFS At of X is called the t-intuitionistic fuzzy (t-IF) PMS-ideal of a PMS-algebra X if(i)
μAt0≥μAtx and νAt0≤νAtx,(ii)
μAty∗x≥minμAtz∗y,μAtz∗x,(iii)
νAty∗x≤maxνAtz∗y,νAtz∗x, for all x,y,z∈X.Example 1.
ConsiderX=0,a,b such that X,∗,0 is a PMS-algebra with Table 1.
Define the intuitionistic fuzzy setA=μA,νA in X by
μAx=0.8,ifx=0,0.5,ifx=a,0.6,ifx=b and νAx,q=0.2,ifx=0,0.4,ifx=a,0.3,ifx=b for all x∈X.
If we taket=0.4, then we have μAtx=minμAx,t=minμAx,0.4=0.4 and νAtx=maxνAx,1−t=maxνAx,1−0.4=maxνAx,0.6=0.6, for all x∈X.
Then by routine calculation, we can see thatAt=μAt,νAt is a t-IF PMS-ideal of X.Table 1
X,∗,0 is a PMS-algebra.
∗0ab00abab0abab0Theorem 1.
Every t-IF PMS-ideal ofX is a t-IF PMS-subalgebra of XProof.
LetAt be a t-IF PMS-ideal of X and x,y∈X. Then by Definition 14ii,iii and Definition 1i, we have
μAtx∗y≥minμAt0∗x,μAt0∗y=minμAtx,μAty and(2)νAtx∗y≤maxνAt0∗x,νAt0∗y=maxνAtx,νAty.
Therefore,At is a t-IF PMS-subalgebra of X.Theorem 2.
LetAt be a t-IF PMS-ideal of X. If x≤y, then μAtx≥μAty and νAtx≤νAty, for all x,y∈XProof.
Letx,y∈X such that x≤y. Then x∗y=0. By Definition 14i, Proposition 1iv, Definition 14, and Theorem 1, we have (3)μAtx=μAt0∗x≥minμAty∗0,μAty∗x=minμAty∗0,μAtx∗y∗0=minμAty∗0,μAt0∗0=minμAty∗0,μAt0=μAty∗0≥minμAty,μAt0=μAty,νAtx=νAt0∗x≤maxνAty∗0,νAty∗x=maxνAty∗0,νAtx∗y∗0=maxνAty∗0,νAt0∗0=maxνAty∗0,νAt0=νAty∗0≤maxνAty,νAt0=νAty,and, therefore, μAtx≥μAty and νAtx≤νAty, for all x,y∈X.Theorem 3.
IfA is an IF PMS-ideal of X, then At is also a t-IF PMS-ideal of X.Proof.
LetA be an IF PMS-ideal of X and x,y,z∈X. Then by the definition of t-IFS and definition of intuitionistic fuzzy PMS-ideal, we have
μAt0=minμA0,t≥minμAx,t=μAtx and νAt0=maxvAx,1−t≤maxvA0,1−t=vAtx
Also,(4)μAty∗x=minμAy∗x,t≥minminμAz∗y,μAz∗x,t=minminμAz∗y,t,minμAz∗x,t=minμAtz∗y,μAtz∗x⇒μAty∗x≥minμAtz∗y,μAtz∗x,and(5)νAty∗x=maxνAy∗x,1−t≤maxmaxνAz∗y,νAz∗x,1−t=maxmaxνAz∗y,1−t,maxνAz∗x,1−t=maxνAtz∗y,νAtz∗x⇒νAty∗x≤maxνAtz∗y,νAtz∗x.
Hence,μAty∗x≥minμAtz∗y,μAtz∗x and νAty∗x≤maxνAtz∗y,νAtz∗x, for all x,y,z∈X. Therefore, At is a t-IF PMS-ideal of X.Remark 5.
The converse of above theorem need not be necessarily true. This fact is shown by the following example:Example 2.
LetX=0,a,b be a set with Table 1 as in Example 1. Define the intuitionistic fuzzy set B=μB,νB in X by
μBx=0.4,ifx=0,0.5,ifx=a,0.3,ifx=b and νBx,q=0.5,ifx=0,0.4,ifx=a,0.6,ifx=b for all x∈X.
SinceμB0=0.4<0.5=μBa and νB0=0.5>0.4=vBa, it follows that B is not an intuitionistic fuzzy PMS-ideal of X as it does not satisfy Definition 14i. If we take t=0.3, then μBtx=0.3 and νBtx=0.7, for all x∈X. Therefore, by routine calculations, we get that(i)
μBt0≥μBtx and νBt0≤μBtx(ii)
μBty∗x≥minμBtz∗y,μBtz∗x, and(iii)
νBty∗x≤maxνBtz∗y,νBtz∗x,∀x,y,z∈X.
Therefore, by Definition14, Bt=μBt,νBt is a t-IF PMS-ideal of X.
The following theorem provides a condition for an intuitionistic fuzzy set in a PMS-algebra to be at-intuitionistic fuzzy PMS-ideal.Theorem 4.
LetA be an IFS in X and t∈0,1 such that t≤minm,1−n, where m=minμAx|x∈X and n=maxνAx|x∈X. Then At is a t-IF PMS-ideal of X.Proof.
Sincet≤minm,1−n, we have m≥t and 1−n≥t
⇒m≥t and n≤1−t, ⇒minμAx|x∈X≥t and maxνAx|x∈X≤1−t, ⇒μAx≥t and νAx≤1−t, for all x∈X.
So,minμAx,t=t and maxνAx,1−t=1−t, ⇒μAtx=t and νAtx=1−t, for all x∈X.
Therefore,μAt0=t=μAtx and νAt0=1−t=νAtx. This satisfies the condition μAt0≥μAtx and νAt0≤νAtx, for all x∈X.
Also,μAty∗x=t≥mint,t=minμAtz∗y,μAtz∗x and
νAty∗x=1−t≤max1−t,1−t=maxνAtz∗y,νAtz∗x, for all x,y,z∈X.
Hence,μAty∗x≥minμAtz∗y,μAtz∗x and νAty∗x≤maxνAtz∗y,νAtz∗x, for all x,y,z∈X.Example 3.
Letℤ be the set of all integers. Let ∗ be a binary operation on ℤ defined by x∗y=y−x for all x,y∈ℤ, where ′−′ is the usual subtraction of integers. Then ℤ,∗,0 is a PMS-algebra since Definition 1 is satisfied as shown below.(1)
0∗x=x−0=x(2)
y∗x∗z∗x=z∗x−y∗x=x−z−x−y=y−z=z∗y.
Define an intuitionistic fuzzy setA=x,μAx,νAx|x∈ℤ by(6)μAx=0.8,ifx∈4,0.5,ifx∈2−4,0.3,otherwise,νAx=0.2,ifx∈4,0.4,ifx∈2−4,0.6,otherwise.
Sincem=0.3 and n=0.6, then minm,1−n=min0.3,1−0.6=0.3. So by Theorem 4  for t≤0.3, At is a t-IF PMS-ideal of ℤ.
The subsequent result shows that the intersection of any twot-IF PMS-ideal is a t-IF PMS-ideal.Theorem 5.
The intersection of any two t-IF PMS-ideals ofX is also a t-IF PMS-ideal of X.Proof.
LetA1t and A2t be any two t-IF PMS-ideals of X and x,y,z∈X. Then by Definition 14 and Definition 10, we have and (7)μA1∩A2t0=minμA1∩A20,t=minminμA10,μA20,t=minminμA10,t,minμA20,t=minμA1t0,μA2t0≥minμA1tx,μA2tx=minminμA1x,t,minμA2x,t=minminμA1x,μA2x,t=minμA1∩A2x,t=μA1∩A2tx,νA1∩A2t0=maxνA1∩A20,1−t=maxmaxνA10,νA20,1−t=maxmaxνA10,1−t,maxνA20,1−t=maxνA1t0,νA2t0≤maxνA1tx,νA2tx=maxmaxνA1x,1−t,minνA2x,1−t=maxmaxνA1x,νA2x,1−t=maxνA1∩A2x,1−t=νA1∩A2tx,thus, μA1∩A2t0≥μA1∩A2tx and νA1∩A2t0≤νA1∩A2tx,∀x∈X(8)μA1∩A2ty∗x=minμA1∩A2y∗x,t=minminμA1y∗x,μA2y∗x,t=minminμA1y∗x,t,minμA2y∗x,t=minμA1ty∗x,μA2ty∗x≥minminμA1tz∗y,μA1tz∗x,minμA2tz∗y,μA2tz∗x=minminμA1tz∗y,μA2tz∗y,minμA1tz∗x,μA2tz∗x=minμA1t∩A2tz∗y,μA1t∩A2tz∗x=minμA1∩A2tz∗y,μA1∩A2tz∗xBy remark2.1,and(9)νA1∩A!ty∗x=maxνA1∩A2y∗x,1−t=maxmaxνA1y∗x,νA2y∗x,1−t=maxmaxνA1y∗x,1−t,maxνA2y∗x,1−t=maxνA1ty∗x,νA2ty∗x≤maxmaxνA1tz∗y,νA1tz∗x,maxνA2tz∗y,νA2tz∗x=maxmaxνA1tz∗y,νA2tz∗y,maxνA1tz∗x,νA2tz∗x=maxνA1t∩A2tz∗y,νA1t∩A2tz∗x=maxνA1∩A2tz∗y,νA1∩A2tz∗xBy remark2.
⇒μA1∩A2ty∗x≥minμA1∩A2tz∗y,μA1∩A2tz∗x and(10)νA1∩A2ty∗x≤maxνA1∩A2tz∗y,νA1∩A2tz∗x.
Therefore,At∩Bt is a t-IF PMS-ideals of X.
The above theorem can also be generalized to any family of t-IF PMS-ideals in PMS-algebra as given in the next corollary.Corollary 1.
The intersection of a family of t-IF PMS-ideals ofX is again a t-IF PMS-ideal of X.Remark 6.
The union of any two t-IF PMS-ideals ofX may not be a t-IF PMS-ideal of X. This is shown by the next example.Example 4.
Letℤ be the set of all integers and ∗ is a binary operation on ℤ defined as in Example 3. Clearly, ℤ,∗,0 is a PMS-algebra. Define the intuitionistic fuzzy sets A=x,μAx,νAx|x∈ℤ and B=x,μBx,νBx|x∈ℤ in ℤ respectively by(11)μAx=0.7,ifx∈2,0.3,otherwise,νAx=0.2,ifx∈2,0.5,otherwise,and(12)μBx=0.5,ifx∈3,0.2,otherwise,νBx=0.3,ifx∈3,0.5,otherwise.
Clearly,A and B are IF PMS-ideals of ℤ. Thus, by Theorem 3At and Bt are t-IF PMS-ideals of ℤ for t∈0,1. If we take t = 0.6, then At and Bt are given by(13)μAtx=0.6,ifx∈2,0.3,otherwise,νAtx=0.4,ifx∈2,0.5,otherwise,and(14)μBtx=0.5,ifx∈3,0.2,otherwise,νBtx=0.4,ifx∈3,0.5,otherwise.
NowμAt∪Btx=maxμAtx,μBtx and νAt∪Btx=minνAtx,νBtx. Therefore,(15)μAt∪Btx=0.6,ifx∈2,0.5,ifx∈3−2,0.3,ifx∉2andx∉3,νA∪Bx=0.4ifx∈2∪3,0.5ifx∉2andx∉3.
Takex=2,y=3, and z=−1, then μAt∪Btx=0.6,μAt∪Bty=0.5,μAt∪Btz=0.3, and νAt∪Btx=0.4,νAt∪Bty=0.4,νAt∪Btz=0.5.
NowμAt∪Bty∗x=μAt∪Btx−y=μAt∪Bt−1=0.3,μAt∪Btz∗y=μAt∪Bty−z=μAt∪Bt4=0.6,μAt∪Btz∗x=μAt∪Btx−z=μAt∪Bt3=0.5 and minμAt∪Btz∗y,μAt∪Btz∗x=min0.6,0.5=0.5.(16)⇒μAt∪Bty∗x=0.3≱0.5=minνAt∪Btz∗y,νAt∪Btz∗x.
Similarly,νAt∪Bty∗x=νAt∪Bt−1=0.5,νAt∪Btz∗y=νAt∪Bt4=0.4,νAt∪Btz∗x=νAt∪Bt3=0.4 and maxνAt∪Btz∗y,νAt∪Btz∗x=max0.4,0.5=0.4.(17)vAt∪Bty∗x=0.5≰0.4=maxvAt∪Btz∗y,vAt∪Btz∗x.
As a result of (16) and(17), we arrive at a contradiction with Definition 14.
Therefore,At∪Bt is not an IF-PMS-ideal of ℤ.Theorem 6.
IfAt is a t-IF PMS-ideal of X, then □At is also a t-IF PMS-ideal of X.Proof.
SupposeAt is a t-IF PMS-ideal of X. Then by Definition 14, we have μAt0≥μAtx and μAty∗x≥minμAtz∗y,μAtz∗x for all x,y,z∈X. So, we need to show that μ¯At0≤μ¯Atx and μ¯Aty∗x≤maxμ¯Atz∗y,μ¯Atz∗x for all x,y,z∈X. Now, μ¯At0=1−μAt0≤1−μAtx=μ¯Atx⇒μ¯At0≤μ¯Atx
Also,(18)μ¯Aty∗x=1−μAty∗x≤1−minμAtz∗y,μAtz∗x=max1−μAtz∗y,1−μAtz∗x=maxμ¯Atz∗y,μ¯Atz∗x⇒μ¯Aty∗x≤maxμ¯Atz∗y,μ¯Atz∗x.
Hence,□At is a t-IF PMS-ideal of X.Theorem 7.
IfAt is a t-IF PMS-ideal of X, then ♢At is also a t-IF PMS-ideal of X.Proof.
SupposeAt is a t-IF PMS-ideal of X. Then by Definition 14, we have
νAt0≤νAtx and νAty∗x≤maxνAtz∗y,νAtz∗x, for all x,y,z∈X. So, we have to show that ν¯At0≥ν¯Atx and ν¯Aty∗x≥minν¯Atz∗y,ν¯Atz∗x, for all x,y,z∈X.
Now,ν¯At0=1−νAt0≥1−νAtx=ν¯Atx⇒ν¯At0≥ν¯Atx.
And,(19)ν¯Atx∗y=1−νAtx∗y≥1−maxνAtx,νAty=min1−νAtx,1−νAty=minν¯Atx,ν¯Aty⇒ν¯Atx∗y≥minν¯Atx,ν¯Aty.
Hence,♢At is a t-IF PMS-ideal of X.Theorem 8.
LetAt be a t-IFS of X. Then At is a t-IF PMS-ideal of X if and only if the nonempty subset Cα,βAt of X is a PMS-ideal of X for all α,β∈0,1 with α+β≤1.Proof.
SinceCα,βAt≠∅, there exist x∈X such that x∈Cα,βAt. Then, μAtx≥α and νAtx≤β. Since At is a t-IF PMS-ideal of X, μAt0≥μAtx and νAt0≤νAtx for all x∈X. Thus, μAt0≥α and νAt0≤β. Therefore, 0∈Cα,βAt.
Letx,y,z∈X such that z∗x,z∗y∈Cα,βAt. Then μAtz∗y≥α,μAtz∗x≥α and νAtz∗y≤β,νAtz∗x≤β. Since At is a t-IF PMS-ideal of X, we have μAty∗x≥minμAtz∗y,μAtz∗x≥α and νAty∗x≤maxνAtz∗y,νAtz∗x≤β. Thus, y∗x∈Cα,βAt. Therefore, Cα,βAt is a PMS-ideal of X.
Conversely, supposeCα,βAt is a PMS-ideal of X. Let x∈X such that μAx=α and νAx=β. Since Cα,βAt is a PMS-ideal of X, we have that 0∈Cα,βAt.
This impliesμA0≥α=μAx and νA0≤β=νAx.
Hence,μA0≥μAx and νA0≤νAx for every x∈X.
Also, letx,y,z∈X such that α=minμAtz∗y,μAtz∗x and β=maxνAtz∗y,νAtz∗x. Then μAtz∗y≥α,νAtz∗y≤β and μAtz∗x≥α,νAtz∗x≤β. Thus, z∗y,z∗x∈Cα,βAt. Since Cα,βAt is a PMS-ideal of X, it follows that y∗x∈Cα,βAt. So that we have μAty∗x≥α=minμAtz∗y,μAtz∗x and νAty∗x≤β=maxνAtz∗y,νAtz∗x. Hence, μAty∗x≥minμAtz∗y,μAtz∗x and νAty∗x≤maxνAtz∗y,νAtz∗x, for all x,y,z∈X. Therefore, At is a t-IF PMS-ideal of X.


## 4. Homomorphism oft-Intuitionistic Fuzzy PMS-Ideals

In this section, we discusst-intuitionistic fuzzy PMS-ideals under homomorphism. We study the homomorphic images and inverse images of t-intuitionistic fuzzy PMS-ideals and find some related results.Definition 15.
LetX and Y be two nonempty sets and f:X⟶Y be a mapping. Let At and Bt be t-IFSs of X and Y, respectively. Then the image of At under f is denoted by fAt and is defined as fAt=y,μfAty,νfAty|y∈Y, where(20)μfAty=supμAtx:,x∈f−1y,0,otherwise,and(21)νfAty=infνAtx:,x∈f−1y,1,otherwise.
Also, the inverse image ofBt under f is denoted by f−1Bt and is defined as
f−1Btx=x,μf−1Btx,νf−1Bx|x∈X, where μf−1Btx=μBtfx and νf−1Btx=νBtfx for all x∈X and t∈0,1.
Note: For anyx∈X , we have μfAtfx≥μAtx and νfAtfx≤νAtx.Theorem 9.
Letf: X⟶Y be an epimorphism of PMS-algebras and t∈0,1. If At is a t-IF PMS-ideal of X, then fAt is a t-IF PMS-ideal of Y.Proof.
Sincef:X⟶Y is an epimorphism of PMS-algebras, for each y∈Y, there exists x∈X, such that fx=y. Then μfAt0=μfAt0=μfAtf0≥μAt0≥μAtx=μfAtfx=μfAty=μfAty and νfAt0=νfAt0=νfAtf0≤νAt0≤νAtx=νfAtfx=νfAty=νfAty.
Hence,μfAt0≥μfAty and νfAt0≤νfAty.
Also, letx,y,z∈Y. Since f:X⟶Y is an epimorphism of PMS-algebras, there exist a,b,c∈X such that fa=x,fb=y and fc=z. So, using Definitions 14 and 15, we have,(22)μfAty∗x=μfAty∗x=μfAtfb∗fa=μfAtfb∗a≥μAtb∗a≥minμAtc∗b,μAtc∗a,∀a,b,c∈Xsuch thatfa=x,fb=yandfc=z=minμfAtfc∗b,μfAtfc∗a=minμfAtfc∗fb,μfAtfc∗fa=minμfAtz∗y,μfAtz∗x=minμfAtz∗y,μfAtz∗x,νfAty∗x=νfAty∗x=νfAtfb∗fa=νfAtfb∗a≤νAtb∗a≤maxνAtc∗b,νAtc∗a,∀a,b,c∈Xsuch thatfa=x,fb=yandfc=z=maxνfAtfc∗b,νfAtfc∗a=maxνfAtfc∗fb,νfAtfc∗fa=maxνfAtz∗y,νfAtz∗x=maxνfAtz∗y,νfAtz∗x,and, hence, fAt is a t-IF PMS-ideal of Y.Theorem 10.
Letf:X⟶Y be a homomorphism of PMS-algebras and t∈0,1. If Bt is a t-IF PMS-ideal of Y, then f−1Bt is a t-IF PMS-ideal of X.Proof.
LetBt be a t-IF PMS-ideal of Y for t∈0,1 and let x∈X. Then
μf−1Bt0=μBtf0≥μBtfx=μf−1Btx and
νf−1Bt0=νBtf0≤νBtfx=νf−1Btx⇒μf−1Bt0≥μf−1Btx and νf−1Bt0≤νf−1Btx,∀x∈X.
Letx,y,z∈X. Then by Definition 14 and Definition 15, we have(23)μf−1Bty∗x=μBtfy∗x=μBtfy∗fx≥minμBtfz∗fy,μBtfz∗fx=minμBtfz∗y,μBtfz∗x=minμf−1Btz∗y,μf−1Btz∗x,νf−1Bty∗x=νBtfy∗x=νBtfy∗fx≤maxνBtfz∗fy,νBtfz∗fx=maxνBtfz∗y,νBtfz∗x=maxνf−1Btz∗y,νf−1Btz∗x.
Hence,f−1Bt is a t-IF PMS-ideal of X.
The converse of the above theorem is true iff is an epimorphism of PMS-algebras.Theorem 11.
Letf:X⟶Y be an epimorphism of PMS-algebras and Bt is a t-IFS in Y. If f−1Bt is a t-IF PMS-ideal of X, then Bt is a t-IF PMS-ideal of Y for t∈0,1.Proof.
Assume thatf:X⟶Y is an epimorphism of PMS-algebras and f−1Bt is a t-IF PMS-ideal of X. Since f is an epimorphism of PMS-algebras for any x∈Y, there exist a∈X such that fa=x. Then,
μBt0=μBtf0=μf−1Bt0≥μf−1Bta=μBtfa=μBtx and(24)νBt0=νBtf0=νf−1Bt0≤νf−1Bta=νBtfa=νBtx.
Also, letx,y,z∈Y. Then there exist a,b,c∈X such that fa=x,fb=y, and fc=z.
Now,(25)μBty∗x=μBtfb∗fa=μBtfb∗a=μf−1Btb∗a≥minμf−1Btc∗b,μf−1Btc∗a=minμBtfc∗b,μBtfc∗a=minμBtfc∗fb,μBtfc∗fa=minμBtz∗y,μBtz∗x,and(26)νBty∗x=νBtfb∗fa=νBtfb∗a=νf−1Btb∗a≤maxνf−1Btc∗b,νf−1Btc∗a=maxνBtfc∗b,νBtfc∗a=maxνBtfc∗fb,νBtfc∗fa=maxνBtz∗y,νBtz∗x.
Hence,Bt is a t-IF PMS-ideal of Y.Theorem 12.
LetAt be a t-IFS of X and f:X⟶Y be an epimorphism of PMS-algebras. Then the homomorphic image of the nonempty subset Cα,βAt of X is a PMS-ideal of Y.Proof.
LetAt be a t-IF PMS-ideal of X and let x,y,z∈Y. Since f is an epimorphism of PMS-algebras, there exist a,b,c∈X such that fa=x,fb=y, and fc=z. By Theorem 9, fAt is a t-IF PMS-ideal of Y, and by Theorem6, Cα,βAt is a PMS-ideal of X, that is 0∈Cα,βAt and c∗b,c∗a∈Cα,βAt⇒b∗a∈Cα,βAt.
Then,μfAt0=μfAt0=μfAtf0≥μAt0≥α⇒μfAt0≥α and(27)νfAt0=νfAt0=νfAtf0≤μAt0≤β⇒νfAt0≤β.
Hence,0∈fCα,βAt.
Also, assume thatz∗y,z∗x∈fCα,βAt, that is fc∗fb,fc∗fa∈fCα,βAt.
So, we have(28)μfAty∗x=μfAty∗x=μfAtfb∗fa=μfAtfb∗a≥μAtb∗a≥α,and(29)νfAty∗x=νfAty∗x=νfAtfb∗fa=νfAtfb∗a≤νAtb∗a≤β.
Thus,μfAtfb∗fa=μfAty∗x≥α and(30)νfAtfb∗fa=νfAty∗x≤β.
Therefore,fb∗fa∈fCα,βAt. Hence, fCα,βAt is a PMS-ideal of Y.Theorem 13.
LetBt be a t-IFS of a PMS-algebra Y and f:X⟶Y be an epimorphism of PMS-algebras. Then the homomorphic inverse image of the nonempty subset Cα,βBt of Y is a PMS-ideal of X.Proof.
LetBt be a t-IF PMS-ideal of Y and let x,y,z∈Y. Since f is an epimorphism of PMS-algebras, there exist a,b,c∈X such that fa=x,fb=y, and fc=z. By Theorem 10, f−1Bt is a t-IF PMS-ideal of Y, and by Theorem 8,   Cα,βBt is a PMS-ideal of Y, that is 0∈Cα,βBt and z∗y,z∗x∈Cα,βBt⇒y∗x∈Cα,βBt.
μf−1Bt0=μBtf0=μBt0≥α and νf−1Bt0=νBtf0=νBt0≤β.
Hence,0∈f−1Cα,βBt.
Now letc∗b,c∗a∈f−1Cα,βBt. So, we have
μf−1Btb∗a=μBtfb∗a=μBtfb∗fa=μBty∗x≥α and(31)νf−1Btb∗a=νBtfb∗a=νBtfb∗fa=νBty∗x≤β.
Hence,μf−1Btb∗a≥α and νf−1Btb∗a≤β⇒b∗a∈f−1Cα,βBt.
Therefore,f−1Cα,βBt is a PMS-ideal of X.


## 5. Cartesian Product oft-Intuitionistic Fuzzy PMS-Ideals

In this section, we consider the Cartesian product of at-intuitionistic fuzzy PMS-ideal and investigate its related properties. We define the strongest t-intuitionistic fuzzy PMS-relation and study its relationship with a t-intuitionistic fuzzy PMS-ideal.Theorem 14.
LetAt and Bt be two t-IF PMS-ideals of X and Y, respectively. Then At×Bt is a t-IF PMS-ideal of X×Y.Proof.
Letx,y∈X×Y. Then by Definition 14, we have
μAt×Bt0,0=minμAt0,μBt0≥minμAtx,μBty=μAt×Btx,y and(32)νAt×Bt0,0=maxνAt0,νBt0≤maxνAtx,νBty=νAt×Btx,y.
Thus,μAt×Bt0,0≥μAt×Btx,y and νAt×Bt0,0≤νAt×Btx,y,∀x,y∈X×Y.
Also, letx1,x2,y1,y2,z1,z2∈X×Y. Again by Definition 14, we have(33)μAt×Bty1,y2∗x1,x2=μAt×Bty1∗x1,y2∗x2=minμAty1∗x1,μBty2∗x2≥minminμAtz1∗y1,μAtz1∗x1,minμBtz2∗y2,μBtz2∗x2=minminμAtz1∗y1,μBtz2∗y2,minμAtz1∗x1,μBtz2∗x2=minμAt×Btz1∗y1,z2∗y2,μAt×Btz1∗x1,z2∗x2=minμAt×Btz1,z2∗y1,y2,μAt×Btz1,z2∗x1,x2⇒μAt×Bty1,y2∗x1,x2≥minμAt×Btz1,z2∗y1,y2,μAt×Btz1,z2∗x1,x2,and(34)νAt×Bty1,y2∗x1,x2=νAt×Bty1∗x1,y2∗x2=maxνAty1∗x1,νBty2∗x2≤maxmaxνAtz1∗y1,νAtz1∗x1,maxνBtz2∗y2,νBtz2∗x2=maxmaxνAtz1∗y1,νBtz2∗y2,maxνAtz1∗x1,νBtz2∗x2=maxνAt×Btz1∗y1,z2∗y2,νAt×Btz1∗x1,z2∗x2=maxνAt×Btz1,z2∗y1,y2,νAt×Btz1,z2∗x1,x2,⇒νAt×Bty1,y2∗x1,x2≤maxνAt×Btz1,z2∗y1,y2,νAt×Btz1,z2∗x1,x2.
Therefore,At×Bt is an intuitionistic fuzzy PMS-ideal of X×Y.Theorem 15.
LetAt and Bt be any two t-IF sets in X and Y, respectively. If At×Bt is a t-IF PMS-ideal of X×Y, then either At is a t-IF PMS-ideal of X or Bt is a t-IF PMS-ideal of Y.Proof.
LetAt and Bt be t-IFSs of X and Y, respectively, such that At×Bt is a t-IF PMS-ideal of X×Y. Then μAt×Bt0,0≥μAt×Btx,y, ∀x,y∈X×Y.
AssumeμAtx>μAt0 and μBty>μBt0 for some x,y∈X×Y. Then,(35)μAt×Btx,y=minμAtx,μBty>minμAt0,μBt0=μAt×Bt0,0,which is a contradiction,Thus,μAt0≥μAtxorμBt0≥μBty,∀x,y∈X×Y.
Similarly,νAt0≤νAtx or νBt0≤νBty,∀x,y∈X×Y.
Letx1,x2,y1,y2,z1,z2∈X×Y. Since At×Bt is a t-IF PMS-ideal of X×Y, we have(36)μAt×Bty1,y2∗x1,x2≥minμAt×Btz1,z2∗y1,y2,μAt×Btz1,z2∗x1,x2=minμAt×Btz1∗y1,z2∗y2,μAt×Btz1∗x1,z2∗x2=minminμAtz1∗y1,μBtz2∗y2,minμAtz1∗x1,μBtz2∗x2.
AsμAt×Bty1,y2∗x1,x2=μAt×Bty1∗x1,y2∗x2, we have(37)μAt×Bty1∗x1,y2∗x2≥minminμAtz1∗y1,μBtz2∗y2,minμAtz1∗x1,μBtz2∗x2⇒minμAty1∗x1,μBty2∗x2≥minminμAtz1∗y1,μBtz2∗y2,minμAtz1∗x1,μBtz2∗x2=minminμAtz1∗y1,μAtz1∗x1,minμBtz2∗y2,μBtz2∗x2.
If we putx2=y2=z2=0 (or resp. x1=y1=z1=0), we have either
μAty1∗x1≥minμAtz1∗y1,μAtz1∗x1 or μBty2∗x2≥minμBtz2∗y2,μBtz2∗x2.
In similar way, we can show that eitherνAty1∗x1≤maxνAtz1∗y1,νAtz1∗x1 or νBty2∗x2≤maxνBtz2∗y2,νBtz2∗x2.
Therefore,At is a t-IF PMS-ideal of X or Bt is a t-IF PMS-ideal of Y.Definition 16.
LetAt and Bt be t-IFSs of X and Y w.r.t IFSs A and B. Then, α,β-cut of At×Bt is a crisp subset Cα,βAt×Bt of X×Y is given by
Cα,βAt×Bt=x,y∈X×Y:μAtx,y≥α,νAtx,y≤β, where α,β∈0,1 with α+β≤1.Theorem 16.
LetAt and Bt be t-IFSs of X and Y respectively. Then At×Bt is a t-IF PMS-ideals of X×Y if and only if the nonempty subset Cα,βAt×Bt of X×Y is a PMS-ideal of X×Y for all α,β∈0,1 with α+β≤1.Proof.
LetAt=μAt,νAt and Bt=μBt,νBt be t-IFSs of X and Y, respectively. Since Cα,βAt×Bt≠∅, there exist x,y∈X×Y such that x,y∈Cα,βAt×Bt. Then, μAt×Btx,y≥α and νAt×Btx,y≤β. Since At×Bt is a t-IF PMS-ideal of X×Y, μAt×Bt0,0≥μAtx,y and νAt×Bt0,0≤νAtx,y for all x,y∈X×Y. Thus, it follows that μAt×Bt0,0≥α and νAt×Bt0,0≤β. Therefore, 0,0∈Cα,βAt
Letx1,x2,y1,y2,z1,z2∈X×Y such that z1,z2∗y1,y2,z1,z2∗x1,x2∈Cα,βAt×Bt for α,β∈0,1.Then, μAt×Btz1,z2∗y1,y2≥α,μAt×Btz1,z2∗x1,x2≥α, and νAt×Btz1,z2∗y1,y2≤β,νAt×Btz1,z2∗x1,x2≤β. Since At×Bt is a t-IF PMS- ideal of X×Y, we have μAt×Bty1,y2∗x1,x2≥minμAt×Btz1,z2∗y1,y2,μAt×Btz1,z2∗y1,y2≥minα,α=α and νAt×Bty1,y2∗x1,x2≤maxνAt×Btz1,z2∗y1,y2,νAt×Btz1,z2∗x1,x2≤maxβ,β=β.
⇒μAt×Bty1,y2∗x1,x2≥α and νAt×Bty1,y2∗x1,x2≤β.
Therefore,y1,y2∗x1,x2∈Cα,βAt×Bt.
Hence,Cα,βAt×Bt  is a PMS-ideal of X×Y.
Conversely, supposeCα,βAt×Bt is a PMS-ideal of X×Y for all α,β∈0,1 with α+β≤1. Assume that At×Bt is not a t-IF PMS-ideal of X×Y. Then there exist x1,x2,y1,y2,z1,z2∈X×Y such that μAt×Bty1,y2∗x1,x2<minμAt×Btz1,z2∗y1,y2,μAt×Btz1,z2∗y1,y2 and νAt×Bty1,y2∗x1,x2>maxνAt×Btz1,z2∗y1,y2,νAt×Btz1,z2∗x1,x2. Then by taking
α0=1/2μAt×Bty1,y2∗x1,x2+minμAt×Btz1,z2∗y1,y2,μAt×Btz1,z2∗y1,y2 and β0=1/2νAt×Bty1,y2∗x1,x2+maxνAt×Btz1,z2∗y1,y2,νAt×Btz1,z2∗x1,x2, μAt×Bty1,y2∗x1,x2<α0<minμAt×Btz1,z2∗y1,y2,μAt×Btz1,z2∗y1,y2 and νAt×Bty1,y2∗x1,x2>β0>maxνAt×Btz1,z2∗y1,y2,νAt×Btz1,z2∗y1,y2.
Hence,y1,y2∗x1,x2∉Cα,βAt×Bt but z1,z2∗y1,y2∈Cα,βAt×Bt and z1,z2∗x1,x2∈Cα,βAt×Bt. This implies Cα,βAt×Bt is not a PMS-ideal of X×Y, which is a contradiction.
Therefore,μAt×Bty1,y2∗x1,x2≥minμAt×Btz1,z2∗y1,y2,μAt×Btz1,z2∗x1,x2 and νAt×Bty1,y2∗x1,x2≤maxνAt×Btz1,z2∗y1,y2,νAt×Btz1,z2∗x1,x2. for all x1,y1,x2,y2∈X×Y.
Hence,At×Bt is an intuitionistic fuzzy PMS-ideal of X×Y.Definition 17.
LetAt=μAt,νAt be a t-intuitionistic fuzzy set in X and Rt=μRt,νRt be a t-intuitionistic fuzzy relation on X. Then the strongest t-intuitionistic fuzzy PMS-relation RAtt on X, that is, a t-intuitionistic fuzzy PMS-relation Rt on At whose membership function μRAtt:X×X⟶0,1 and whose nonmembership function νRAtt:X×X⟶0,1 is given by μRAttx,y=minμAtx,μAty and νRAttx,y=maxνAtx,νAty, for all x,y∈X.Theorem 17.
LetAt=μAt,νAt be a t-intuitionistic fuzzy subset of PMS-algebra X and let RAtt be the strongest t-intuitionistic fuzzy PMS-relation on X, then At is a t-intuitionistic fuzzy PMS-ideal of X if and only if RAtt is a t-intuitionistic fuzzy PMS-ideal of X×X.Proof.
Assume thatAt is a t-intuitionistic fuzzy PMS-ideal of X. Let x,y∈X×X. Then
μRAtt0,0=minμAt0,μAt0≥minμAtx,μAty=μRAttx,y and
νRAtt0,0=maxνAt0,νAt0≤maxνAtx,νAty=νRAttx,y, for all x,y∈X, ⇒μRAtt0,0≥μRAttx,y and νRAtt0,0≤νRAttx,y, for all x,y∈X.
Also, letx1,x2,y1,y2,z1,z2∈X×X. Then(38)μRAtty1,y2∗x1,x2=μRAtty1∗x1,y2∗x2=minμAty1∗x1,μAy2∗x2≥minminμAtz1∗y1,μAtz1∗x1,minμAtz2∗y2,μAtZ2∗x2=minminμAtz1∗y1,μAtz2∗y2,minμAtz1∗x1,μAtZ2∗x2=minμRAttz1∗y1,z2∗y2,μRAttz1∗x1,z2∗x2=minμRAttz1,z2∗y1,y2,μRAttz1,z2∗x1,x2⇒μRAtty1,y2∗x1,x2≥minμRAttz1,z2∗y1,y2,μRAttz1,z2∗x1,x2,and(39)νRAtty1,y2∗x1,x2=νRAtty1∗x1,y2∗x2=maxνAty1∗x1,νAy2∗x2≤maxmaxνAtz1∗y1,νAtz1∗x1,maxνAtz2∗y2,νAtZ2∗x2=maxmaxνAtz1∗y1,νAtz2∗y2,maxνAtz1∗x1,νAtZ2∗x2=maxνRAttz1∗y1,z2∗y2,νRAttz1∗x1,z2∗x2=maxνRAttz1,z2∗y1,y2,νRAttz1,z2∗x1,x2⇒νRAtty1,y2∗x1,x2≤maxνRAttz1,z2∗y1,y2,νRAttz1,z2∗x1,x2.
Therefore,RAtt is a t-intuitionistic fuzzy PMS-ideal of X×X.
Conversely, assume thatRAtt is a t-intuitionistic fuzzy PMS-ideal of X×X. Let x1,x2,y1,y2,z1,z2∈X×X. Then we have(i)
minμAt0,μAt0=μRAtt0,0≥μRAttx1,x2=minμAtx1,μAtx2(40)⇒minμAt0,μAt0≥minμAtx1,μAtx2⇒μAt0≥μAtx1 or μAt0≥μAtx2and(41)maxνAt0,νAt0=νRAtt0,0≤νRAttx1,x2=maxνAtx1,νAtx2⇒maxνAt0,νAt0≤maxνAtx1,νAtx2.⇒νAt0≤νAtx1 or νAt0≤νAx2(ii)
minμAty1∗x1,μAty2∗x2=μRAtty1∗x1,y2∗x2=μRAtty1,y2∗x1,x2≥minμRAttz1,z2∗y1,y2,μRAttz1,z2∗x1,x2=minμRAtz1∗y1,z2∗y2,μRAz1∗x1,z2∗x2=minminμAtz1∗y1,μAtz2∗y2,minμAtz1∗x1,μAtz2∗x2=minminμAtz1∗y1μAtz1∗x1,minμAtz2∗y2μAtz2∗x2If we putx2=y2=z2=0 (or resp. x1=y1=z1=0), then we getμAty1∗x1≥minμAtz1∗y1,μAtz1∗x1 (or resp. μAty2∗x2≥minμAtz2∗y2,μAtz2∗x2)(iii)
maxνAty1∗x1,νAty2∗x2=νRAtty1∗x1,y2∗x2=νRAtty1,y2∗x1,x2≤maxνRAttz1,z2∗y1,y2,νRAttz1,z2∗x1,x2=maxνRAttz1∗y1,z2∗y2,νRAttz1∗x1,z2∗x2=maxmaxνAtz1∗y1,νAtz2∗y2,maxνAtz1∗x1,νAtz2∗x2=maxmaxνAtz1∗y1,νAtz1∗x1,maxνAtz2∗y2νAtz2∗x2
If we putx2=y2=z2=0 (or resp. x1=y1=z1=0), then we get
νAty1∗x1≤maxνAtz1∗y1,νAtz1∗x1 (or resp. νAty2∗x2≤maxμAtz2∗y2,μAtz2∗x2).
Hence,At is a t-intuitionistic fuzzy PMS-ideal of X.


## 6. Conclusion

In this article, we used the concept of at-intuitionistic fuzzy set to PMS-ideals in a PMS-algebra. We studied the notion of a t-intuitionistic fuzzy PMS-ideal of a PMS-algebra and explored some related properties. We provided the relationships between a t-intuitionistic fuzzy PMS-ideal and a t-intuitionistic fuzzy PMS-subalgebra of a PMS-algebra, as well as the relationships between an intuitionistic fuzzy PMS-ideal and a t-intuitionistic fuzzy PMS-ideal of a PMS-algebra. We established a condition for an intuitionistic fuzzy set in a PMS-algebra to be a t-intuitionistic fuzzy PMS-ideal of a PMS-algebra. We described the t-intuitionistic fuzzy PMS-ideals of PMS-algebra by their α,β level cuts. We studied a t-intuitionistic fuzzy PMS-ideal of a PMS-algebra under homomorphism and explored the homomorphic image and inverse image of the t-intuitionistic fuzzy PMS-ideal. Furthermore, we discussed the Cartesian product of any two t-intuitionistic fuzzy PMS-ideals of PMS-algebra and obtained some interesting results. We characterized the Cartesian product of the t-intuitionistic fuzzy PMS-ideals by their α,β level cuts. Finally, we defined the strongest t-intuitionistic fuzzy PMS-relation on a PMS-algebra and studied the relationship between the strongest t-intuitionistic fuzzy PMS-relation and a t-intuitionistic fuzzy PMS-ideal. We hope that the findings of this work will add other dimensions to the structures of t-intuitionistic fuzzy PMS-ideals based on t-intuitionistic fuzzy sets and serve as the foundation for further studies. In our future works, we will extend this concept to t-Q intuitionistic fuzzy PMS-ideals, t-intuitionistic multi-fuzzy, and anti-multi-fuzzy PMS-ideals to obtain additional novel results. Moreover, we will develop the neutro-algebraic structures with respect to the PMS-ideals of PMS-algebra.



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*Source: 5101293-2022-09-23.xml*