# Approximations of Intuitionistic Fuzzy Ideals over Dual Spaces by Soft Binary Relations

Authors: Muhammad Zishan Anwar; Shahida Bashir; Muhammad Aslam; Muhammad Shabir
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The major advantage of this proposed work is to investigate roughness of intuitionistic fuzzy subsemigroups (RIFSs) by using soft relations. In this way, two sets of intuitionistic fuzzy (IF) soft subsemigroups, named lower approximation and upper approximation regarding aftersets and foresets, have been introduced. In RIFSs, incomplete and insufficient information is handled in decision-making problems like symptom diagnosis in medical science. In addition, this new technique is more effective as compared to the previous literature because we use intuitionistic fuzzy set (IFS) instead of fuzzy set (FS). Since the FS describes the membership degree only but often in real-world problems, we need the description of nonmembership degree. That is why an IFS is a more useful set due to its nonmembership degree and hesitation degree. The above technique is applied for left (right) ideals, interior ideals, and bi-ideals in the same manner as described for subsemigroups.


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## Body

## 1. Introduction

We are facing critical problems involving uncertainty and impreciseness in everyday life. Inexact and incomplete information data has not complete and precise reasoning possibly. Nowadays, the gap between the world full of vagueness and traditional mathematics with precise concepts is going to reduce with highly appreciated mode. Researchers are keen to deal with different nature of uncertainty with different methods. In this regard, they studied many newly defined theories such as RSs, FSs, IFSs, and SSs. The FSs are very important to manage the uncertainties in real-world problems. They have numerous applications in several fields such as vacuum cleaner, control of subway systems, unmanned helicopters, transmission systems, models for new pricing, and weather forecasting systems. This logic has given new life to scientific fields and has been used in many fields such as electronics, image processing, and optimization [1]. The IFS theory introduced by Attanassov [2] is one of the generalizations of FS theory. In IFS, membership degree, degree of nonmembership, and degree of hesitation of every element are expressed whose sum must be equal to 1. As we know, IFSs have degree of membership and degree of nonmembership which are more valuable in medical field. Intuitionistic fuzzy environment is more suitable to diagnose disease than FS due to its nonmembership degree. The FSs have only degree of membership but IF soft sets (IFSSs) keep a controlled degree of the vagueness, and they transform an imprecise pattern classification problem into a well-defined and precise optimization problem because an IFS gives out the uncertainty by a nonmembership degree. In real-life problems, IF logic has more effective use to control and overcome the uncertainty than fuzzy logic [3–6].The rough sets (RSs) were introduced by Pawlak [7, 8]. They attract many researchers and specialists because they handle vagueness, uncertainty, and impreciseness in a noncustomary manner and discover relationship of structures with FSs. They also help to make decisions in an eccentric way in daily life problems. The abdominal pain of children is a very serious issue in medical field which is due to different reasons, and this is a challenging issue for researchers to diagnose the reasons correctly. Researchers can use the RS theory to diagnose it and give further consultations. This theory is also used for feature extraction, decision rule generation, data reduction, and feature selection and also applied successfully to pattern recognition, intelligent systems, machine learning, mereology, expert systems, signal analysis, decision analysis, and more other fields.In RS theory, we define approximation spaces as set with multiple memberships, but in FS theory, it concerns only with partial membership. In RS theory, we use indiscernibility to remove useless data without using basic knowledge-based data. Usually, we approximate a crisp set into two formal approximations, lower and upper approximation. Upper approximation is a set which has elements having possible belonging with target set and lower approximated set has objectives having positive belonging with the target set. In simple words, we can say that a RS is a nonempty set regarding its boundary region, and in other case, this set is crisp set [7–20].The soft set (SS) theory is presented by Molodtsov [21], and this is a very suitable approach to remove those associated difficulties which the other theories were unable to tackle, like FS theory, interval mathematics, and probability theory. In this theory, parameters can be chosen by researchers with their form according to their needs. In industrialized countries, more visible reason for cancer death in men is prostate cancer, which depends on many factors such as age, family cancer history, the level of prostate-specific antigen in blood and ethnic background, etc. Many researchers are working on finding the risks of prostate cancer with the help of FSs and SSs [22]. Nowadays, the SS theory has many study work by different authors rapidly. Aktas and Cagman described soft groups [23]. Jun connected SSs with ideal theory ([24, 25]). Maji et al. [26] worked on IFSSs. Razak and Mohamad [27] worked on decision-making regarding fuzzy soft sets (FSSs) in connection with SSs [28].Al-shami et al. [29, 30] introduced extended form of FS called square-root FS and contrast square-root FS with IFS and Pythagorean FS. Hariwan et al. [31] proposed a useful concept of (3,2)-fuzzy sets with connection of other types of fuzzy sets and discussed its basic properties and operations. (3,2)-Fuzzy sets are more useful than IFSs and Pythagorean FSs due to their larger range of membership grades. Since we know RSs are useful to deal with incompleteness and IFSs are useful to deal vagueness, SSs have rich operations due to its parameters. So the combination of IFSs, RSs, and SSs is a valuable combination to deal with impreciseness [32–36].


### 1.1. Related Works

Binary relations are always very important in information sciences and mathematics both. Extended form of ordinary binary relations is soft binary relation which is a family of parameters of binary relation to a universe. Anwar et al. [37] presented a suitable model of IFRS in terms of soft relations and algorithm for real-world problems. Later on, Anwar et al. [38, 39] worked on [37] based on multisoft relations and its algebraic properties with useful algorithms and introduced the optimistic multigranulation intuitionistic fuzzy rough set (OMGIFRS) and pessimistic multigranulation intuitionistic fuzzy rough set (PMGIFRS) in terms of multisoft relations.


### 1.2. Innovative Contribution

For several practical applications in real world, equivalence relation is much restrictive. Skowron and Stepaniuk [40] replaced equivalence relations by tolerance relations. The soft covering has also been discussed by Li et al. [41]. The rough approximation by soft binary relations handles multiple binary relations. Ali [42] presented the conceptual theory about soft binary relation, and he discussed the soft lower and soft upper approximation operations regarding soft equivalence relations. In rough approximations, only binary relations are addressed, but in any other case, there are different several binary relations in connection with rough approximations in terms of soft binary relations. Shabir et al. presented prime and semiprime L-fuzzy hyperideals in terms of soft sets. They also discussed applications of semihypergroups in terms of L-fuzzy soft sets and fuzzy ideals in connection with rough fuzzy ternary subsemigroups and 3-dimensional congruence relation in their previous publications [41–49]. In 2019, Kanwal and Shabir discussed the fuzzy set of semigroup in terms of rough approximation with soft relations [50]. In this research, we have studied the rough approximation of IF set in semigroups based on soft relations.


### 1.3. Motivation

The main motivation of this research study is to extend the concept of fuzzy ideals into IF ideals based on soft relations in terms of semigroups. Its related properties are also discussed. The semigroup attracts many algebraists due to their applications to formal languages, automata theory and network analogy, etc. The connection of semigroup theory and theory of machines increases the importance of both theories during the past few decades. In association with the study of machines and automata, other areas of applications have been improved such as formal languages, and the software uses the language of modern algebra in terms of Boolean algebra, semigroups, and others. The semigroup theory contributes in biology, psychology, biochemistry, and sociology [51]. The FS describes only degree of membership of each element which is insufficient to tackle uncertainty in several real-world problems. Here, we need IFS to describe degree of membership as well as degree of nonmembership of each element to control vagueness and impreciseness in real-world problems. In [52], Bashir et al. discussed a useful model of regular ternary semirings based on bipolar fuzzy sets and its algebraic properties. In Kanwal and Shabir’s paper [50], fuzzy ideals have been discussed, but in this paper, we discuss our proposed model in intuitionistic environment which is a more suitable environment to deal with incompleteness and uncertainty.


### 1.4. Organization of the Paper

The setting of the paper is as follows. In Section2, a few basic concepts with SSs, soft binary relations, IFSs, and IF ideals are presented. In Section 3, we discussed an approximation of an IFS in terms of soft binary relations. We made approximations of IF set by the foresets and aftersets, and we get two IFSSs, called the upper approximation and lower approximation regarding foresets and aftersets. After applying these concepts, approximations of IF subsemigroups, IF left (right) ideals, IF interior ideals, and IF bi-ideals of semigroups are discussed with examples. In Section 4, we present the comparison with the previous work. In Section 5, conclusion is described to present this research work and future work.See Table1 for the acronyms.Table 1
List of acronyms.
AcronymsRepresentationsFSsFuzzy setsIFSsIntuitionistic fuzzy setsRSsRough setsSSsSoft setsFSSsFuzzy soft setsIFSSsIntuitionistic fuzzy soft setsIFRSsRough intuitionistic fuzzy setsMGRSMultigranulation rough setOMGRSOptimistic multigranulation rough setPMGRSPessimistic multigranulation rough setRIFsRough intuitionistic fuzzy subsemigroups


## 1.1. Related Works

Binary relations are always very important in information sciences and mathematics both. Extended form of ordinary binary relations is soft binary relation which is a family of parameters of binary relation to a universe. Anwar et al. [37] presented a suitable model of IFRS in terms of soft relations and algorithm for real-world problems. Later on, Anwar et al. [38, 39] worked on [37] based on multisoft relations and its algebraic properties with useful algorithms and introduced the optimistic multigranulation intuitionistic fuzzy rough set (OMGIFRS) and pessimistic multigranulation intuitionistic fuzzy rough set (PMGIFRS) in terms of multisoft relations.


## 1.2. Innovative Contribution

For several practical applications in real world, equivalence relation is much restrictive. Skowron and Stepaniuk [40] replaced equivalence relations by tolerance relations. The soft covering has also been discussed by Li et al. [41]. The rough approximation by soft binary relations handles multiple binary relations. Ali [42] presented the conceptual theory about soft binary relation, and he discussed the soft lower and soft upper approximation operations regarding soft equivalence relations. In rough approximations, only binary relations are addressed, but in any other case, there are different several binary relations in connection with rough approximations in terms of soft binary relations. Shabir et al. presented prime and semiprime L-fuzzy hyperideals in terms of soft sets. They also discussed applications of semihypergroups in terms of L-fuzzy soft sets and fuzzy ideals in connection with rough fuzzy ternary subsemigroups and 3-dimensional congruence relation in their previous publications [41–49]. In 2019, Kanwal and Shabir discussed the fuzzy set of semigroup in terms of rough approximation with soft relations [50]. In this research, we have studied the rough approximation of IF set in semigroups based on soft relations.


## 1.3. Motivation

The main motivation of this research study is to extend the concept of fuzzy ideals into IF ideals based on soft relations in terms of semigroups. Its related properties are also discussed. The semigroup attracts many algebraists due to their applications to formal languages, automata theory and network analogy, etc. The connection of semigroup theory and theory of machines increases the importance of both theories during the past few decades. In association with the study of machines and automata, other areas of applications have been improved such as formal languages, and the software uses the language of modern algebra in terms of Boolean algebra, semigroups, and others. The semigroup theory contributes in biology, psychology, biochemistry, and sociology [51]. The FS describes only degree of membership of each element which is insufficient to tackle uncertainty in several real-world problems. Here, we need IFS to describe degree of membership as well as degree of nonmembership of each element to control vagueness and impreciseness in real-world problems. In [52], Bashir et al. discussed a useful model of regular ternary semirings based on bipolar fuzzy sets and its algebraic properties. In Kanwal and Shabir’s paper [50], fuzzy ideals have been discussed, but in this paper, we discuss our proposed model in intuitionistic environment which is a more suitable environment to deal with incompleteness and uncertainty.


## 1.4. Organization of the Paper

The setting of the paper is as follows. In Section2, a few basic concepts with SSs, soft binary relations, IFSs, and IF ideals are presented. In Section 3, we discussed an approximation of an IFS in terms of soft binary relations. We made approximations of IF set by the foresets and aftersets, and we get two IFSSs, called the upper approximation and lower approximation regarding foresets and aftersets. After applying these concepts, approximations of IF subsemigroups, IF left (right) ideals, IF interior ideals, and IF bi-ideals of semigroups are discussed with examples. In Section 4, we present the comparison with the previous work. In Section 5, conclusion is described to present this research work and future work.See Table1 for the acronyms.Table 1
List of acronyms.
AcronymsRepresentationsFSsFuzzy setsIFSsIntuitionistic fuzzy setsRSsRough setsSSsSoft setsFSSsFuzzy soft setsIFSSsIntuitionistic fuzzy soft setsIFRSsRough intuitionistic fuzzy setsMGRSMultigranulation rough setOMGRSOptimistic multigranulation rough setPMGRSPessimistic multigranulation rough setRIFsRough intuitionistic fuzzy subsemigroups


## 2. Preliminaries and Basic Concepts

In this section, basic notions about IFSs, SSs, IFSSs, and some background materials are given. Throughout this paper,U,· is semigroup and U1 and U2 represent two nonempty finite sets unless stated otherwise. Here, we recall some ideas and results which are useful for this paper.We will denote the product of two elementsx,y∈U by xy instead of x.y. In what follows by subsets, we always means nonempty ones. Let us consider two subsets A and B of U, and then, the product AB is defined as AB=ab:a∈A,b∈B.Definition 1 (see [53]).
A binary relationT from U1 to U2 is a subset of U1×U2, and a subset of U1×U1 is said to be a binary relation on U1. If T is a binary relation on U1, then T is said to be reflexive if u,u∈T for all u∈U1, symmetric if u,v∈T implies v,u∈T for all u,v∈U1, and transitive if u,v∈T and v,w∈T implies u,w∈T for all u,v,w∈U1. If a binary relation T is reflexive, symmetric, and transitive, then it is called an equivalence relation. A set is partitioned into disjoint classes by an equivalence relation.Definition 2 (see [54]).
For any subsetA of U if ab∈A for all a,b∈A, then A is called a subsemigroup of U. A left (right) ideal of U is a subset A of U such that UA⊆AAU⊆A. A two-sided ideal is a subset A of U which is a left as well as right ideal of U. A subsemigroup A of U is said to be an interior ideal of U if UAU⊆A. A subsemigroup A of U is said to be a bi-ideal of U if AUA⊆A.Definition 3 (see [2]).
An intuitionistic fuzzy set (IFS)B in U1 is an object having the shape B=x,μBx,γBx:x∈U1, where μB:U1⟶0,1 and γB:U1⟶0,1 satisfying 0≤μBx+γBx≤1 for all x∈U1. The values μBx and γBx are called membership degree and nonmembership degree of x∈U1 to B, respectively. The number πBx=1−μBx−γBx is called the hesitancy degree of x∈U1 to B. The collection of all IFS in U1 is denoted by IFU1. In the remaining paper, we shall write an IFS by B=μB,γB instead of B=x,μBx,γBx:x∈U1. Let B=μB,γB and B1=μB1,γB1 be two IFSs in U1. Then, B⊆B1 if and only if μBx≤μB1x and γB1x≤γBx for all x∈U1. Two IFSs B and B1 are said to be equal if and only if B⊆B1 and B1⊆B. The union and intersection of the two IFSs B and B1 in U1 are denoted and defined by B∪B1=μB∪μB1,γB∩γB1 and B∩B1=μB∩μB1,γB∪γB1, where μB∪μB1x=supμBx,μB1x, γB∩γB1x=infγBx,γB1x,μB∩μB1x=infμBx,μB1x, and γB∪γB1x=supγBx,γB1x.Next, we define two special types of IFSs. The IF universe setU=1U=<1,0> and IF empty set ∅=0U=<0,1>, where 1x=1 and 0x=0 for all x∈U. The complement of an IFS A=<μ,γ> is denoted and defined as Ac=<γ,μ>.Definition 4 (see [21]).
A pairσ,A is said to be a soft set (SS) over U if σ:A⟶PU, where A⊆E, E is the set of parameters and the set U has power set PU. Thus, σe is a subset of U for all e∈A. Hence, a SS over U is a parametrized collection of subsets of U.Definition 5 (see [26, 55]).
A pairσ,A is called an intuitionistic fuzzy soft set (IFSS) over U if σ:A⟶IFU and A⊆E where E is the set of parameters. Thus, σe is an IFS in U for all e∈A. Hence, an IFSS over U is a parametrized collection of IFSs in U.Ifσ,A and ρ,B are two IFSSs over U, we say that σ,A is an IF soft subset of ρ,B if 1A⊆B and 2σe is an IF subset of ρe for all e∈A. If σ,A and ρ,B over U are IFSSs, then they called IF soft equal if σ,A is an IF soft subset of ρ,B and ρ,B is an IF soft subset of σ,A. The union of two IFSSs σ,A and ρ,A over the common universe U is the IFSS λ,A, where λe=σe∪ρe for all e∈A. Over the common universe U, the intersection of two IFSSs σ,A and ρ,A is the IFSS π,A, where πe=σe∩ρe for all e∈A.Definition 6 (see [21]).
An IFSB=μB,γB in U is called an IF subsemigroup of U if it satisfies the following:
(1)
μBxy≥μBx∧μBy(2)
γBxy≤γBx∨γBy for all x,y∈UAn IFSB=μB,γB in a semigroup U is called an IF left (resp., right) ideal of U if it satisfies μBxy≥μByresp.μBxy≥μBx and γBxy≤γByresp.γBxy≤γBx for all x,y∈U. An IF left ideal and IF right ideal are called fuzzy ideals. An IF subsemigroup B=μB,γB in U is called an IF interior ideal of U if it satisfies μBxay≥μBa and γBxay≤γBa for all x,y,a∈U.An IF subsemigroupB=μB,γB in U is called an IF bi-ideal of U if it satisfies the following:
(1)
μBxwy≥μBx∧μBy(2)
γBxwy≤γBx∨γBy for all x,y,w∈U [2, 26, 56–58]Definition 7 (see [59]).
An IFSSσ,A over U is called IF soft subsemigroup (left ideal, right ideal, interior ideal, and bi-ideal) over U if each σe is IF subsemigroup (left ideal, right ideal, interior ideal, and bi-ideal) of U for all e∈A.Definition 8 (see [60]).
A soft binary relationσ,A from U1 to U2 is a SS over U1×U2, that is, σ:A⟶PU1×U2, where A⊆E.
Of course,σ,A is a parameterized collection of binary relations from U1 to U2; that is, for each e∈A, we have a binary relation σe from U1 to U2.Definition 9 (see [61]).
A soft binary relationσ,A from a semigroup U1 to a semigroup U2 is said to be soft compatible if p,q,r,s∈σe implies pr,qs∈σe for all p,r∈U1 and q,s∈U2.
In general, ifσ,A is a soft compatible relation from U1 to U2, then pσe.qσe⊆pqσe; indeed if a∈pσe and b∈qσe, then p,a∈σe and q,b∈σe. By compatibility of σ,A,pq,ab∈σe; that is, ab∈pqσe. Similarly, σep.σeq⊆σepq. The following example shows that in general, pσe.qσe≠pqσe and σep.σeq≠σepq.Example 1.
LetU1=p,q,r,s and U2=1,2,3,4 be two semigroups, and their multiplication tables are as shown in Tables 2 and 3, respectively).Table 2
Multiplication table forU1.
.pqrsppppsqpqpsrpprssssssTable 3
Multiplication table forU2.
.123411234222223333344321LetA=e1,e2. Define σ:A⟶PU1×U2 by
(1)σe1=p,1,q,2,r,3,s,4,p,2,p,3,q,4,p,4,q,1,s,1,s,2,s,3,σe2=p,2,p,3,q,2,q,3,r,2,r,3,s,2,s,3.Then,σ,A is a soft compatible relation from U1 to U2.
(2)pσe1=1,2,3,4,qσe1=1,2,4,rσe1=3,sσe1=1,2,3,4pσe2=2,3,qσe2=2,3,rσe2=2,3,sσe2=2,3.But(3)pσe1.rσe1=2,3≠1,2,3,4=prσe1.Now,(4)σe11=p,q,s,σe12=p,q,s,σe13=p,r,s,σe14=p,q,sσe21=∅,σe22=p,q,r,s,σe23=p,q,r,s,sσe2=∅.But(5)σe13.σe11=∅≠p,q,r,s=σe131.Definition 10 (see [61]).
A soft compatible relationσ,A from a semigroup U1 to a semigroup U2 is said to be soft complete regarding aftersets if pσe.qσe=pqσe for all p,q∈U1 and e∈A and is said to be soft complete relation regarding foresets if σer.σes=σers for all r,s∈U2 and e∈A.Definition 11 (see [37]).
Letσ,A be a soft binary relation from U1 to U2 and B=μB,γB be an IFS in U2. Then, the lower approximation σ¯B=σ¯μB,σ¯γB and the upper approximation σ¯B=σ¯μB,σ¯γB of B=μB,γB are IFSSs over U1 and defined as
(6)σ¯μBeu1=∧a∈u1σeμBa,ifu1σe≠∅,1,ifu1σe=∅,σ¯γBeu1=∨a∈u1σeγBa,ifu1σe≠∅,0,ifu1σe=∅,σ¯μBeu1=∨a∈u1σeμBa,ifu1σe≠∅,0,ifu1σe=∅,σ¯γBeu1=∧a∈u1σeγBa,ifu1σe≠∅,1,ifu1σe=∅,for all u1∈U1, where u1σe=a∈U2:u1,a∈σe and is called the afterset of u1 for u1∈U1 and e∈A.Definition 12 (see [37]).
Letσ,A be a soft binary relation from U1 to U2 and B=μB,γB be an IFS in U1. Then, the lower approximation Bσ¯=μBσ¯,γBσ¯ and the upper approximation Bσ¯=μBσ¯,γBσ¯ of B=μB,γB are IFSSs over U2 and defined as
(7)μBσ¯eu2=∧a∈σeu2μBa,ifσeu2≠∅,1,ifσeu2=∅,γBσ¯eu2=∨a∈σeu2γBa,ifσeu2≠∅,0,ifσeu2=∅,μBσ¯eu2=∨a∈σeu2μBa,ifσeu2≠∅,0,ifσeu2=∅,γBσ¯eu2=∧a∈σeu2γBa,ifσeu2≠∅,1,ifσeu2=∅,for all u2∈U2 where σeu2=a∈U1:a,u2∈σe and is called the foreset of u2 for u2∈U2 and e∈A.
Of course,σ¯B:A⟶IFU1, σ¯B:A⟶IFU1 and Bσ¯:A⟶IFU2, Bσ¯:A⟶IFU2.Theorem 13 (see [37]).
Letσ,A be a soft binary relation from U1 to U2 that is σ:A⟶PU1×U2. For any IFSs D=μD,γD,D1=μD1,γD1 and D2=μD2,γD2 of U2, the following hold.
(1)
D1≤D2 implies σ¯D1≤σ¯D2(2)
D1≤D2 implies σ¯D1≤σ¯D2(3)
σ¯D1∩σ¯D2=σ¯D1∩D2(4)
σ¯D1∩σ¯D2⊇σ¯D1∩D2(5)
σ¯D1∪σ¯D2⊆σ¯D1∪D2(6)
σ¯D1∪σ¯D2=σ¯D1∪D2(7)
σ¯1U2=1U1 if uσe≠∅(8)
σ¯1U2=1U1 if uσe≠∅(9)
σ¯D=σ¯DCC if uσe≠∅(10)
σ¯D=σ¯DCC if uσe≠∅(11)
σ¯0U2=0U1=σ¯0Wif uσe≠∅Theorem 14 (see [37]).
Letσ,A be a soft binary relation from U1 to U2; that is, σ:A⟶PU1×U2. For any IFSs D=μD,γD,D1=μD1,γD1 and D2=μD2,γD2 of U1, the following are true.
(1)
IfD1≤D2D1 implies σ¯≤D2σ¯(2)
IfD1≤D2 implies D1σ¯≤D2σ¯(3)
D1σ¯∩D2σ¯=D1∩D2σ¯(4)
D1σ¯∩D2σ¯⊇D1∩D2σ¯(5)
D1σ¯∪D2σ¯⊆D1∪D2σ¯(6)
D1σ¯∪D2σ¯=D1∪D2σ¯(7)
σ¯1U1=1U2 if uσe≠∅(8)
σ¯1U1=1U2 if uσe≠∅(9)
Dσ¯=DCσ¯C if uσe≠∅(10)
Dσ¯=DCσ¯C if uσe≠∅(11)
0U1σ¯=0U2=0U1σ¯


## 3. Approximations of IF Ideals in Semigroups by Soft Binary Relation

This is our major section of the paper. Our related work is narrated in this section. Here, we discuss the rough approximations of IF subsemigroup (IF left (right) ideal, IF interior ideal, and IF bi-ideal) in a semigroup regarding aftersets as well as regarding foresets by using soft compatible relation. We show that upper approximation of an IF subsemigroup (IF left (right), IF interior ideal, and IF bi-ideal) in a semigroup is an IF soft subsemigroup (IF soft left (right) ideal, IF soft interior ideal, and IF soft bi-ideal) and discuss examples which shows that its converse is not true. Similar results for lower approximation are also proved.Theorem 15.
Letσ,A be a soft compatible relation from a semigroup U1 to a semigroup U2.
(1)
IfB2 is an IF subsemigroup of U2, then σ¯B2,A is an IF soft subsemigroup of U1(2)
IfB2 is an IF left (right, two-sided) ideal of U2, then σ¯B2,A is an IF soft left (right, two-sided) ideal of U1Proof.
(1)
We assume thatB2 is an IF subsemigroup of U2. Now for u,v∈U1,(8)σ¯μB2eu∧σ¯μB2ev=∨x∈uσeμB2x∧∨y∈vσeμB2y=∨x∈uσe∨y∈vσeμB2x∧μB2y≤∨x∈uσe∨y∈vσeμB2xy≤∨xy∈uvσeμB2xy=∨a′∈uvσeμB2a′=σ¯μB2euv.
Similarly fors,t∈U1,(9)σ¯γB2es∨σ¯γB2et=∧a∈sσeγB2a∨∧b∈tσeγB2b=∧a∈sσe∧b∈tσeγB2a∨γB2b≥∧a∈sσe∧b∈tσeγB2ab≥∧ab∈stσeγB2ab=∧b′∈stσeγB2b′=σ¯γB2est.
Hence,σ¯B2e is an IF subsemigroup of U1 for all e∈A, so σ¯B2,A is an IF soft subsemigroup of U1.
(2)
Assume thatB2 is an IF left ideal of U2. Now for x,y∈U1,(10)σ¯μB2ey=∨q∈yσeμB2q≤∨p∈xσe∨q∈yσeμB2pq≤∨pq∈xyσeμB2pq=∨a′∈xyσeμB2a′=σ¯μB2exy.
Similarly forx,y∈U1,(11)σ¯γB2ey=∧q∈yσeγB2q≥∧p∈xσe∧q∈yσeγB2pq≥∧pq∈xyσeγB2pq=∧b′∈xyσeγB2b′=σ¯γB2exy.
Hence,σ¯B2e is an IF left ideal of U1 for each e∈A, so σ¯B2,A is an IF soft left ideal of U1.In Theorem15 from part 1, soft compatible relations from U1 to U2 are given, and B2 is an IF subsemigroup in U2. After combining theme, we get generalized IF soft subsemigroups in U1. Similarly, we take an IF left (right, two-sided) ideal B2 of U2, and we get generalized IF soft left (right, two-sided) ideal of U1.Theorem 16.
Letσ,A be a soft compatible relation from a semigroup U1 to a semigroup U2:
(1)
IfB1 is an IF subsemigroup of U1, then B1σ¯,A is an IF soft subsemigroup of U2(2)
IfB1 is an IF left (right, two-sided) ideal of U1, then B1σ¯,A is an IF soft left (right, two-sided) ideal of U2Proof.
It follows from Theorem15.In Theorem16 from part 1, soft compatible relations from U1 to U2 are given, and B1 is an IF subsemigroup in U1. After combining them, we get generalized IF soft subsemigroups in U2. Similarly, we take an IF left (right, two-sided) ideal B1 of U1, and we get generalized IF soft left (right, two-sided) ideal of U2.Now, we show that the converses of parts of the above theorem do not hold in general.Example 2.
LetU1=p,q,r,s,t and U2=1,2,3,4,5 be two semigroups, and their multiplication tables are as follows in Tables 4 and 5, respectively.Table 4
Multiplication table forU1.
.pqrstpqqsssqqqsssrssrsrsssssstssrsrTable 5
Multiplication table forU2.
.12345115345212345315345415345515345LetA=e1,e2. Define σ:A⟶PU1×U2 by
(12)σe1=p,1,q,2,r,3,s,4,t,5,q,1,r,5,q,5,s,3,s,5,s,1,σe2=p,1,q,2,r,3,s,4,t,5,q,1,r,5,q,5,s,3,s,5,s,1,q,3.Then,σ,A is a soft compatible relation from U1 to U2.
(13)pσe1=1,qσe1=1,2,5,rσe1=3,5,sσe1=1,3,4,5,tσe1=5pσe2=1,qσe2=1,2,3,5,rσe2=3,5,sσe2=1,3,4,5,tσe2=5,σe11=p,q,s,σe12=q,σe13=r,s,σe14=s,σe15=q,r,s,tσe21=p,q,s,σe22=q,σe23=q,r,s,σe24=s,σe25=q,r,s,t.(1)
DefineB1:U2⟶0,1 (given in Table 6)Table 6
Intuitionistic fuzzy setB1.
B112345μB10.50.40.310.1γB10.40.50.700.8Then,B1 is not an IF subsemigroup of U2 because if we take x=1,y=2, then μB112≱μB11∧μB12 and γB112≰γB11∨γB122. Upper approximation of B1 is given in Table 7.Table 7
Upper approximation ofB1.
pqrstσ¯μB1e10.50.50.310.1σ¯γB1e10.40.40.700.8σ¯μB1e20.50.50.310.1σ¯γB1e20.40.40.700.8Clearly,σ¯B1e1 and σ¯B1e2 are IF subsemigroups of U1, so σ¯B1,A is an IF soft subsemigroup of U1.
(2)
DefineB2:U1⟶0,1 (given in Table 8)Table 8
Intuitionistic fuzzy setB2.
B2pqrstμB20.20.70.800.9γB20.70.30.210.1Then,B2 is not an IF subsemigroup of U1 because if we take x=t,y=t, then μB2tt≱μB2t∧μB2t and γB2tt≰γB2t∨γB2t. Upper approximation of B2 is given in Table 9. Clearly, B2σ¯e1 and B2σ¯e2 are IF subsemigroups of U2, so B2σ¯,A is an IF soft subsemigroup of U2.
(3)
DefineB:U2⟶0,1 (given in Table 10)Table 9
Upper approximation ofB2.
12345σ¯μB2e10.70.70.800.9σ¯γB2e10.30.30.210.1σ¯μB2e20.70.70.800.9σ¯γB2e20.30.30.210.1Table 10
Intuitionistic fuzzy setB.
B12345μB0.50.40.310.1γB0.40.60.600.8Then,B is not an IF left ideal of U2 because if we take x=1,y=2, then μB12≱μB2 and γB12≰γB2. Upper approximation of B is given in Table 11. Clearly, σ¯Be1 and σ¯Be2 are IF left ideals of U1, so σ¯B,A is an IF soft left ideal of U1.
(4)
DefineH:U1⟶0,1 (given in Table 12)Table 11
Upper approximation ofB.
pqrstσ¯μBe10.50.50.310.1σ¯γBe10.40.40.600.8σ¯μBe20.50.50.310.1σ¯γBe20.40.40.600.8Table 12
Intuitionistic fuzzy setH.
HpqrstμH0.20.70.800.9γH0.70.20.110.1Then,H is not an IF left ideal of U1 because if we take x=t,y=t, then μHtt≱μHt and γHtt≰γHt. Upper approximation of H is given in Table 13.Table 13
Upper approximation ofH.
12345σ¯μHe10.70.70.800.9σ¯γHe10.20.20.110.1σ¯μHe20.70.70.800.9σ¯μHe20.20.20.110.1Clearly,Hσ¯e1 and Hσ¯e2 are IF left ideals of U2, so Hσ¯,A is an IF soft left ideal of U2.Example 3.
Consider the semigroups and soft binary relation of Example2. Define B:U2⟶0,1 (given in Table 14).Table 14
Intuitionistic fuzzy setB.
B12345μB0.70.70.800.9γB0.20.30.110.1Then,B is an IF left ideal of U2. Lower approximation of B is given in Table 15.Table 15
Lower approximation ofB.
pqrstσ¯μBe10.70.70.800.9σ¯γBe10.20.30.910.1Butσ¯Be1 is not an IF left ideal of U1 because if we take x=r,y=t, then μBrt≱μBt and γBrt≰γBt.This example shows that if the soft relation is compatible, then lower approximation of an IF left ideal is not IF soft left ideal. However, the following result is true.Theorem 17.
Letσ,A be a soft complete relation regarding aftersets from a semigroup U1 to a semigroup U2.
(1)
IfB is an IF subsemigroup of U2, then σ¯B,A is an IF soft subsemigroup of U1(2)
IfB is an IF left (right, two-sided) ideal of U2, then σ¯B,A is an IF soft left (right, two-sided) ideal of U1Proof.
(1)
Assume thatB is an IF subsemigroup of U2. Now for x,y∈U1,(14)σ¯μBexy=∧a′∈xyσeμBa′=∧a′∈xσe.yσeμBa′=∧p∈xσe.q∈yσeμBpq≥∧p∈xσe∧q∈yσeμBp∧μBq≥∧p∈xσeμBp∧∧q∈yσeμBq=σ¯μBex∧σ¯μBey.
Similarly, forx,y∈U1,(15)σ¯γBexy=∨a′∈xyσeγBa′=∨a′∈xσe.yσeγBa′=∨p∈xσe.q∈yσeγBpq≤∨p∈xσe∨q∈yσeγBp∨γBq≤∨p∈xσeγBp∨∨q∈yσeγBq=σ¯γBex∨σ¯γBey.
Hence,σ¯Be is an IF subsemigroup of U1 for each e∈A, so σ¯B,A is an IF soft subsemigroup of U1.
(2)
SupposeB is an IF left ideal of U2. Now for x,y∈U1,(16)σ¯μBexy=∧a′∈xyσeμBa′=∧a′∈xσe.yσeμBa′=∧p∈xσe.q∈yσeμBpq≥∧q∈yσeμBq=σ¯μBey.
Similarly, forx,y∈U1,(17)σ¯γBexy=∨a′∈xyσeγBa′=∨a′∈xσe.yσeγBa′=∨p∈xσe.q∈yσeγBpq≤q∈yσeγBq=σ¯γBey.
Hence,σ¯Be is an IF left ideal of U1 for all e∈A, so σ¯B,A is an IF soft left ideal of U1.Theorem 18.
Supposeσ,A is a soft complete relation regarding foresets from a semigroup U1 to a semigroup U2. Then, the following are true:
(1)
IfB is an IF subsemigroup of U1, then Bσ¯,A is an IF soft subsemigroup of U2(2)
IfB is an IF left (right, two-sided) ideal of U1, then Bσ¯,A is an IF soft left (right, two-sided) ideal of U2Proof.
It follows from Theorem17.Example 4.
Consider the semigroups of Example1.
LetA=e1,e2. Define σ:A⟶PU1×U2 by
(18)σe1=p,2,p,3,q,2,q,3,r,2,r,3,s,2,s,3σe2=p,2,q,2,r,2,s,2.Then,σ,A is a soft complete relation from U1 to U2 with respect to the aftersets.
(19)pσe1=2,3,qσe1=2,3,rσe1=2,3,sσe1=2,3pσe2=2,qσe2=2,rσe2=2,sσe2=2.(1)
DefineB:U2⟶0,1 (given in Table 16)Table 16
Intuitionistic fuzzy setB.
B1234μB0.20.40.60.8γB0.70.50.20.1Then,B is not an IF subsemigroup of U2 because if we take x=4,y=4, then μB44≱μB4∧μB4 and γB44≰γB4∨γB4. Lower approximation of B is given in Table 17.Table 17
Lower approximation ofB.
pqrsσ¯μBe10.40.40.40.4σ¯γBe10.50.50.50.5σ¯μBe20.40.40.40.4σ¯γBe20.50.50.50.5Clearly,σ¯Be1 and σ¯Be2 are IF subsemigroups of U1, so σ¯B,A is an IF soft subsemigroup of U1.
(2)
DefineB1:U2⟶0,1 (given in Table 18)Table 18
Intuitionistic fuzzy setB1.
B11234μB10.20.40.60.8γB10.70.50.20.1Then,B1 is not an IF left ideal of U2 because if we take x=3,y=4, then μB134≱μB14 and γB134≰γB14. Lower approximations of B1 are given in Table 19.Table 19
Lower approximation ofB1.
pqrsσ¯μB1e10.40.40.40.4σ¯γB1e10.50.50.50.5σ¯μB1e20.40.40.40.4σ¯γB1e20.50.50.50.5Clearly,σ¯B1e1 and σ¯B1e2 are IF left ideals of U1, so σ¯B1,A is an IF soft left ideal of U1.Now defineσ:A⟶PU1×U2 by
(20)σe1=s,1,s,2,s,3,s,4,σe2=p,1,p,2,p,3,p,4,s,1,s,2,s,3,s,4.These are soft complete relations fromU1 to U2 with respect to the foresets.
(21)σe11=s,σe12=s,σe13=s,σe14=sσe21=p,s,σe22=p,s,σe23=p,s,σe24=p,s.(1)
DefineB:U1⟶0,1 (given in Table 20)Table 20
Intuitionistic fuzzy setB.
pqrsμB0.10.30.50.7γB0.90.60.40.2Then,B is not an IF subsemigroup of U1 because if we take x=r,y=q, then μBrq≱μBr∧μBq and γBrq≰γBr∨γBq. Lower approximation of B is given in Table 21.Table 21
Lower approximation ofB.
1234σ¯μBe10.70.70.70.7σ¯γBe10.20.20.20.2σ¯μBe20.10.10.10.1σ¯γBe20.90.90.90.9Clearly,σ¯Be1 and σ¯Be2 are IF subsemigroups of U2, so σ¯B,A is an IF soft subsemigroup of U2.
(2)
DefineB1:U1⟶0,1 (given in Table 22)Table 22
Intuitionistic fuzzy setB1.
pqrsμB10.10.30.50.7γB10.80.60.50.2Then,B1 is not an IF left ideal of U1 because if we take x=r,y=q, then μB1rq≱μB1q and γB1rq≰γB1q. Lower approximation of B1 is given in Table 23.Table 23
Lower approximation ofB1.
1234σ¯μB1e10.70.70.70.7σ¯γB1e10.20.20.20.2σ¯μB1e20.10.10.10.1σ¯γB1e20.80.80.80.8Clearly,σ¯B1e1 and σ¯B1e2 are IF left ideals of U2, so σ¯B1,A is an IF soft left ideal of U2.The next theorem shows that upper approximation of product of right and left IF ideals is contained in the intersection of their upper approximations.Theorem 19.
Supposeσ,A is a soft binary relation from a semigroup U1 to a semigroup U2; that is, σ:A⟶PU1×U2. Then, for an IF right ideal B1=μB1,γB1 and for an IF left ideal B2=μB2,γB2 of U2, σ¯B1B2⊆σ¯B1∩σ¯B2.Proof.
Assume thatB1 is an IF right ideal and B2 is IF left ideal of U2, so by definition, B1B2⊆B1∩B2.
It follows from Theorem13,
(22)σ¯μB1μB2e⊆σ¯μB1∩μB2e⊆σ¯μB1e∩σ¯μB2e.
Hence,(23)σ¯μB1μB2⊆σ¯μB1e∩σ¯μB2.
Also,(24)σ¯γB1γB2e⊇σ¯γB1∩γB2e⊇σ¯γB1e∩σ¯γB2e.
Hence,(25)σ¯γB1γB2⊇σ¯γB1e∩σ¯γB2.Theorem 20.
Supposeσ,A is a soft bianry relation from a semigroup U1 to a semigroup U2; that is, σ:A⟶PU1×U2. Then, for an IF right ideal B1=μB1,γB1 and for an IF left ideal B2=μB2,γB2 of U1, B1B2σ¯⊆B1σ¯∩B2σ¯.Proof.
It follows from Theorem19.Theorem 21.
Letσ,A be a soft binary relation from a semigroup U1 to a semigroup U2; that is, σ:A⟶PU1×U2. Then, for an IF right ideal B1=μB1,γB1 and an IF left ideal B2=μB2,γB2 of U2, σ¯B1B2⊆σ¯B1∩σ¯B2.Proof.
Assume thatB1 is an IF right ideal and B2 is IF left ideal of U2, so by definition B1B2⊆B1∩B2.
It follows from Theorem13,
(26)σ¯μB1μB2e⊆σ¯μB1∩μB2e⊆σ¯μB1e∩σ¯μB2e.
Hence,(27)σ¯μB1μB2⊆σ¯μB1e∩σ¯μB2.
Also,(28)σ¯γB1γB2e⊇σ¯γB1∩γB2e⊇σ¯γB1e∩σ¯γB2e.
Hence,(29)σ¯γB1γB2⊇σ¯γB1e∩σ¯γB2.Theorem 22.
Supposeσ,A is a soft binary relation from a semigroup U1 to a semigroup U2; that is, σ:A⟶PU1×U2. Then for IF right ideal B1=μB1,γB1 and IF left ideal B2=μB2,γB2 of U1, B1B2σ¯⊆B1σ¯∩B2σ¯.Proof.
It follows from Theorem21.Now for IF interior ideals of a semigroup, we discuss a few properties.Theorem 23.
Letσ,A be a soft compatible relation from a semigroup U1 to a semigroup U2. If B is an IF interior ideal of U2, then σ¯B,A is an IF soft interior ideal of U1.Proof.
Suppose thatB is an IF interior ideal of U2. Thus, B is an IF subsemigroup of U2, so by Theorem 15, σ¯B,A is an IF soft subsemigroup of U1. Now for x,a,y∈U1,(30)σ¯μBea=∨q∈aσeμBq≤∨p∈xσe∨q∈aσe∨r∈yσeμBpqr≤∨pqr∈xayσeμBpqr=a′∈xayσeμBa′=σ¯μBexay.
Similarly, fors,b,t∈U1,(31)σ¯γBeb=∧u∈sσeγBu≤∧u∈sσe∧v∈bσe∧w∈tσeγBuvw≤∧uvw∈sbtσeγBuvw=∧b′∈sbtσeγBb′=σ¯γBesbt.Hence,σ¯Be is an IF interior ideal of U1 for all e∈A, so σ¯B,A is an IF soft interior ideal of U1.For converse of above result, we show the following example.Example 5.
LetU1=1,2,3 and U2=p,q,r be two semigroups, and their multiplication tables are as shown in Tables 24 and 25, respectively.Table 24
Multiplication table forU1.
.123112321233123Table 25
Multiplication table forU2.
.pqrppprqpqrrprrAndA=e1,e2. Define σ:A⟶PU1×U2 by
(32)σe1=1,p,2,q,3,r,1,q,2,p,1,r,3,pσe2=1,p,1,q,1,r,2,p,2,q,3,r.Then,σ,A is a soft compatible relation from U1 to U2.
(33)1σe1=p,q,r,2σe1=p,q,3σe1=p,r1σe2=p,q,r,2σe2=p,q,3σe2=r.DefineB:U2⟶0,1 (given in Table 26).Table 26
Intuitionistic fuzzy setB.
pqrμB00.10.1γB0.90.80.9Then,B is not an IF interior ideal of U2 because if we take x=q,a=r,y=p, then μBqrp≱μBr and γBqrp≰γBr. Upper approximation of B is given in Table 27.Table 27
Upper approximation ofB.
123σ¯μBe10.10.10.1σ¯γBe10.80.80.9σ¯μBe20.10.10.1σ¯γBe20.80.80.9Clearly,σ¯Be1 and σ¯Be2 are IF interior ideals of U1, so σ¯B,A is an IF soft interior ideal of U1.Theorem 24.
Letσ,A be a soft compatible relation from a semigroup U1 to a semigroup U2. If B is an IF interior ideal of U1, then Bσ¯,A is an IF soft interior ideal of U2.Proof.
It follows from Theorem23.For converse of above result, we show the following example.Example 6.
Consider the semigroups of Example5.
AndA=e1,e2. Define σ:A⟶PU1×U2 by
(34)σe1=1,p,2,q,3,r,1,q,2,p,1,r,3,pσe2=1,p,1,q,1,r,2,p,2,q,3,q.
Then,σ,A is a soft compatible relation from U1 to U2.
(35)σe1p=1,2,3,σe1q=1,2,σe1r=1,3σe2p=1,2,σe2q=2,σe2r=2,3.DefineB:U1⟶0,1 (given in Table 28).Table 28
Intuitionistic fuzzy setB.
123μB00.10.1γB0.90.80.9Then,B is not an IF interior ideal of U1 because if we take x=2,a=3,y=1, then μB231≱μB3 and γB231≰γB3. Upper approximation of B is given in Table 29.Table 29
Upper approximation ofB.
pqrμBσ¯e10.10.10.1γBσ¯e10.80.80.8μBσ¯e20.10.10.1γBσ¯e20.80.80.8Clearly,Bσ¯e1 and Bσ¯e2 are IF interior ideals of U2, so Bσ¯,A is an IF soft interior ideal of U2.Now for lower approximations of an IF interior ideal of a semigroup, we discuss some results.Theorem 25.
Supposeσ,A is a soft complete relation regarding aftersets from a semigroup U1 to a semigroup U2. If B is an IF interior ideal of U2, then σ¯B,A is an IF soft interior ideal of U1.Proof.
Suppose thatB is an IF interior ideal of U2. Thus, B is an IF subsemigroup of U2, so by Theorem 17, σ¯B,A is an IF soft subsemigroup of U1. Now for x,a,y∈U1,(36)σ¯μBexay=∧a′∈xayσeμBa′=∧a′∈xσe.aσe.yσeμBa′=∧p∈xσe.q∈aσe.r∈yσeμBpqr≥∧q∈aσeμBq=σ¯μBea.
Similarly, fors,b,t∈U1,(37)σ¯γBesbt=∨b′∈sbtσeγBb′=∨b′∈sσe.bσe.tsσeγBb′=∨u∈sσe.v∈bσe.w∈tσeγBuvw≤∨u∈sσeγBu=σ¯γBeb.Hence,σ¯Be is an IF interior ideal of U1 for all e∈A, so σ¯B,A is an IF soft interior ideal of U1.From following example, we show the converse of above theorem which is not true.Example 7.
Consider the semigroups of Example1.
LetA=e1,e2. Define σ:A⟶PU1×U2 by
(38)σe1=p,2,p,3,q,2,q,3,r,2,r,3,s,2,s,3σe2=p,2,q,2,r,2,s,2.Then,σ,A is a soft complete relation from U1 to U2 with respect to the aftersets.
(39)pσe1=2,3,qσe1=2,3,rσe1=2,3,sσe1=2,3pσe2=2,qσe2=2,rσe2=2,sσe2=2.(1)
DefineB:U2⟶0,1 such that B=μB,γB (given in Table 30)Table 30
Intuitionistic fuzzy setB.
1234μB0.40.60.81γB0.50.30.10Then,B is not an IF interior ideal of U2, because if we take x=2,a=3,y=1, then μB231≱μB3 and γB231≰γB3. Lower approximation of B is given in Table 31.Table 31
Lower approximation ofB.
pqrsσ¯μBe10.60.60.60.6σ¯γBe10.30.30.30.3σ¯μBe20.60.60.60.6σ¯γBe20.30.30.30.3Clearly,σ¯Be1 and σ¯Be2 are IF interior ideals of U1, so σ¯B,A is an IF soft interior ideal of U1.Theorem 26.
Supposeσ,A is a soft complete relation regarding foresets from a semigroup U1 to a semigroup U2. If B is an IF interior ideal of U1, then Bσ¯,A is an IF soft interior ideal of U2.Proof.
It follows from Theorem25.Example 8.
Consider the semigroups of Example1.
Defineσ:A⟶PU1×U2 by
(40)σe1=s,1,s,2,s,3,s,4,σe2=p,1,p,2,p,3,p,4,s,1,s,2,s,3,s,4.Then,σ,A is a soft complete relation from U1 to U2 with respect to the foresets.
(41)σe11=s,σe12=s,σe13=s,σe13=sσe21=p,s,σe22=p,s,σe23=p,s,σe14=p,s.DefineB:U1⟶0,1 (given in Table 32).Table 32
Intuitionistic fuzzy setB.
pqrsμB00.30.50.7γB10.70.40.2Then,B is not an IF interior ideal of U1 because if we take x=q,a=r,y=p, then μBqrp≱μBr and γBqrp≰γBr. Lower approximation of B is given in Table 33.Table 33
Lower approximation ofB.
1234μBσ¯e10.70.70.70.7γBσ¯e10.20.20.20.2μBσ¯e20000γBσ¯e21111Clearly,Bσ¯e1 and Bσ¯e2 are IF interior ideals of U2, so Bσ¯,A is an IF soft interior ideal of U2.Now, we discuss properties for IF bi-ideals of a semigroup.Theorem 27.
Supposeσ,A is a soft compatible relation from a semigroup U1 to a semigroup U2. If B1 is an IF bi-ideal of U2 then σ¯B1,A is an IF soft bi-ideal of U1.Proof.
Suppose thatB1 is an IF bi-ideal of U2. Thus, B1 is an IF subsemigroup of U2, so by Theorem 15, σ¯B1,A is an IF soft subsemigroup of U1. Now for x,a,y∈U1,(42)σ¯μB1ex∧σ¯μB1ez=∨p∈xσeμB1p∧∨q∈zσeμB1q=∨p∈xσe∨q∈zσeμB1p∧μB1q≤∨p∈xσe∨r∈yσe∨q∈zσeμB1prq≤∨prq∈xyzσeμB1prq=∨a′∈xyzσeμB1a′=σ¯μB1exay.
Similarly, forx,a,y∈U1,(43)σ¯γB1e∨xσ¯γB1ez=∧p∈xσeγB1p∨∧q∈zσeγB1q=∧p∈xσe∧q∈zσeγB1p∨γB1q≥∧p∈xσe∧r∈yσe∧q∈zσeγB1prq≥∧prq∈xyzσeγB1prq=∧b′∈xyzσeγB1b′=σ¯γB1exay.Hence,σ¯B1e is an IF bi-ideal of U1 for all e∈A, so σ¯B1,A is an IF soft bi-ideal of U1.Theorem 28.
Supposeσ,A is a soft compatible relation from a semigroup U1 to a semigroup U2. If B1 is an IF bi-ideal of U1, then B1σ¯,A is an IF soft bi-ideal of U2.Proof.
It follows from Theorem27.Example 9.
Consider the semigroups and soft relations of Example2.DefineB1:U1⟶0,1 (given in Table 34).Table 34
Intuitionistic fuzzy setB1.
pqrstμB10.50.70.90.2γB00.40.30.10.7Then,B1 is not an IF bi-ideal of U1 because if we take x=p,w=s,y=p, then μBpsp≱μBp∧μBp and γBpsp≰γBp∨μBp. Upper approximation of B1 is given in Table 35.Table 35
Upper approximation ofB1.
12345σ¯μB1e110.50.90.90.9σ¯γB1e100.40.10.10.1σ¯μB1e210.50.90.90.9σ¯γB1e200.40.10.10.1Clearly,σ¯B1e1 and σ¯B1e2 are IF bi-ideals of U2, so σ¯B1,A is an IF soft bi-ideal of U2.Theorem 29.
Letσ,A be a soft complete relation regarding aftersets from a semigroup U1 to a semigroup U2. If B1 is an IF bi-ideal of U2, then σ¯B1,A is an IF soft bi-ideal of U1.Proof.
Suppose thatB1 is an IF bi-ideal of U2. Thus, B1 is an IF subsemigroup of U2, so by Theorem 17, σ¯B1,A is an IF soft subsemigroup of U1. Now for x,a,y∈U1,(44)σ¯μB1exay=∧a′∈xayσeμB1a′=∧a′∈xσe.aσe.yσeμB1a′=∧p∈xσe.q∈aσe.r∈yσeμB1pqr≥∧p∈xσeμB1p∧∧r∈yσeμB1r=σ¯μB1ex∧σ¯μB1ey.
Similarly, forx,a,y∈U1,(45)σ¯γB1exay=∨b′∈xayσeγB1b′=∨b′∈xσe.aσeyσeγB1b′=∨p∈xσe.q∈aσe.r∈yσeγB1pqr≤∨p∈xσeγB1p∨∨r∈yσeγB1r=σ¯γB1ex∨σ¯γB1ey.Hence,σ¯B1e is an IF bi-ideal of U1 for each e∈A, so σ¯B1,A is an IF soft bi-ideal of U1.Example 10.
Taking the semigroups of Example1.
Defineσ:A⟶PU1×U2 by
(46)σe1=p,2,p,3,q,2,q,3,r,2,r,3,s,2,s,3,σe2=p,2,q,2,r,2,s,2.Then,σ,A is a soft complete relation from U1 to U2 with respect to the aftersets.
(47)pσe1=2,3,qσe1=2,3,rσe1=2,3,sσe1=2,3pσe2=2,qσe2=2,rσe2=2,sσe1=2.DefineB1:U2⟶0,1 (given in Table 36).Table 36
Intuitionistic fuzzy setB1.
1234μB10.70.20.10.4γB10.20.70.90.5Then,B1 is not an IF bi-ideal of U2 because if we take x=1,w=4,y=1, then μB1141≱μB11∧μB11 and γB1141≰γB11∨μB11. Lower approximation of B1 is given in Table 37.Table 37
Lower approximation ofB1.
pqrsσ¯μB1e10.10.10.10.1σ¯γB1e10.90.90.90.9σ¯μB1e20.20.20.20.2σ¯γB1e20 : 70.70.70.7Clearly,σ¯B1e1 and σ¯B1e2 are IF bi-ideals of U1, so σ¯B1,A is an IF soft bi-ideal of U1.Theorem 30.
Letσ,A be a soft complete relation with respect to the foresets from a semigroup U1 to a semigroup U2. If B2 is an IF bi-ideal of U1, then B2σ¯,A is an IF soft bi-ideal of U2.Proof.
It follows from Theorem29.Example 11.
Taking the semigroups of Example1.
Defineσ:A⟶PU1×U2 by
(48)σe1=s,1,s,2,s,3,s,4,σe2=p,1,p,2,p,3,p,4,s,1,s,2,s,3,s,4.Then,σ,A is a soft complete relation from U1 to U2 with respect to the foresets.
(49)σe11=s,σe12=s,σe13=s,σe13=s,σe21=p,s,σe22=p,s,σe23=p,s,σe14=p,s.DefineB2:U1⟶0,1 (given in Table 38).Table 38
Intuitionistic fuzzy setB2.
pqrsμB20.10.80.60.7γB20.90.10.30.3Then,B2 is not an IF bi-ideal of U1 because if we take x=q,w=r,y=q, then μB2qrq≱μB2q∧μB2q and γB2qrq≰γB2q∨μB2q. Lower approximation of B2 is given in Table 39.Table 39
Lower approximation ofB2.
1234σ¯μB2e10.70.70.70.7σ¯γB2e10.30.30.30.3σ¯μB2e20.10.10.10.1σ¯γB2e20.90.90.90.9Clearly,σ¯B2e1 and σ¯B2e2 are IF bi-ideals of U2, so σ¯B2,A is an IF soft bi-ideal of U2.The flow chart of our proposed model is shown in Figure1.Figure 1
Flow chart of proposed model.


## 4. Comparison with Previous Work

Our research work is a generalized form of rough approximation of an FS in semigroups based on soft relation which is discussed by Kanwal and Shabir in [50]. In this paper, we use an IFS in semigroups instead of only an FS. As we know, an IFS is better than an FS to control the vagueness and uncertainty in real-life problems. In FS theory, only membership function is discussed, but in IFS theory, functions of membership and nonmembership are also discussed. For example, if a person wants to buy a shirt with his choice, then he can buy his favourite shirt with specific design and colour by using this RIFS model. His preference will be calculated more effectively than Kanwal and Shabir’s technique [50] which is about an FS. Cagman and Engino [62] redefined the operations of SSs and proposed the products of SSs and uni-int decision function and their corresponding decision-making algorithm. Maji et al. [63] proposed a decision-making algorithm using SSs with RSs. In comparison with these two above methods, our proposed model is more reliable due to IFSs. Similarly, IFSs play an important role in medical field to detect intestinal bacteria such as Shigella and Salmonella which cause dysentery and typhoid, respectively. This is a challenging situation for microbiologists to diagnose these such deceases. Khatibi and Montazer (2009) presented value approach towards the bacteria classification problem to examine capabilities of FSs and IFSs in facing vagueness and uncertainty in the medical pattern recognition by using their five similarity measures. After comparing and illustrating the measures by experimental results, they obtained better detection rates of IFS Hausdorff and Mitchell similarity measures with valuable results 95.27% and 94.48% detection rate, respectively. On the other hand, the FS Euclidean distance yielded 85% detection rate only [64]. We can say that we make better decision by using RIFS model.


## 5. Conclusion

This paper has developed RIFS model in terms of soft binary relations. By using this approach, we obtain the upper approximation and lower approximation in the form of two pairs of IFSSs regarding aftersets and foresets. After applying these concepts, we discuss rough approximations of IF subsemigroups, IF left (right) ideals, IF interior ideals, and IF bi-ideals of semigroups. Earlier, rough approximation of an FS in semigroups based on soft binary relations has been discussed [50]. In an FS, only membership function is discussed, but in some critical situations of real world, we need nonmembership function to discuss uncertainty in good manner. Actually, the membership function alone is insufficient to deal vagueness. In such critical situations, an IFS is more useful set because its elements have membership function, nonmembership function, and the hesitation margin function. Since IFS has degree of membership, degree of nonmembership and hesitant degree with their sum of 1 but pythagorean FSs are more useful sets than IFSs because they cover more points that were not covered by IFSs. So Pythagorean FSs can describe the vagueness and impreciseness related to membership degree and nonmembership degree in a better way than IFSs.


### 5.1. Future Study

For future study, it is more considerable to make granulation roughness of an IFS and the roughness in interval valued set (IFS) under different several environments of soft binary relations. Study of this research work might be expanded to more outcomes in soft computing. For more research study, by taking on IF rough sets (IFRS), we can expand knowledge addition in IF information systems.


## 5.1. Future Study

For future study, it is more considerable to make granulation roughness of an IFS and the roughness in interval valued set (IFS) under different several environments of soft binary relations. Study of this research work might be expanded to more outcomes in soft computing. For more research study, by taking on IF rough sets (IFRS), we can expand knowledge addition in IF information systems.



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*Source: 3996256-2022-06-02.xml*