# Rough Fuzzy Ideals Induced by Set-Valued Homomorphism in Ternary Semigroups

Authors: Shahida Bashir; Muhammad Aslam; Rabia Mazhar; Junaid Asghar
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The main objective of this paper is to characterize rough approximations of fuzzy ideals in ternary semigroups. Rough fuzzy ideals are used to deal with vague and incomplete information in decision-making problems. In this research, approximations for fuzzy prime ideals in ternary semigroups are studied. It is proved that generalized lower approximations and generalized upper approximations of∈,∈∨q-fuzzy prime (resp., semiprime) ideals of ternary semigroups are ∈,∈∨q-fuzzy prime (resp., semiprime) ideals. For this, the concept of SSVH (strong set-valued homomorphism) and SVH (set-valued homomorphism) is used. Also, it is shown by examples that the lower approximations of fuzzy subsemigroups and fuzzy ideals are not fuzzy subsemigroups and fuzzy ideals, respectively, for SVH.


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## Body

## 1. Introduction and Motivation

In classical mathematics, the membership of components in a set is calculated by crisp set. According to this, a component either belongs or does not belong to the set, and a statement is true or false. In real life, there are many midway situations of a complicated problem. After a long effort, many theories are introduced by researchers to handle such problems such as fuzzy set theory, rough set theory, and soft set theory etc. Historically, Zadeh in 1965 [1] initiated the concept of fuzzy set theory, and he provided the idea to study ambiguity complications. After that, this concept has been used in different fields of our world. Fuzzy mathematics has stayed as a vital branch of mathematics until now. One of the foremost vital directions in fuzzy mathematics is the study of fuzzy algebras. Fuzzy set theory has many applications in robotics, artificial intelligence, control engineering, and many more branches of applied and pure mathematics.In 1982, Pawlak [2] initiated the theory of rough set. Rough set is characterized using the lower and upper approximations. Based on the known attributes, a rough set contains all the information. Let C be any nonempty subset of universal set W, and then, lower and upper approximations of C are shown in Figure 1, where each rectangular region represents an equivalence class, since, in same equivalence class, elements are indiscernible from each other and equivalence classes are the small granularity in the whole data. Then, to define exactly C, a pair of approximation is used. The lower approximation consists of those equivalence classes that are completely enclosed in C, while the upper approximation (grey rectangles) consists of all equivalence classes that are partly inside the C and in the lower approximation.Figure 1
Upper and lower approximations.


### 1.1. Applications of the Proposed Model

In mathematics, algebraic structures play very important role. In different fields, these structures have solid applications, for example, in computer science, theoretical physics, information sciences, control engineering, topological spaces, and coding theory. Lehmer [3] was the first who furnished the idea of ternary algebraic systems. In the 19th century, cubic associations and ternary algebraic operations were measured by numerous mathematicians. Similarly, rough set theory is used to determine the generalized rules that describe the connection between acoustical parameters of concert halls and sound processing approaches. This theory is a useful tool in the areas of artificial intelligence, such as pattern recognition, learning algorithms, inductive reasoning, and automatic classification. Also, it has many applications in measurement theory, classification theory, taxonomy, cluster analysis, etc. In medical field, a serious challenge is the abdominal pain in children. There are many reasons of this disease, and it is hard to detect the main cause. This theory helps the doctors to detect by discharge observations (for more applications, see [4–6]).


### 1.2. Innovative Contribution

There are many structures which are not closed under binary multiplication. For example, the subsetℝ+ of real numbers ℝ is closed with respect to the binary multiplication of semigroup, while in ℝ−, the closure law does not hold, but it is closed with respect to ternary product. To handle such kind of problems in algebraic structures, we study ternary structures (for more applications of ternary operation, see [7, 8]). Ternary semigroups along with their algebraic structures are deliberated through few other authors as well, including Sioson, who examined ideals along with ternary semigroup [9]. In ternary semigroups, bi-ideals and quasi-ideals were examined by Dixit and Dewan [10]. Shabir and Bashir studied prime ideals in ternary semigroups [11]. Ternary semigroups in the form of minimal and maximal lateral ideals were investigated by Iampan in 2007 [12]. Kar and Maity in [13] deal with the study of congruences of ternary semigroups. Bashir et al. studied three-dimensional congruence relation in rough fuzzy ternary semigroups [14].


### 1.3. Related Works

Rosenfeld [15] studied fuzzification of algebraic structures and then approached the fuzzy groups. P. Ming and L. Ming in 1980 studied the fuzzy point [16]. This fundamental notion of fuzzy sets applied to semigroup was initially considered by Kuroki in 1991 [17]. Bhakat and Das studied the fuzzy subgroups [18]. Among these ∈,∈∨q-fuzzy ideals in semigroups are most significant (see [19]). The idea of rough semigroups was popularized by Biswas and Nanda [20]. Petchkhaew and Chinram studied rough fuzzy ideals in ternary semigroups in 2009 [21]. Rameez et al. worked on the generalization of roughness in ∈,∈∨q-fuzzy ideals of hemirings [22]. In 2021, Anwar et al. worked on roughness and multigranulation roughness of intuitionistic fuzzy sets in terms of soft relations [23, 24]. In 2022, Anwar et al. studied pessimistic multigranulation roughness of intuitionistic fuzzy sets based on soft relations [25]. Recently, Shabir et al. approximate fuzzy ideals of semirings using bipolar techniques [26]. In this paper, we generalize the structure of [22] to ternary semigroups.


### 1.4. Organization of the Paper

This paper is arranged as follows: in Section2, some basic material related to ternary semigroups, fuzzy sets, rough sets, and ∈,∈∨q-fuzzy ideals in ternary semigroup is given. Section 3> is about to our main work in which the lower approximation and upper approximation of fuzzy ternary subsemigroups, fuzzy ideals, fuzzy semiprime ideals, and fuzzy prime ideals are studied. Approximations of ∈,∈∨q-fuzzy ternary subsemigroups and ∈,∈∨q-fuzzy ideals are studied in Section 4. Lastly, comparative study and conclusions are given. The list of acronyms used in the research article is given in Abbreviations.


## 1.1. Applications of the Proposed Model

In mathematics, algebraic structures play very important role. In different fields, these structures have solid applications, for example, in computer science, theoretical physics, information sciences, control engineering, topological spaces, and coding theory. Lehmer [3] was the first who furnished the idea of ternary algebraic systems. In the 19th century, cubic associations and ternary algebraic operations were measured by numerous mathematicians. Similarly, rough set theory is used to determine the generalized rules that describe the connection between acoustical parameters of concert halls and sound processing approaches. This theory is a useful tool in the areas of artificial intelligence, such as pattern recognition, learning algorithms, inductive reasoning, and automatic classification. Also, it has many applications in measurement theory, classification theory, taxonomy, cluster analysis, etc. In medical field, a serious challenge is the abdominal pain in children. There are many reasons of this disease, and it is hard to detect the main cause. This theory helps the doctors to detect by discharge observations (for more applications, see [4–6]).


## 1.2. Innovative Contribution

There are many structures which are not closed under binary multiplication. For example, the subsetℝ+ of real numbers ℝ is closed with respect to the binary multiplication of semigroup, while in ℝ−, the closure law does not hold, but it is closed with respect to ternary product. To handle such kind of problems in algebraic structures, we study ternary structures (for more applications of ternary operation, see [7, 8]). Ternary semigroups along with their algebraic structures are deliberated through few other authors as well, including Sioson, who examined ideals along with ternary semigroup [9]. In ternary semigroups, bi-ideals and quasi-ideals were examined by Dixit and Dewan [10]. Shabir and Bashir studied prime ideals in ternary semigroups [11]. Ternary semigroups in the form of minimal and maximal lateral ideals were investigated by Iampan in 2007 [12]. Kar and Maity in [13] deal with the study of congruences of ternary semigroups. Bashir et al. studied three-dimensional congruence relation in rough fuzzy ternary semigroups [14].


## 1.3. Related Works

Rosenfeld [15] studied fuzzification of algebraic structures and then approached the fuzzy groups. P. Ming and L. Ming in 1980 studied the fuzzy point [16]. This fundamental notion of fuzzy sets applied to semigroup was initially considered by Kuroki in 1991 [17]. Bhakat and Das studied the fuzzy subgroups [18]. Among these ∈,∈∨q-fuzzy ideals in semigroups are most significant (see [19]). The idea of rough semigroups was popularized by Biswas and Nanda [20]. Petchkhaew and Chinram studied rough fuzzy ideals in ternary semigroups in 2009 [21]. Rameez et al. worked on the generalization of roughness in ∈,∈∨q-fuzzy ideals of hemirings [22]. In 2021, Anwar et al. worked on roughness and multigranulation roughness of intuitionistic fuzzy sets in terms of soft relations [23, 24]. In 2022, Anwar et al. studied pessimistic multigranulation roughness of intuitionistic fuzzy sets based on soft relations [25]. Recently, Shabir et al. approximate fuzzy ideals of semirings using bipolar techniques [26]. In this paper, we generalize the structure of [22] to ternary semigroups.


## 1.4. Organization of the Paper

This paper is arranged as follows: in Section2, some basic material related to ternary semigroups, fuzzy sets, rough sets, and ∈,∈∨q-fuzzy ideals in ternary semigroup is given. Section 3> is about to our main work in which the lower approximation and upper approximation of fuzzy ternary subsemigroups, fuzzy ideals, fuzzy semiprime ideals, and fuzzy prime ideals are studied. Approximations of ∈,∈∨q-fuzzy ternary subsemigroups and ∈,∈∨q-fuzzy ideals are studied in Section 4. Lastly, comparative study and conclusions are given. The list of acronyms used in the research article is given in Abbreviations.


## 2. Preliminaries

Here, we introduce some basic notions about ternary semigroups, fuzzy ternary semigroups, and rough sets.A nonempty set is called a ternary semigroup, if there exists a ternary operationA×A×A⟶A written as d,e,f⟶def satisfying the following identity defgh=defgh=defgh for any d,e,f,g,h∈A. By a subset, we always mean a nonempty subset. A subset M of a ternary semigroup A is known as a left ideal of A if AAM⊆M, a lateral ideal of A if AMA⊆M, and a right ideal of A if MAA⊆M. A proper ideal P of a ternary semigroup A is said to be a prime ideal of A if LMN⊆P implies L⊆P or M⊆P or N⊆P, for all ideals L,M,N of A, and it is semiprime if I3⊆P implies I⊆P for all ideal I of A [11].A fuzzy set defined on ternary semigroup such thatμ:A⟶0,1 is known as fuzzy ternary subsemigroup; here, 0,1 represents a unit section of a real number line. In this paper, A stands for ternary semigroup and μ is fuzzy subset (FS) of A, unless stated otherwise.Definition 1 (see [16]).
A fuzzy point is defined as(1)μd=tifd=e0ifd≠e,where d is support of an FS μ of A and t∈0,1. It is denoted as et.Definition 2 (see [27]).
Ifdi is a fuzzy point, then di∈μdiqμ, respectively, as follows.
(1)
Ifμd≥i, then di belongs to μ and is denoted as di∈μ(2)
Ifμd+i>1, then di is said to be quasi-coincident with μ and is written as diqμ(3)
Ifμd≥i or μd+i>1, then di∈∨qμ
When any ofdi∈μ, diqμ, or di∈∨qμ does not hold, then we write di∈¯μ, diq¯μ or di∈∨q¯μ, respectively.Definition 3 (see [27]).
The fuzzy ternary subsemigroupμ of A is defined as for every d,e,f∈A, μdef≥minμd,μe,μf.Definition 4 (see [27]).
The fuzzy left (resp., right, lateral) idealμ of A is defined as μdef≥μf (resp. μdef≥μd and μdef≥μe for all d,e,f∈A.Definition 5 (see [21]).
The fuzzy prime idealμ of A is defined as for all d,e,f∈A,μdef=μd or μdef=μe or μdef=μf and fuzzy semiprime if μddd=μd.Definition 6 (see [27]).
The∈,∈∨q-fuzzy ternary subsemigroup μ of A is defined as for every d,e,f∈A and t,r,s∈0,1, dt,er,fs∈μ⟶defmint,r,s∈∨qμ.Definition 7 (see [27]).
The∈,∈∨q-fuzzy left (resp., right, lateral) ideal μ of A is defined as for di∈μ and e,f∈A⟶efdi∈∨qμresp.defi∈∨qμandedfi∈∨qμ for alld,e,f∈A and i∈0,1.Definition 8.
The∈,∈∨q-fuzzy ideal μ of A is said to be a semiprime if d3i∈μ implies di∈∨qμ, for all d∈A and i∈0,1.Definition 9.
Ifμ is ∈,∈∨q-fuzzy ideal of A, then it is said to be a prime if for all d,e,f∈A, i∈0,1,defi∈μ implies di∈∨qμorei∈∨qμ or fi∈∨qμ and semiprime if d3i∈μ implies di∈∨qμ.Theorem 10 (see [27]).
Ifμ is a ∈,∈∨q-fuzzy ternary subsemigroup of A, then μdef≥∧μd,μe,μf,0.5 for all d,e,f∈A.Theorem 11 (see [27]).
Ifμ is a ∈,∈∨q-fuzzy left ideal of A. Then, μdef≥∧μf,0.5 for every d,e,f∈A.Theorem 12.
Ifμ is a ∈,∈∨q-fuzzy semiprime ideal of A, then for any d∈A,μd≥μd3,0.5.Proof.
Proof is same as Theorem11.Theorem 13.
Ifμ is a ∈,∈∨q-fuzzy prime ideal of A, then for any d,e,f∈A,∨μd,μe,μf≥∧μdef,0.5.Proof.
Proof is same as Theorem11.Now, we give some concepts on rough set. LetL be an equivalence relation on universe W, and then, for any s∈W, an equivalence class of s is the collection of elements of W that are related to s and is written as sL. A pair W,L is said to be an approximation space.Definition 14.
Supposeμ is FS of A. Then, there are the following sets: L¯μ=∧μtt∈sL and L¯μ=∨μtt∈sL as a lower approximation and upper approximation, respectively.Definition 15.
Letμ be any FS of A. Then, for every d∈A, we define fuzzy subsets Jμ¯d=∧m∈Jdμm and Jμ¯d=∨m∈Jdμm. Jμ¯ is called the lower approximation and Jμ¯ is called the upper approximation of the FS μ with respect to J. The combination of Jμ¯,Jμ¯ is called the rough FS of AifJμ¯≠Jμ¯.Definition 16.
LetA and A′ be two ternary semigroups, and then, set valued map J:A⟶P∗A′ is called an SVH if for all d,e,f∈A,JdJeJf⊆Jdef.Definition 17.
LetA and A′ be two ternary semigroups, and then, set valued map J:A⟶P∗A′ is called an SSVH if for all d,e,f∈A,JdJeJf=Jdef.Here,P∗A′ means the collection of all nonempty subsets of A′.In this paper, for alld∈A, the image Jd is always a nonempty subset of A. This mapping is natural. In case of groups, canonical maps are set-valued maps in which image of an element is a coset. Also, J is a function J:A⟶P∗A, unless stated otherwise.


## 3. Lower and Upper Approximations of Fuzzy Ideals

Initially in this section, approximations of fuzzy ternary subsemigroups are studied. Then, it is shown that the approximations of fuzzy semiprime ideals and fuzzy prime ideals are fuzzy semiprime ideals and fuzzy prime ideals, respectively.Theorem 18.
IfJ is SSVH and μ is fuzzy ternary subsemigroup ofA, J¯μ is fuzzy ternary subsemigroup of A.Proof.
Sinceμ is a fuzzy ternary subsemigroup of A, then μdef≥∧μd,μe,μf for all d,e,f∈A.
Consider(2)J¯μdef=∧m∈Jdefμm,=∧m∈JdJeJfμm,asJisSSVH,=∧rst∈JdJeJfμrst,takem=rstwherer∈Jd,s∈Je,andt∈Jf,≥∧r∈Jd,s∈Je,t∈Jfμr∧μs∧μt,=∧r∈Jdμr∧∧s∈Jeμs∧∧t∈Jfμt,=J¯μd∧J¯μe∧J¯μf.
This impliesJ¯μdef≥J¯μd∧J¯μe∧J¯μf.
Hence,J¯μ is a fuzzy ternary subsemigroup of A.
In general, the lower approximation of a fuzzy ternary subsemigroup is not a fuzzy ternary subsemigroup for SVH.Example 1.
LetH=d,e,f,ζ be a ternary semigroup under ternary multiplication defined as def=def in Table 1.Table 1
Ternary multiplication.
·defζdfffζefffζffffζζζζζζLetJ:A⟶P∗A be defined as Jd=e,Je=d, and Jf=Jζ=f,ζ, and then, J is SVH. Now, let μ be an FS of A given by μd=0.8=μe,μf=0.9, and μζ=0.5, and then, μ is a fuzzy ternary subsemigroup of A. Then, J¯μd=0.8=J¯μe; also, J¯μf=0.5=J¯μζ. Then, J¯μ is not a fuzzy ternary subsemigroup of A because J¯μdef≥J¯μd∧J¯μe∧J¯μf does not hold, as J¯μdee=0.5 and J¯μd∧J¯μe∧J¯μe=0.8.Theorem 19.
IfJ is SVH and μ is fuzzy ternary subsemigroup of A, J¯μ is a fuzzy ternary subsemigroup of A.Proof.
Asμ is a fuzzy ternary subsemigroup of A, then μdef≥∧μd,μe,μf for all d,e,f∈A. Consider
(3)J¯μdef=∨m∈Jdefμm,≥∨m∈JdJeJfμm,sinceJisSVH,=∨rst∈JdJeJfμrst,takem=rstwherer∈Jd,s∈Je,andt∈Jf,≥∨r∈Jd,s∈Je,t∈Jfμr∧μs∧μt,=∨r∈Jdμr∧∨s∈Jeμs∧∨t∈Jfμt,=J¯μd∧J¯μe∧J¯μf.
This impliesJ¯μdef≥J¯μd∧J¯μe∧J¯μf.
Hence, provedJ¯μ is a fuzzy ternary subsemigroup of A.Theorem 20.
IfJ is SSVH and μ is fuzzy left (resp., right, lateral) ideal of A, J¯μ is a fuzzy left (resp., right, lateral) ideal of A.Proof.
Letμ be a fuzzy left ideal of A. We have to show that J¯μdef≥J¯μf. Consider
(4)J¯μdef=∧m∈Jdefμm,=∧m∈JdJeJfμm,asJisSSVH,=∧rst∈JdJeJfμrst,bytakingm=rstsuchthatr∈Jd,s∈Je,andt∈Jf,≥∧t∈Jfμt,=J¯μf.
So,J¯μdef≥J¯μf.Proofs for right and lateral ideals are in similar way.In general, the lower approximation of a fuzzy ideal is not a fuzzy ideal ofA for SVH.Example 2.
LetA=0,m,n be a ternary semigroup under ternary multiplication defined in Table 2.Table 2
Ternary multiplication.
·0mn0000m0mnn0nnLetJ:A⟶P∗A be defined as J0=0,n,Jm=n, and Jn=m,n, and then, J is SVH. Let μ be an FS of A given as μ0=1,μm=0.4, and μn=0.7, and then, μ is fuzzy left ideal of A.So, by the definition of lower approximation,J¯μ0=0.7,J¯μm=0.7, and J¯μn=0.4. This implies J¯μ is not a fuzzy ideal of A.J¯μnnm≥J¯μm is not satisfied because J¯μnnm=J¯μn=0.4 and J¯μm=0.7.Theorem 21.
IfJ is an SVH and μ is a fuzzy left (resp., right, lateral) ideal of A,J¯μ is a fuzzy left (resp., right, lateral) ideal of A.Proof.
Letμ be a fuzzy left ideal of A; we have to prove that J¯μdef≥J¯μf. Consider
(5)J¯μdef=∨m∈Jdefμm,≥∨m∈JdJeJfμm,asJisSVH,=∨rst∈JdJeJfμrst,bytakingm=rstsuchthatr∈Jd,s∈Je,andt∈Jf,≥∨t∈Jfμt,=J¯μf.
So,J¯μdef≥J¯μf.Proofs for right and lateral ideals are in same way.Theorem 22.
IfJ is an SSVH and μ is a fuzzy semiprime ideal of A, J¯μ is a fuzzy semiprime ideal.Proof.
Sinceμ is a fuzzy semiprime ideal of A, then by Theorem 20, J¯μ is fuzzy ideal. Also, μd3=μd for all d∈A. To prove that J¯μ is fuzzy semiprime, we have to show that for all d∈A,J¯μd3=J¯μd. Consider
(6)J¯μd3=∧m∈Jd3μm,=∧m∈JdJdJdμmasJisSSVH,=∧rrr∈JdJdJdμr3,form=r3suchthatr∈Jd,=∧r∈Jdμr,=J¯μd.Theorem 23.
IfJ is an SSVH and μ is a fuzzy semiprime ideal of A, then J¯μ is fuzzy semiprime ideal.Proof.
. Same as Theorem22.Theorem 24.
LetJ be an SSVH and μ be a fuzzy prime ideal of A, J¯μ is a fuzzy prime ideal.Proof.
Asμ is fuzzy prime ideal A, so μdef=μd or μdef=μe or μdef=μf for all d,e,f∈A. Also, by Theorem 20, J¯μ is a fuzzy ideal. Consider
(7)J¯μdef=∧m∈Jdefμm,=∧m∈JdJeJfμmasJisSSVH,=∧rst∈JdJeJfμrst,form=rstsuchthatr∈Jd,s∈Je,andt∈Jf,=∧r∈Jd,s∈Je,t∈Jfμrst,=∧r∈Jdμror∧s∈Jeμsor∧t∈Jfμt,=J¯μdorJ¯μeorJ¯μf.Theorem 25.
IfJ is an SSVH and μ is a fuzzy prime ideal of A, J¯μ is a fuzzy prime ideal.Proof.
Same as Theorem24.


## 4. Approximation of∈,∈∨q-Fuzzy Ideals

This section is generalization of Section3 because ∈,∈∨q-fuzzy ideals are actually generalization of fuzzy ideals. Here, we have discussed the lower and upper approximation of ∈,∈∨q-fuzzy ternary subsemigroup and ∈,∈∨q-fuzzy ideals (prime and semiprime) of ternary semigroups.Theorem 26.
IfJ is an SSVH and μ is a ∈,∈∨q-fuzzy ternary subsemigroup of A,J¯μ is a ∈,∈∨q-fuzzy ternary subsemigroup of A.Proof.
Letdr,es,ft∈J¯μ, where d,e,f∈A and r,s,t∈0,1, and then, J¯μd≥r, J¯μe≥s, and J¯μf≥t. Consider
(8)J¯μdef=∧u∈Jdefμu,=∧u∈JdJeJfμu,=∧jkl∈JdJeJfμjkl,hereu=jklsuchthatj∈Jd,k∈Je,andl∈Jf,≥∧j∈Jd,k∈Je,l∈Jfminμj,μk,μl,0.5,usingTheorem10,=min∧j∈Jdμj,∧k∈Jdμk,∧l∈Jdμl,0.5,=minJ¯μd,J¯μe,J¯μf,0.5,≥minr,s,t,0.5.
Then,J¯μdef≥minr,s,t when minr,s,t≤0.5. So, defminr,s,t∈J¯μ⟶1.
When minr,s,t>0.5, we have, J¯μdef≥0.5.
Then,J¯μdef+minr,s,t>1, so defminr,s,tqJ¯μ⟶2. From (1) and (2), we get J¯defminr,s,t∈∨qJ¯μ.Theorem 27.
IfJ is an SVH and μ is a ∈,∈∨q-fuzzy ternary subsemigroup of A,J¯μ is a ∈,∈∨q-fuzzy ternary subsemigroup of A.Proof.
Takedr,es,ft∈J¯μ, where d,e,f∈A and r,s,t∈0,1, and then, J¯μd≥r, J¯μe≥s, and J¯μf≥t. Consider
(9)J¯μdef=∧u∈Jdefμu,≥∧u∈JdJeJfμu,=∧jkl∈JdJeJfμjkl,hereu=jklsuchthatj∈Jd,k∈Je,andl∈Jf,≥∧j∈Jd,k∈Je,l∈Jfminμj,μk,μl,0.5,=min∧j∈Jdμj,∧k∈Jdμk,∧l∈Jdμl,0.5,=minJ¯μd,J¯μe,J¯μf,0.5,≥minr,s,t,0.5.
Now,J¯μdef≥minr,s,t when minr,s,t≤0.5. So, defminr,s,t∈J¯μ⟶1.
When minr,s,t>0.5 we have, J¯def≥0.5.
Also,J¯def+minr,s,t>1, so defminr,s,tqJ¯μ⟶2. From (1) and (2), we get J¯defminr,s,t∈∨qJ¯μ.Theorem 28.
IfJ is SSVH and μ is a ∈,∈∨q-fuzzy left (resp., right, lateral) ideal of A, J¯μ is a ∈,∈∨q fuzzy left (resp., right, lateral) ideal of A.Proof.
Letμ be a ∈,∈∨q-fuzzy left ideal of A. Take fi∈J¯μ and d,e∈A, and then, J¯μf≥i. Consider
(10)J¯μdef=∨u∈Jdefμu,=∨u∈JdJeJfμu,=∨jkl∈JdJeJfμjkl,foru=jklwherej∈Jd,k∈Je,andl∈Jf,≥∨l∈Jfminμl,0.5,byTheorem11,=minJ¯μf,0.5,≥mini,0.5.
Wheni≤0.5, then we have J¯μdef≥i.This impliesdefi∈J¯μ⟶1.Wheni>0.5, we have J¯μdef+i>1. This implies defi∈∨qJ¯μ.Similarly, we can prove this for right and lateral ideals in same way.Theorem 29.
IfJ is SSVH and μ is a ∈,∈∨q-fuzzy left resp., (right, lateral) ideal of A, J¯μis a fuzzy left (resp., right, lateral) ideal of A.Proof.
Same as Theorem28.Theorem 30.
IfJ is an SSVH and μ is a ∈,∈∨q-fuzzy semiprime ideal of A, J¯μ is a ∈,∈∨q-fuzzy semiprime ideal of A.Proof.
Supposed3r∈J¯μ for d∈A and r∈0,1, and then, J¯μd3≥r. Consider
(11)J¯μd=∨u∈Jdμu,≥∨uuu∈JdJdJdminμu3,0.5,byTheorem12,=∨u3∈Jd3minμu3,0.5,=minJ¯μd3,0.5,≥minr,0.5.
Whenr≤0.5, then J¯μd≥r. So, dr∈J¯μ. When r>0.5, we have J¯μd≥0.5, so J¯μd+r>r and then drqJ¯μ. Therefore, we get dr∈∨qJ¯μ.Theorem 31.
IfJ is SSVH and μ is a ∈,∈∨q-fuzzy semiprime ideal of A, J¯μ is a ∈,∈∨q-fuzzy semiprime ideal of A.Proof.
Similar to Theorem30.Theorem 32.
IfJ is SSVH and μ is a ∈,∈∨q-fuzzy prime ideal of A, J¯u is ∈,∈∨q-fuzzy prime ideal of A.Proof.
By Theorem28, J¯μ is a ∈,∈∨q-fuzzy ideal. Further by Theorem 13, maxμd,μe,μf≥minμdef,0.5. Let defr∈J¯μ, and then, J¯μdef≥r for d,e,f∈A and r∈0,1. Consider
(12)J¯μd∨J¯μe∨J¯μf=∨j∈Jdμj∨∨k∈Jeμk∨∨l∈Jfμl,≥∨j∈Jd,k∈Je,l∈Jfμjkl∧0.5,=∨jkl∈JdJeJfμjkl∧0.5,=∨jkl∈Jdefμjkl∧0.5,=J¯μdef∧0.5,≥minr,0.5.Whenr≤0.5, we have J¯μd∨J¯μe∨J¯μf≥r, and then, J¯μd≥r or J¯μe≥r or J¯μf≥r. Therefore, dr∈J¯μ or er∈J¯μ or fr∈J¯μ.Whenr>0.5, we have J¯μd∨J¯μe∨J¯μf≤0.5, so J¯μd≥0.5 or J¯μe≥0.5 or J¯μf≥0.5. This implies that J¯μd+r>1 or J¯μe+r>1 or J¯μf+r>1. Hence, drqJ¯μ or erqJ¯μ or frqJ¯μ. Therefore, dr∈∨qJ¯μ or er∈∨qJ¯μ or fr∈∨qJ¯μ.Theorem 33.
IfJ is an SSVH and μ is a fuzzy prime ideal of A,J¯μ is a ∈,∈∨q-fuzzy prime ideal of A.Proof.
Similar to Theorem32.


## 5. Comparative Study

In this paper, we have extended the work of [22] to ternary structure, which is more generalized form than [22]. This work is better than [22] because there are many structures in which binary operation does not hold. For example, W=0,i,−i is not closed under binary operation, while it is closed under ternary product. Also, the set of negative integers, set of negative rational numbers, and set of negative real numbers are closed under ternary multiplication but not closed under usual multiplication. Therefore, it is a need to study the ternary operation which is generalization of binary operation.


## 6. Conclusions

Here, we observe that the lower approximation of fuzzy ternary subsemigroups (resp., fuzzy ideals) using SSVH is fuzzy ternary subsemigroups (resp., fuzzy ideals). The upper approximation of fuzzy ternary subsemigroups (resp., fuzzy ideals) using SVH is fuzzy ternary subsemigroups (resp., fuzzy ideals). By examples, we show that the lower approximation of a fuzzy ternary subsemigroup (resp., fuzzy ideal) is not fuzzy ternary subsemigroup (resp., fuzzy ideal) for SVH. Also, it is proved that the approximations of fuzzy semiprime (resp., prime) ideals using SSVH is fuzzy semiprime (resp., prime) ideals.The lower approximation of an∈,∈∨q-fuzzy ternary subsemigroup using SSVH and the upper approximation of an ∈,∈∨q-fuzzy ternary subsemigroup using SVH are ∈,∈∨q-fuzzy ternary subsemigroups. It is also shown that the approximations of ∈,∈∨q-fuzzy ideals using SSVH are ∈,∈∨q-fuzzy ideals.In fuzzy set, membership degree is limited to [0,1], but there was a difficulty to deliberate the irrelevancy of data. It is noticed that people may have bipolar responses at a time for the same qualities of an item or a plan. Bipolarity is also a significant theory, used in our life. In the future, we will study its bipolar andm-polar part also; we will extend this work to other algebraic structures such as ternary hemiring and LA-semigroup.



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*Source: 6247354-2022-05-26.xml*