# Generalized Ideals of BCK/BCI-Algebras Based on Fuzzy Soft Set Theory

Authors: G. Muhiuddin; D. Al-Kadi; K. P. Shum; A. M. Alanazi
Journal: Advances in Fuzzy Systems
 (2021)

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## Abstract

In the present paper, using Lukaswize triple-valued logic, we introduce the notion ofα,β-intuitionistic fuzzy soft ideal of BCK/BCI-algebras, where αandβ are the membership values between an intuitionistic fuzzy soft point and intuitionistic fuzzy set. Moreover, intuitionistic fuzzy soft ideals with thresholds are introduced, and their related properties are investigated.


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## Body

## 1. Introduction

In the real world, there are several difficult problems that cannot be solved by usual mathematical methods. As a result, several theories were introduced to solve the existing problems. One of them is the theory of fuzzy sets proposed by Zadeh [1]. After the introduction of fuzzy sets by Zadeh, fuzzy set theory has become an active area of research in various fields up to now (see, for example, [2, 3]), and many generalizations of fuzzy sets have been defined and studied.Atanassov defined a new generalization of the fuzzy set, namely, intuitionistic fuzzy set [4]. Kim considered intuitionistic fuzziness on subalgebras and ideals in BCK-algebras [5]. All these theories have their weaknesses as pointed out in [6]; thus, Molodstov introduced the idea of soft sets which have been a useful tool for dealing with uncertainties. Maji et al. defined soft binary operations [7]. Also, Maji et al. presented the concept of fuzzy soft set [8]. Later on, Maji et al. introduced and studied intuitionistic fuzzy soft sets (see [9–11]), and more specifically, Akram et al. studied intuitionistic fuzzy soft K-algebras (see [12]). In recent years, a number of research papers have been devoted to the study of soft set theory applied to different algebraic structures (see, for example, [13–18]). Jun et al. applied soft set and fuzzy soft set theories to BCK/BCI-algebras in [19, 20], respectively, and Akram et al. applied the same theories on K-algebras in [21]. Larimi and Jun introduced the concepts of ∈,∈∨q-intuitionistic fuzzy h-ideals of hemiring [22]. Various attributes of BCK/BCI-algebra are considered in [23–31].Based on fuzzy points, Jana et al. studied different types of ideals [32, 33]. Moreover, the same authors studied generalized intuitionistic fuzzy ideals of BCK/BCI-algebras and Lukaswize intuitionistic fuzzy BCK/BCI-subalgebras based on 3-valued logic (see [34, 35]).This motivated us to study intuitionistic fuzzy soft ideals ofBCK/BCI-algebra X using cut sets and the degree of existence of fuzzy soft point x,a in an intuitionistic fuzzy soft set R˜,A of X. Also, an α,β-intuitionistic fuzzy soft ideal of X is introduced by applying the Lukaswize triple-valued logic, where α,β∈∈,q,∈∧q,∈∨q, with α≠∈∧q. Moreover, intuitionistic fuzzy soft ideals of BCK/BCI-algebras with thresholds are investigated, and related results are obtained.


## 2. Preliminaries

The algebraic structures ofBCK- and BCI-algebras were introduced by K. Iseki.An algebraX,∗,0 of type 2,0 with 0 as the identity element is called a BCI-algebra if for every x,y,z∈X, the following conditions are satisfied:(1)K1x∗y∗x∗z∗z∗y=0,K2x∗x∗y∗y=0,K3x∗x=0,K4x∗y=0,y∗x=0⟹x=y.The partial ordering is defined asx≤y⟺x∗y=0.IfBCI-algebra X satisfies 0∗x=0 for every x∈X, then X is a BCK-algebra.A nonempty subsetI of X is called an ideal of X if it satisfies the following:K110∈IK12 for every x,y∈X, x∗y∈I, and y∈I⇒x∈IUnless or otherwise mentioned,X denotes a BCK/BCI-algebra.Definition 1.
(see [12]). For an initial set X and a set of parameters A, a pair R,A is said to be a soft set over X⇔R which is a mapping of A into the set of all subsets of the set X.Definition 2.
(see [14]). Let E be a collection of parameters, and let ℝX denote the collection of all fuzzy sets in X. Then, R˜,A is called a fuzzy soft set over X, where A is a subset of E and R˜ is a mapping given by R˜:A⟶ℝX.
It is easy to see that every classical soft set may be considered as a fuzzy soft set. In general, for everyδ∈A, R˜δ is a fuzzy subset of X, and it is called a fuzzy value set. If for every δ∈A, R˜δ is a crisp subset of X, then R˜,A is generated as a standard soft set. Let φR˜δx denote the degree of existence function; then, R˜δ can be written as a fuzzy set such that R˜δ=x,φR˜δx|x∈Xandδ∈A.Definition 3.
(see [15]). Let E be a collection of parameters, and let IℝX denote the set of all intuitionistic fuzzy sets in X. Then, R˜,A is called an intuitionistic fuzzy soft set (IFSS) over X, where A is a subset of E and R˜ is a mapping given by R˜:A⟶IℝX.
In general, for everyδ∈A, R˜δ is an intuitionistic fuzzy subset of X, and it is called an intuitionistic fuzzy value set. Clearly, R˜δ can be written as an intuitionistic fuzzy set (IFS) such that R˜δ=x,φR˜δx,ψR˜δx|x∈Xandδ∈A, where φR˜δx and ψR˜δx represent the degree of existence and nonexistence functions, respectively. If for every x∈X, φR˜δx=1−ψR˜δx, then R˜δ will be generated as a standard fuzzy set, and then R˜,A will be generated as a traditional fuzzy soft set.


## 3.α,β-Intuitionistic Fuzzy Soft Ideals of BCK/BCI-Algebras

Definition 4.
An IFSSR˜,A in X is called an intuitionistic fuzzy soft ideal (IFSID) of X if R˜δ=x,φR˜δx,ψR˜δx|x∈Xandδ∈A satisfies the following:(1)
φR˜δ0≥φR˜δx and ψR˜δ0≤ψR˜δx,(2)
φR˜δx≥φR˜δx∗y∨φR˜δy and ψR˜δx≤ψR˜δx∗y∧ψR˜δy,for each x,y∈X and δ∈A.Definition 5.
LetR˜,A be an IFSS of X and a∈0,1.(1)
The sets(2)R˜δax=1,ifφR˜δx≥a,12,ifφR˜δx<a≤1−ψR˜δx,0,ifa>1−ψR˜δx,(3)R˜δa¯x=1,ifφR˜δx>a,12,ifφR˜δx≤a<1−ψR˜δx,0,ifa≥1−ψR˜δx,are called thea-upper cut and a-stronger upper cut of the IFSS R˜,A, respectively.(2)
The sets(4)R˜δax=1,ifψR˜δx≥a,12,ifψR˜δx<a≤1−φR˜δx,0,ifa>1−φR˜δx,(5)R˜δa¯x=1,ifψR˜δx>a,12,ifψR˜δx≤a<1−φR˜δx,0,ifa≥1−φR˜δx,are called thea-lower cut and a-stronger lower cut of the IFSS R˜,A, respectively.(3)
The sets(6)R˜δax=1,ifφR˜δx+a≥1,12,ifψR˜δx≤a<1−φR˜δx,0,ifa<ψR˜δx,(7)R˜δa¯x=1,ifφR˜δx+a>1,12,ifψR˜δx<a≤1−φR˜δx,0,ifa≤ψR˜δx,are called thea-upper Q-cut and a-stronger upper Q-cut of the IFSS R˜,A, respectively.(4)
The sets(8)R˜δax=1,ifψR˜δx+a≥1,12,ifφR˜δx≤a<1−ψR˜δx,0,ifa<φR˜δx,(9)R˜δa¯x=1,ifψR˜δx+a>1,12,ifφR˜δx<a≤1−ψR˜δx,0,ifa≤φR˜δx,are called the a-lower Q-cut and a-stronger lower Q-cut of the IFSS R˜,A, respectively.Definition 6.
(1)
The degree of existence ofx,a∈R˜δ is x,a∈R˜δ, and the degree of existence of x,aqR˜δ is x,aqR˜δ if it satisfies the following relations: x,a∈R˜δ=R˜δax and x,aqR˜δ=R˜δa¯x(2)
The degree of existence ofx,a∈R˜δ and x,aqR˜δ is x,a∈∧qR˜δ, and the degree of existence of x,a∈R˜δ or x,aqR˜δ is x,a∈∨qR˜δ if it satisfies the following relations: x,a∈∧qR˜δ=x,a∈R˜δ∧x,aqR˜δ=R˜δax∧R˜δa¯x and x,a∈∨qR˜δ=x,a∈R˜δ∨x,aqR˜δ=R˜δax∨R˜δa¯x(3)
The degree of nonexistence ofx,a∈R˜δ is x,a∈¯R˜δ, and the degree of nonexistence of x,aqR˜δ is x,aq¯R˜δ if it satisfies the following relations: x,a∈¯R˜δ=R˜δax and x,aq¯R˜δ=R˜δa¯x(4)
The degree of nonexistence ofx,a∈R˜δ and x,aqR˜δ is x,a∈∧q¯R˜δ, and the degree of nonexistence of x,a∈R˜δ or x,aqR˜δ is x,a∈∨q¯R˜δ if it satisfies the following relations: x,a∈∧q¯R˜δ=x,a∈¯∨q¯R˜δ=x,a∈¯R˜δ∨x,aq¯R˜δ=R˜δax∨R˜δa¯x and x,a∈∨q¯R˜δ=x,a∈¯R˜δ∧x,aq¯R˜δ=x,a∈¯R˜δ∧x,aq¯R˜δ=R˜δax∧R˜δa¯x
Let⟶ denote the implication of Lukaswize triple-valued logic. Lukaswize truth table is presented in Table 1.
Letα,β∈∈,q,∈∧q,∈∨q. Then, for a∈0,1,x∈X, x,a is a fuzzy soft point, and x,aαR˜δ,x,aβR˜δ∈0,1/2,1.Table 1
Lukaswize truth table.
⟶01/2101111/21/211101/21Definition 7.
LetR˜,A be an IFSS in X. If for every α,β∈∈,q,∈∧q,∈∨q and s,t∈0,1 such that R˜,A satisfies for every x,y∈X and δ∈A,(1)
x,sαR˜δ⟶0,sβR˜δ=1,(2)
x∗y,sαR˜δ∧y,tαR˜δ⟶x,s∧tβR˜δ=1,then R˜,A is called an α,β-intuitionistic fuzzy soft ideal of X.
LetX be a set. Then, define a triple-valued fuzzy set mapping R˜δ:X⟶0,1/2,1.Definition 8.
LetR˜,A be an IFSS in X. If for every α,β∈∈,q,∈∧q,∈∨q and s,t∈0,1, the IFS R˜δ satisfies for every x,y∈X,(1)
0,sβR˜δ≥x,sαR˜δ,(2)
x,s∧tβR˜δ≥x∗y,sαR˜δ∧y,tαR˜δ,then R˜,A is an α,β-intuitionistic fuzzy soft ideal of X.Example 1.
Consider aBCK/BCI-algebra X=0,1,2,3,4 with Table 2.
Let us consider the following IFS ofX:(10)φR˜δx=0.5,ifx=0,0.3,ifx=1,3,0.2,ifx=2,4,ψR˜δx=0.3,ifx=0,0.5,ifx=1,3,0.6,ifx=2,4.
Then, it can be easily shown thatR˜δ=x,φR˜δx,ψR˜δx is an ∈,∈∨q−IFID of X. Hence, R˜,A is an ∈,∈∨q−IFSID of X.
Also, if we consider the IFS ofX,(11)φR˜δx=0.6ifx=0,0.2ifx=1,3,0.1ifx=2,4,ψR˜δx=0.1ifx=0,0.2ifx=1,3,0.5ifx=2,4,then it can be easily proved that R˜δ=x,φR˜δx,ψR˜δx is an ∈∧q,∈-IFID of X. Hence, R˜,A is an ∈,∈∧q−IFSID of X.Table 2
: Cayley table of the binary operation∗.
∗01234000000110101222020331303444240Theorem 1.
LetR˜,A be an α,β-intuitionistic fuzzy soft ideal of X. If α∉∧q, then R˜,A0¯ is a fuzzy soft ideal of X.Proof.
We shall prove that(1)
R˜δ0¯0≥R˜δ0¯x(2)
R˜δ0¯x≥R˜δ0¯x∗y∧R˜δ0¯y for every x,y∈X(1)
We have to showR˜δ0¯x=1⇒R˜δ0¯0=1. If R˜δ0¯x=1, then φR˜δx>0. Let φR˜δx>t and t>0 for all x∈P and for all t∈0,1. Then, there exists s∈0,1 such that 0<s−1<t=φR˜δx. Thus, x,t∈R˜δ=R˜δtx=1 and x,sqR˜δ=R˜δs¯x=1.Ifα=∈ or α=∈∨q, then x,tαR˜δ=1, and β∈∈,q,∧q,∨q. Now, from Definition 8, 1≥0,tβR˜δ≥x,tαR˜δ=1. Therefore, 0,tβR˜δ=1⇒ either R˜δt0=1 or R˜δt¯0=1⇒ either φR˜δ0≥t>0 or φR˜δ0>1−t≥0⇒φR˜δ0>0⇒R˜δ0¯0=1Ifα=q, then x,sαR˜δ=1, and β∈∈,q,∧q,∨q. Now, from Definition 8, 1≥0,sβR˜δ≥x,sαR˜δ=1. Therefore, 0,sβR˜δ=1⇒ either R˜δs0=1 or R˜δs¯0=1⇒ either φR˜δ0≥s>0 or φR˜δ0>1−s≥0⇒φR˜δ0>0⇒R˜δ0¯0=1Next, we have to show thatR˜δ0¯x=1/2⇒R˜δ0¯0≥1/2. Suppose R˜δ0¯x=1/2. Then, ψR˜δx<1. Then, there exist s,t∈0,1 such that ψR˜δx<1−t<s<1. Then, 0<t<1−ψR˜δx, and so, R˜δtx≥1/2. Thus, x,t∈R˜δ≥1/2 and R˜δs¯x≥1/2⇒x,sqR˜δ≥1/2.Ifα=∈ or α=∈∨q, then x,tαR˜δ≥1/2, and β∈∈,q,∧q,∨q. Now, from Definition 8, 1≥0,tβR˜δ≥x,tαR˜δ≥1/2. Therefore, 0,tβR˜δ≥1/2⇒ either R˜δt0≥1/2 or R˜δt¯0≥1/2⇒ either ψR˜δ0≤1−t<1 or ψR˜δ0<t<1⇒ψR˜δ0<1⇒R˜δ0¯0≥1/2Ifα=q, then x,sαR˜δ≥1/2, and so, β∈∈,q,∧q,∨q. Now, from Definition 8, 0,sβR˜δ≥x,tαR˜δ≥1/2. Therefore, 0,sβR˜δ≥1/2⇒ either R˜δs0≥1/2 or R˜δs¯0≥1/2⇒ either ψR˜δ0≤1−s<1 or ψR˜δ0<s<1⇒ψR˜δ0<1⇒R˜δ0¯0≥1/2Hence,R˜δ0¯0≥R˜δ0¯x for all x∈X and δ∈A.(2)
First, we show thatR˜δ0¯x∗y∧R˜δ0¯y=1⇒R˜δ0¯x=1. If R˜δ0¯x∗y∧R˜δ0¯y=1, then R˜δ0¯x∗y=1 and R˜δ0¯y=1, and so, φR˜δx∗y>0 and φR˜δy>0. Let φR˜δx∗y∧φR˜δy=t>0. Then, there exists s∈0,1 such that 0<1−s<t=φR˜δx∗y∧φR˜δy. Thus, x∗y,t∈R˜δ=R˜δtx∗y=1, y,t∈R˜δ=R˜δty=1,x∗y,sqR˜δ=R˜δs¯x∗y=1, and y,sqR˜δ=R˜δs¯y=1.
Ifα=∈ or α=∈∨q, then x∗y,tαR˜δ=1,y,tαR˜δ=1, and β∈∈,q,∧q,∨q. Now, from Definition 8, 1≥x,tβR˜δ≥x∗y,tαR˜δ∧y,tαR˜δ=1. Therefore, x,tβR˜δ=1⇒ either R˜δtx=1 or R˜δt¯x=1⇒ either φR˜δx≥t>0 or φR˜δx>1−t≥0⇒φR˜δx>0⇒R˜δ0¯x=1
Ifα=q, then x∗y,sαR˜δ=y,sαR˜δ=1, and β∈∈,q,∧q,∨q. Now, from Definition 8, 1≥x,sβR˜δ≥x∗y,sαR˜δ∧y,sαR˜δ=1. Therefore, x,sβR˜δ=1⇒ either R˜δsx=1 or R˜δs¯x=1⇒ either φR˜δx≥s>0 or φR˜δx>1−s≥0⇒φR˜δx>0⇒R˜δ0¯x=1
Next, we show thatR˜δ0¯x∗y∧R˜δ0¯y=1/2⇒R˜δ0¯x≥1/2. Suppose R˜δ0¯x∗y∧R˜δ0¯y=1/2. Then, R˜δ0¯x∗y≥1/2 and R˜δ0¯y≥1/2⇒ψR˜δx∗y<1 and ψR˜δy<1⇒ψR˜δx∗y∨ψR˜δy<1. Let s,t∈0,1 such that ψR˜δx∗y∨ψR˜δy<1−t<s<1. Then, 0<t<1−ψR˜δx∗y∨ψR˜δy=1−ψR˜δx∗y∧1−ψR˜δy⇒1−ψR˜δx∗y>t and 1−ψR˜δy>t. Thus, x∗y,t∈R˜δ=R˜δtx∗y≥1/2 and y,t∈R˜δ=R˜δty≥1/2. Again, ψR˜δx∗y∨ψR˜δy<s<1 such that ψR˜δx∗y<s and ψR˜δy<s. Thus, x∗y,sqR˜δ=R˜δs¯x∗y≥1/2 and y,sqR˜δ=R˜δs¯y≥1/2.
Ifα=∈ or α=∈∨q, then x∗y,tαR˜δ=1/2,y,tαR˜δ=1/2, and β∈∈,q,∧q,∨q. Now, from Definition 8, x,tβR˜δ≥x∗y,tαR˜δ∧y,tαR˜δ≥1/2. Therefore, x,tβR˜δ≥1/2⇒x,t∈R˜δ=R˜δtx≥1/2 or x,tqR˜δ=R˜δt¯x≥1/2⇒ψR˜δx≤1−t<1 or ψR˜δx<t<1⇒ψR˜δx<1⇒R˜δ0¯x≥1/2
Ifα=q, then x∗y,sαR˜δ∧y,sαR˜δ≥1/2, and β∈∈,q,∧q,∨q. Now, from Definition 8, x,sβR˜δ≥1/2⇒x,s∈R˜δ=R˜δsx≥1/2 or x,sqR˜δ=R˜δs¯x≥1/2⇒ψR˜δx≤1−s<1 or ψR˜δx<s<1⇒ψR˜δx<1⇒R˜δ0¯x≥1/2
Hence,R˜δ0¯x≥R˜δ0¯x∗y∧R˜δ0¯y for every x,y∈X and δ∈A. Therefore, R˜δ0¯ is a fuzzy ideal of X. Thus, R˜,A0¯ is a fuzzy soft ideal of X.


## 4. Intuitionistic Fuzzy Soft Ideals of BCK/BCI-Algebras with Thresholds

Definition 9.
LetR˜,A be an IFSS of X. Then, R˜,A is an intuitionistic fuzzy soft ideal (IFSID) of X with thresholds s,t⇔R˜δ=x,φR˜δx,ψR˜δx is an intuitionistic fuzzy ideals (IFIDs) of X which satisfies the following for every x,y∈X and δ∈A:(1)
φR˜δ0∨s≥φR˜δx∧t and ψR˜δ0∧1−s≤ψR˜δx∨1−t(2)
φR˜δx∨s≥φR˜δx∗y∧φR˜δy∧t and ψR˜δx∧1−s≤ψR˜δx∗y∨ψR˜δy∨1−tExample 2.
Let us consider aBCI-algebra X=0,1,2,3 with the Cayley table given in Table 3.
Let us supposeφR˜δ0=φR˜δ1=m, φR˜δ2=φR˜δ3=s, and ψR˜δ0=ψR˜δ1=1−t,ψR˜δ2=ψR˜δ3=w, where 0<s<t<1, m∈0,s, and w∈0,1−t. Henceforth, R˜δ is an IFID of X with thresholds s,t. Consequently, R˜,A is an IFSID of X with thresholds s,t.Table 3
Cayley table of the binary operation∗.
∗012300123110322230133210Theorem 2.
An IFSSR˜,A of X is an ∈,∈∨q-IFSID of X⇔R˜,A is an IFSID of X with thresholds 0,1/2.Proof.
⇒ Suppose R˜,A is an ∈,∈∨q-IFSID of X. We have to show that R˜,A is an IFSID of X with thresholds 0,1/2. It is enough to show that(1)
φR˜δ0≥φR˜δx∧1/2 and φR˜δ0≤φR˜δx∨1/2(2)
φR˜δx≥φR˜δx∗y∧φR˜δy∧1/2 and ψR˜δx≤ψR˜δx∗y∨ψR˜δy∨1/2 for every x,y∈X and δ∈A(1)
Letx,y∈X, and let t=φR˜δx∧1/2. Then, φR˜δx≥t, and so, x,t∈R˜δ=1. Therefore, from Definition 8,1≥0,t∈∨qR˜δ≥x,t∈R˜δ=1⇒0,t∈∨qR˜δ=1⇒0,t∈R˜δ=1 or 0,t∈qR˜δ=1⇒φR˜δ0≥torφR˜δ0+t>1⇒φR˜δ0≥t or φR˜δ0>1−t≥1/2≥tThus,φR˜δ0≥t=φR˜δx∧1/2 for all x∈X. Now, let ψR˜δx∨1/2=1−s⇒1−ψR˜δx∧1/2=s; then, x,s∈R˜δ≥1/2. Therefore, from Definition 8, 0,s∈∨qR˜δ≥x,s∈R˜δ≥1/2⇒0,s∈∨qR˜δ≥1/2⇒0,s∈R˜δ≥1/2 or 0,sqR˜δ≥1/2⇒s≤1−ψR˜δx or ψR˜δ0<1−s≤sSince1−s≥1/2, s<1/2. Thus, ψR˜δ0≤1−s=ψR˜δx∨1/2.(2)
Letx,y∈X, and let t=φR˜δx∗y∧φR˜δy∧1/2. Then, φR˜δx∗y≥t and φR˜δy≥t, and so, x∗y,t∈R˜δ=1 and y,t∈R˜δ=1. Therefore, from Definition 8,1≥x,t∈∨qR˜δ≥x∗y,t∈R˜δ∧y,t∈R˜δ=1⇒x,t∈∨qR˜δ=1⇒x,t∈R˜δ∨x,tqR˜δ=1⇒x,t∈R˜δ=1 or x,tqR˜δ=1⇒φR˜δx≥t or φR˜δx+t>1⇒φR˜δx≥t or φR˜δx>1−t≥1/2≥t
Thus,φR˜δx≥t=φR˜δx∗y∧φR˜δy∧1/2 for all x,y∈X, and let ψR˜δx∗y∨ψR˜δy∨1/2=1−s; then, ψR˜δx∗y≤1−s and ψR˜δy≤1−s⇒s≤1−ψR˜δx∗y and s≤1−ψR˜δy⇒x∗y,s∈R˜δ≥1/2 and y,s∈R˜δ≥1/2. Therefore, from Definition 8,x,s∈∨qR˜δ≥x∗y,s∈R˜δ∨y,s∈R˜δ≥1/2⇒x,s∈∨qR˜δ≥1/2⇒x,s∈R˜δ≥1/2 or x,sqR˜δ≥1/2⇒s≤1−ψR˜δx or ψR˜δx<s≤1−s
Since1−s≥1/2, s<1/2. Hence, ψR˜δx≤1−s=ψR˜δx∗y∨ψR˜δy∨1/2.
⇐ Let R˜,A be an IFSID of X with thresholds 0,1/2. Let s,t∈0,1, and for x∈X and δ∈A, we have to prove 0,s∈∨qR˜δ≥x,s∈R˜δ.(1)
Leta=x,s∈R˜δ.Case 1: whena=1, x,s∈R˜δ=1⇒φR˜δx≥s. If 0,s∈∨qR˜δ≤1/2, then φR˜δ0<s and φR˜δ0≤1−s. Thus, 1/2>φR˜δ0≥φR˜δx∧1/2; so, φR˜δ0≥φR˜δx≥s. This is a contradiction to φR˜δx<s. Hence, 0,s∈∨qR˜δ=1.Case 2: fora=1/2, we have s≤1−ψR˜δx. If 0,s∈∨qR˜δ=0, then 0,s∈R˜δ=0 and 0,sqR˜δ=0⇒s>1−ψR˜δ0 and s≤ψR˜δ0⇒ψR˜δ0>1−s and s≤ψR˜δ0. Thus, 1/2<ψR˜δ0≤ψR˜δx; so, ψR˜δ0≤ψR˜δx⇒1−ψR˜δ0≥1−ψR˜δx≥s, which is a contradiction to s>1−ψR˜δ0. Therefore, 0,s∈∨qR˜δ≥1/2=x,s∈R˜δ. Hence, 0,s∈∨qR˜δ≥x,s∈R˜δ.(2)
Letx,y∈X, and let b=x∗y,s∈R˜δ∧y,t∈R˜δ.Case 1: atb=1⇒x∗y,s∈R˜δ=1 and y,t∈R˜δ=1⇒φR˜δx∗y≥s and φR˜δy≥t, if x,s∧t∈∨qR˜δ≤1/2, then φR˜δx<s∧t and φR˜δx≤1−s∧t. Thus,1/2>φR˜δx≥φR˜δx∗y∧φR˜δy∧1/2, and so, φR˜δx≥φR˜δx∗y∧φR˜δy≥s∧t. This is a contradiction to φR˜δx<s∧t. Thus, x,s∧t∈∨qR˜δ=1.Case 2: whenb=1/2, x∗y,s∈R˜δ≥1/2 and y,t∈R˜δ≥1/2⇒1−ψR˜δx∗y≥s and 1−ψR˜δy≥t. Now, 1−ψR˜δx∗y∨ψR˜δy=1−ψR˜δx∗y∧1−ψR˜δy≥s∧t.
Ifx,s∧t∈∨qR˜δ=0, then 1−ψR˜δx<s∧t and ψR˜δx≥s∧t. Now, 1/2<ψR˜δx≤ψR˜δx∗y∨ψR˜δy∨1/2⇒ψR˜δx≤ψR˜δx∗y∨ψR˜δy⇒1−ψR˜δx≥1−ψR˜δx∗y∨ψR˜δy=1−ψR˜δx∗y∧1−ψR˜δy≥s∧t which is a contradiction to 1−ψR˜δx<s∧t. Thus, x,s∧t∈∨qR˜δ≥1/2=x∗y,s∈R˜δ∧y,t∈R˜δ. Hence, x,s∧t∈∨qR˜δ≥x∗y,s∈R˜δ∧y,t∈R˜δ. Therefore, R˜δ is an ∈,∈∨q−IFID of X. Hence, R˜,A is an ∈∧q,∈-IFSID of X.Theorem 3.
An IFSSR˜,A is an ∈∧q,∈-IFSID of X⇔R˜,A is an IFSID of X with thresholds 1/2,1.Proof.
⇒ Suppose R˜,A of X is an ∈∧q,∈−IFSID of X. We prove that R˜,A is an IFSID of X with thresholds 1/2,1. It is enough to show that(1)
φR˜δ0∨1/2≥φR˜δx and ψR˜δ0∧1/2≤ψR˜δx(2)
φR˜δx∨1/2≥φR˜δx∗y∧φR˜δy and ψR˜δx∧1/2≤ψR˜δx∗y∨ψR˜δy for all x,y∈X and δ∈A(1)
Letx∈X and t=φR˜δx. If φR˜δ0∨1/2<t, then φR˜δx=t>1/2, which indicates that x,t∈∧qR˜δ=1. Now, 0,t∈R˜δ≥x,t∈∧qR˜δ=1⇒0,t∈R˜δ=1⇒φR˜δ0≥t. This is a contradiction to φR˜δ0<t. Hence, φR˜δ0∨1/2≥φR˜δx. Let t=1−s=ψR˜δx; then, x,s∈R˜δ≥1/2. If ψR˜δ0∧1/2>t and t<1/2<s, then ψR˜δx=1−s=t<s. This indicates x,s∈qR˜δ≥1/2. Thus, x,s∈R˜δ≥1/2 and x,sqR˜δ≥1/2⇒x,s∈∧qR˜δ≥1/2. Now, 0,s∈R˜δ≥x,s∈∧qR˜δ≥1/2⇒s=1−ψR˜δ0. Therefore, ψR˜δ0≤1−s=t. This contradicts the assumption ψR˜δ0>t. Hence, ψR˜δ0∧1/2≤t=ψR˜δx.(2)
Letx,y∈X and t=φR˜δx∗y∧φR˜δy. Now, if φR˜δx∨1/2<t=φR˜δx∗y∧φR˜δy, we have, φR˜δx∗y≥t>1/2 and φR˜δy≥t>1/2⇒x∗y,t∈R˜δ=1,x∗y,tqR˜δ=1 and y,t∈R˜δ=1,y,tqR˜δ=1⇒x∗y,t∈∧qR˜δ=1 and y,t∈∧qR˜δ=1⇒x∗y,t∈∧qR˜δ∧y,t∈∧qR˜δ=1. Therefore, x,t∈R˜δ≥x∗y,t∈∧qR˜δ∧y,t∈∧qR˜δ⇒x,t∈R˜δ=1⇒φR˜δx≥t. This contradicts the assumption φR˜δx∨1/2<t=φR˜δx∗y∧φR˜δy. Hence, φR˜δx∨1/2≥t=φR˜δx∗y∧φR˜δy.
LetψR˜δx∗y∨ψR˜δy=t=1−s; then, 1−s≥ψR˜δx∗y and 1−s≥ψR˜δy, but if ψR˜δx∧1/2≥t=1−s=ψR˜δx∗y∨ψR˜δy, s≤1−ψR˜δx∗y,s≤1−ψR˜δy and ψR˜δx>t, and then s>1/2>t. Thus, ψR˜δx∗y≤t<s and ψR˜δy≤t<s⇒x∗y,s∈R˜δ≥1/2 and x∗y,sqR˜δ≥1/2,y,s∈R˜δ≥1/2 and y,sqR˜δ≥1/2. Therefore, x,s∈R˜δ≥x∗y,s∈∧qR˜δ∧y,s∈∧qR˜δ≥1/2⇒x,s∈R˜δ≥1/2, which indicate that s≤1−ψR˜δx. This contradicts the assumption ψR˜δx>t. Hence, ψR˜δx∧1/2≤t=ψR˜δx∗y∨ψR˜δy.
⇐ Suppose R˜,A is an IFSID of X with thresholds 1/2,1. Let x,y∈X,s,t∈0,1 and a=φR˜δx.Case 1: leta=1. Then, φR˜δx≥s,φR˜δx+s>1⇒φR˜δx≥1/2. Thus, φR˜δ0≥φR˜δx≥s⇒0,s∈R˜δ=1.Case 2: leta=1/2. Then, s≤1−ψR˜δx and ψR˜δx<s⇒1−ψR˜δx>s>ψR˜δx. Thus, ψR˜δx<1/2. Therefore, ψR˜δ0∧1/2≤ψR˜δx⇒ψR˜δ0≤ψR˜δx⇒1−ψR˜δ0≥1−ψR˜δx≥s. Thus, 0,s∈R˜δ≥1/2. Hence, 0,s∈R˜δ≥x,s∈∧qR˜δ.
Next, we prove thatx,s∧t∈R˜δ≥x∗y,s∈∧qR˜δ∧y,t∈∧qR˜δ. Let b=φR˜δx∗y∧φR˜δy.Case 1: letb=1. Then, φR˜δx∗y≥s,φR˜δx∗y+s>1 and φR˜δy≥t,φR˜δy+t>1⇒φR˜δx∗y≥1/2 and φR˜δy≥1/2. Thus, φR˜δx∨1/2≥φR˜δx∗y∧φR˜δy≥s∧t; we get x,s∧t∈R˜δ=1.Case 2: letb=1/2. Then, s≤1−ψR˜δx∗y,ψR˜δx∗y<s and t≤1−ψR˜δy,ψR˜δy<t⇒1−ψR˜δx∗y≥s>ψR˜δx∗y and 1−ψR˜δy≥t>ψR˜δy⇒ψR˜δx∗y<1/2 and ψR˜δy<1/2. Thus, ψR˜δx∧1/2≤ψR˜δx∗y∨ψR˜δy⇒ψR˜δx≤ψR˜δx∗y∨ψR˜δy. Therefore, 1−ψR˜δx≥1−ψR˜δx∗y∧1−ψR˜δy≥s∧t. Thus, x,s∧t∈R˜δ≥1/2. Hence, x,s∧t∈R˜δ≥x∗y,s∈∧qR˜δ∧y,t∈∧qR˜δ. Therefore, R˜δ is an ∈∧q,∈−IFID of X. Hence, R˜,A is an ∈∧q,∈−IFSID of X.Theorem 4.
An IFSSR˜,A of X is an ∈,∈-IFSID of X⇔R˜,Ap is a FSID of X for any p∈0,1.Proof.
⇒ Suppose R˜,A is an ∈,∈-IFSID of X. Let x,y∈X,δ∈A, and for p∈0,1, 0,p∈R˜δ≥x,p∈R˜δ and x,p∈R˜δ≥x∗y,p∈R˜δ∧y,p∈R˜δ⇒R˜δp0≥R˜δpx and R˜δpx≥R˜δpx∗y∧R˜δpy. Therefore, R˜δp is a FID of X. Hence, R˜,Ap is a FSID of X.
⇐ Assume for any p∈0,1, R˜,Ap is a FSID of X. Let x,y∈X,δ∈A, and for s,t∈0,1, we have to show that(1)
0,s∈R˜δ≥x,s∈R˜δ(2)
x,s∧t∈R˜δ≥x∗y,s∈R˜δ∧y,t∈R˜δ(1)
Letc=x,s∈R˜δ.Case 1: forc=1, R˜δsx=1 and R˜δs0≥R˜δsx=1; we get R˜δs0=1⇒φR˜δ0≥s. Thus, 0,s∈R˜δ=1.Case 2: forc=1/2, R˜δsx=1/2. So, R˜δs0≥R˜δsx=1/2, and we get R˜δs0=1/2⇒s≤1−ψR˜δ0. Thus, 0,s∈R˜δ≥1/2. Therefore, 0,s∈R˜δ≥x,s∈R˜δ.Next, to prove(2)
Letc=x∗y,s∈R˜δ∧y,t∈R˜δ.Case 1: atc=1, x∗y,s∈R˜δ=R˜δsx∗y=1 and y,t∈R˜δ=R˜δty=1; then, x,s∧t∈R˜δ=R˜δs∧tx≥R˜δs∧tx∗y∧R˜δs∧ty≥R˜δsx∗y∧R˜δty=x∗y,s∈R˜δ∧y,t∈R˜δ=1⇒R˜δs∧tx=1⇒φR˜δx∗y≥s∧t⇒x,s∧t∈R˜δ=1.Case 2: atc=1/2, x∗y,s∈R˜δ=R˜δsx∗y=1/2 and y,t∈R˜δ=R˜δty=1/2. Therefore, x,s∧t∈R˜δ=R˜δs∧tx≥R˜δs∧tx∗y∧R˜δs∧ty≥R˜δsx∗y∧R˜δty=x∗y,s∈R˜δ∧y,t∈R˜δ=1/2, and we get R˜δs∧tx≥1/2⇒s∧t≤1−ψR˜δx. Thus, x,s∧t∈R˜δ≥1/2. Hence, x,s∧t∈R˜δ≥x∗y,s∈R˜δ∧y,t∈R˜δ. Therefore, R˜δ is an ∈,∈-IFID of X. Hence, R˜,A is an ∈,∈-IFSID of X.Theorem 5.
An IFSSR˜,A of X is an ∈,∈∨q-IFSID of X⇔ for any p∈0,1/2,R˜,Ap is a FSID of X.Proof.
⇒ Suppose R˜,A is an ∈,∈∨q−IFSID of X. Then, for every x,y∈X,δ∈A and for p∈0,1/2, we have to show that(1)
0,p∈∨qR˜δ≥x,p∈R˜δ⇒R˜δp0∨R˜δp¯0≥R˜δpx(2)
x,p∈∨qR˜δ≥x∗y,p∈R˜δ∧y,p∈R˜δ⇒R˜δpx∨R˜δp¯x≥R˜δpx∗y∧R˜δpy
As0<p≤1/2,p≤1/2≤1−p. Then, R˜δp¯0=R˜δ1−p¯0≤R˜δp¯0≤R˜δp0 and R˜δp¯x=R˜δ1−p¯x≤R˜δp¯x≤R˜δpx. Therefore, R˜δpx≤R˜δp0∨R˜δp¯0≤R˜δp0∨R˜δp0=R˜δp0 and R˜δpx∗y∧R˜δpy≤R˜δpx∨R˜δp¯x≤R˜δpx∨R˜δpx=R˜δpx. Thus, R˜δp0≥R˜δpx and R˜δpx≥R˜δpx∗y∧R˜δpy. Therefore, for any p∈0,1/2,R˜δp is a FID of X. Hence, R˜,A is a FSID of X.
⇐ Let R˜,Ap be a FSID of X, for any p∈0,1/2. Let x,y∈X,δ∈A and s,t∈0,1.(1)
Ifs≤1/2, then let a=φR˜δx.Case 1: fora=1, R˜δsx=1,R˜δs0≥R˜δsx=1⇒R˜δs0=1⇒φR˜δ0≥s⇒0,s∈R˜δ=1. Hence, 0,s∈∨qR˜δ=0,s∈R˜δ∨0,sqR˜δ=1.Case 2: leta=1/2. Then, R˜δsx=1/2,R˜δs0≥R˜δsx=1/2⇒s≤1−φR˜δ0. Thus, 0,s∈R˜δ≥1/2, from which 0,s∈∨qR˜δ=0,s∈R˜δ∨0,sqR˜δ=1/2. Hence, 0,s∈∨qR˜δ≥x,s∈R˜δ. If s>1/2, then let p∈0,1 be such that 1−s<p<1/2<s. Then, R˜δs¯0=R˜δ1−s¯0≥R˜δs0 and R˜δs¯0=R˜δ1−s¯0≥R˜δp0. Therefore, 0,s∈∨qR˜δ=0,s∈R˜δ∨0,sqR˜δ=R˜δs0∨R˜δs¯0=R˜δs¯0≥R˜δp0≥R˜δpx≥R˜δsx=x,s∈R˜δ. Hence, 0,s∈∨qR˜δ≥x,s∈R˜δ.(2)
x,s∧t∈∨qR˜δ≥x∗y,s∈R˜δ∧y,t∈R˜δ. If s∧t≤1/2, then let a=x∗y,s∈R˜δ∧y,t∈R˜δ.Case 1: fora=1, x∗y,s∈R˜δ=R˜δsx∗y=1 and y,t∈R˜δ=R˜δty=1, and we have R˜δs∧tx≥R˜δs∧tx∗y∧R˜δs∧ty≥R˜δsx∗y∧R˜δty⇒R˜δs∧tx=1⇒x,s∧t∈R˜δ=1. Thus, x,s∧t∈∨qR˜δ=x,s∧t∈R˜δ∨x,s∧tqR˜δ=1.Case 2: ata=1/2, R˜δsx∗y≥1/2 and R˜δty≥1/2; so, R˜δs∧tx≥R˜δs∧tx∗y∧R˜δs∧ty≥R˜δsx∗y∧R˜δty≥1/2⇒R˜δs∧tx≥1/2⇒x,s∧t∈R˜δ≥1/2. Therefore, x,s∧t∈∨qR˜δ=x,s∧t∈R˜δ∨x,s∧tqR˜δ≥1/2. Hence, x,s∧t∈∨qR˜δ≥x∗y,s∈R˜δ∧y,t∈R˜δ. If s∧t>1/2, then let p∈0,1 be such that 1−s∧t<p<1/2<s∧t. Now, R˜δs∧t¯x=R˜δ1−s∧t¯x≥R˜δs∧tx and R˜δs∧t¯x=R˜δ1−s∧t¯x≥R˜δpx. Thus, x,s∧t∈∨qR˜δ=x,s∧t∈R˜δ∨x,s∧tqR˜δ=R˜δs∧tx∨R˜δs∧t¯x=R˜δs∧t¯x≥R˜δpx≥R˜δpx∗y∧R˜δpy≥R˜δsx∗y∧R˜δty=x∗y,s∈R˜δ∧y,t∈R˜δ. Thus, x,s∧t∈∨qR˜δ=x∗y,s∈R˜δ∧y,t∈R˜δ. Therefore, R˜δ is an ∈,∈∨q -IFID of X. Hence, R˜,A is an ∈,∈∨q-IFSID of X.Theorem 6.
An IFSSR˜,A of X is an ∈∧q,∈-IFSID of X⇔ for any p∈1/2,1,R˜,Ap is a FSID of X.Proof.
⇒ Suppose R˜,A is an ∈∧q,∈-IFSID of X. For every x,y∈X,δ∈A and for p∈1/2,1, R˜δp¯x≥R˜δpx. Then, R˜δp0=0,p∈R˜δ≥x,p∈∧qR˜δ=R˜δpx∧R˜δp¯x=R˜δpx and R˜δpx=x,p∈R˜δ≥x∗y,p∈∧qR˜δ∧y,p∈∧qR˜δ=R˜δpx∗y∧R˜δp¯x∗y∧R˜δpy∧R˜δp¯y=R˜δpx∗y∧R˜δpy. Therefore, R˜δp0≥R˜δpx and R˜δpx≥R˜δpx∗y∧R˜δpy. Thus, R˜δp is a FID of X. Hence, R˜,Ap is a FSID of X.
⇐ We claim that, for any p∈1/2,1,R˜,Ap is a FSID of X. Let x,y∈X,δ∈A and for s,t∈0,1. Let c=x,s∈∧qR˜δ.Case 1: letc=1. Then, R˜δsx≥s and φR˜δx>1−s. Let p=φR˜δx. For φR˜δx>1/2, we have p>1/2 and R˜δpx=1. Then, R˜δp0≥R˜δpx=1⇒φR˜δ0=1⇒φR˜δ0≥p=φR˜δx=s. Hence, 0,s∈R˜δ=1.Case 2: letc=1/2. Then, 1−ψR˜δx≥s and s>ψR˜δx. Thus, 1−ψR˜δx>1/2 and ψR˜δx<1/2. Let p=1−ψR˜δx. Then, p>1/2. Therefore, R˜δp0≥R˜δpx≥1/2⇒1−ψR˜δ0≥p=1−ψR˜δx=s. Hence, 0,s∈R˜δ≥1/2=x,s∈∧qR˜δ.
Next, we provex,s∧t∈R˜δ≥x∗y,s∈∧qR˜δ∧y,t∈∧qR˜δ. Let c=x∗y,s∈∧qR˜δ∧y,t∈∧qR˜δ.Case 1: letc=1. Then, φR˜δx∗y≥s,φR˜δx∗y>1−s and φR˜δy≥t,φR˜δy>1−t. Therefore, φR˜δx∗y>1/2,φR˜δy>1/2. Let p=φR˜δx∗y∧φR˜δy; then, p>1/2, and we get φR˜δx∗y≥p and φR˜δy≥p⇒R˜δpx∗y=1 and R˜δpy=1. Thus, R˜δpx≥R˜δpx∗y∧R˜δpy=1⇒R˜δpx=1, and so, φR˜δx≥p=φR˜δx∗y∧φR˜δy=s∧t. Hence, x,s∧t∈R˜δ=1.Case 2: letc=1/2. Then, 1−ψR˜δx∗y≥s,s>ψR˜δx∗y and 1−ψR˜δy≥t,t>ψR˜δy⇒ψR˜δx∗y<1/2 and ψR˜δy<1/2. Then, 1−ψR˜δx∗y>1/2 and 1−ψR˜δy>1/2. Let p=1−ψR˜δx∗y∧1−ψR˜δy. Then, p>1/2. Therefore, R˜δpx≥R˜δpx∗y∧R˜δpy=1/2 as 1−ψR˜δx∗y≥p and 1−ψR˜δy≥p. Thus, 1−ψR˜δx≥p=1−ψR˜δx∗y∧1−ψR˜δy≥s∧t. Hence, x,s∧t∈R˜δ≥1/2=x∗y,s∈∧qR˜δ∧y,t∈∧qR˜δ. Thus, R˜δ is an ∈∧q,∈-IFID of X. Hence, R˜,A is an ∈∧q,∈-IFSID of X.Theorem 7.
An IFSSR˜,A of X is an ∈∧q,∈∨q-IFSID of X⇔ for any x,y∈X and δ∈A:(1)
φR˜δ0≥φR˜δx∧1/2 or φR˜δ0∨1/2≥φR˜δx(2)
φR˜δx≥φR˜δx∗y∧φR˜δy∧1/2 or φR˜δx∨1/2≥φR˜δx∗y∧φR˜δy(3)
ψR˜δ0≤ψR˜δx∨1/2 or ψR˜δ0∧1/2≤ψR˜δx(4)
ψR˜δx≤ψR˜δx∗y∨ψR˜δy∨1/2 or ψR˜δx∧1/2≤ψR˜δx∗y∨ψR˜δyProof.
⇒ We prove only (2) and (4), and (1) and (3) can be similarly proved.(2)
IfφR˜δx∨1/2<t=φR˜δx∗y∧φR˜δy, then φR˜δx∗y≥t>1/2,φR˜δy≥t>1/2. Then, x∗y,1/2∈∧qR˜δ=1 and y,1/2∈∧qR˜δ=1. Thus, x,1/2∧1/2∈∨qR˜δ≥x∗y,1/2∈∧qR˜δ∧y,1/2∈∧qR˜δ=1⇒x,1/2∈∧qR˜δ=1⇒φR˜δx≥1/2 or φR˜δx+1/2>1. Hence, φR˜δx≥1/2≥φR˜δx∗y∧φR˜δy∧1/2.(4)
IfψR˜δx∧1/2>t=1−s=ψR˜δx∗y∨ψR˜δy, then s≤1−ψR˜δx∗y,s≤1−ψR˜δy, and s>1/2 as x∗y,1/2∈∧qR˜δ≥1/2 and y,1/2∈∧qR˜δ≥1/2. Thus, x,1/2∧1/2∈∨qR˜δ≥x∗y,1/2∈∧qR˜δ∧y,1/2∈∧qR˜δ≥1/2⇒x,1/2∈∨qR˜δ≥1/2⇒1/2≤1−ψR˜δx or ψR˜δx<1/2. Hence, ψR˜δx≤1/2≤ψR˜δx∗y∨ψR˜δy∨1/2.
⇐ For every x,y∈X,δ∈A and s,t∈0,1, let c=x∗y,s∈∧qR˜δ∧y,t∈∧qR˜δ.Case 1: letc=1. Then, φR˜δx∗y≥s,φR˜δx∗y>1−s and φR˜δy≥t,φR˜δy>1−t⇒φR˜δx∗y∧φR˜δy>1/2. Suppose x,s∧t∈∨qR˜δ≤1/2; then, φR˜δx<s∧t and φR˜δx<1−s∧t. Then, φR˜δx<1/2<φR˜δx∗y∧φR˜δy,φR˜δx<φR˜δx∗y∧φR˜δy∧1/2 and φR˜δx∨1/2<φR˜δx∗y∧φR˜δy. This is a contradiction to (2). Hence, x,s∧t∈∨qR˜δ=1.Case 2: letc=1/2. Then, 1−ψR˜δx∗y≥s>ψR˜δx∗y,1−ψR˜δy≥t>ψR˜δy. Thus, ψR˜δx∗y∨ψR˜δy<1/2. If x,s∧t∈∨qR˜δ=0, then ψR˜δx≥s∧t>1−ψR˜δx. Thus, ψR˜δx>1/2, and so, ψR˜δx∧1/2=1/2>ψR˜δx∗y∨ψR˜δy and ψR˜δx>ψR˜δx∗y∨ψR˜δy∨1/2 which is a contradiction to (4). Hence, x,s∧t∈∨qR˜δ≥1/2. Therefore, x,s∧t∈∨qR˜δ≥x∗y,s∈∧qR˜δ∧y,t∈∧qR˜δ. This shows that R˜δ is an ∈∧q,∈∨q-IFID of X. Hence, R˜,A is an ∈∧q,∈∨q-IFSID of X.Theorem 8.
An IFSSR˜,A of X is an IFSID of X with thresholds s,t⇔ for every p∈s,t, R˜,Ap is a FSID of X.Proof.
⇒ Suppose R˜,A is an IFSID with thresholds s,t of X. Let x,y∈X and p∈s,t. Let c=R˜δpx.Case 1: letc=1. Then, φR˜δx≥p>s⇒φR˜δ0∨s≥φR˜δx∧t≥p∧t=p. Thus, φR˜δ0≥p⇒R˜δp0=1.Case 2: letc=1/2. Then, 1−ψR˜δx≥p. Thus, ψR˜δx≤1−p<1−s. As t≥p, now, ψR˜δ0∧1−s≤ψR˜δx∨1−t≤1−p∧1−t=1−p. Therefore, 1−ψR˜δ0≥p⇒R˜δp0≥1/2=R˜δpx. Hence, R˜δp0≥R˜δpx.
Next, we have to proveR˜δpx≥R˜δpx∗y∧R˜δpy. Let c=R˜δpx∗y∧R˜δpy.Case 1: letc=1. Then, R˜δpx∗y=1 and R˜δpy=1⇒φR˜δx∗y≥p>s and φR˜δy≥p>s. Now, φR˜δx∨s≥φR˜δx∗y∧φR˜δy∧t≥p∨p∧t=p⇒φR˜δx≥p. Thus, R˜δpx=1.Case 2: letc=1/2. Then, R˜δpx∗y=1/2 and R˜δpy=1/2⇒1−ψR˜δx∗y≥p or 1−ψR˜δy≥p. Thus, ψR˜δx∗y∧ψR˜δy≤1−p<1−s, from which we get ψR˜δx∧1−s≤ψR˜δx∗y∧ψR˜δy∨1−t≤1−p∨1−t=1−p. Since t≥p and 1−s>1−p, therefore, 1−ψR˜δx≥p, and so, ψR˜δx≥1/2=ψR˜δx∗y∧ψR˜δy. Thus, R˜δpx≥R˜δpx∗y∧R˜δpy. Therefore, R˜δp is a FID of X. Hence, R˜,Ap is a FSID of X.
⇐ We assume for every p∈s,t,R˜,Ap is a FSID of X. We show φR˜δ0∨s≥φR˜δx∧t.
IfφR˜δ0∨s<p=φR˜δx∧t, then p∈s,t and φR˜δx≥p. Thus, R˜δp0≥R˜δpx=1, and we have φR˜δ0=1, and so, φR˜δ0≥p. This is a contradiction to φR˜δ0<p. Hence, φR˜δ0∨s≥φR˜δx∧t.
We have to showψR˜δ0∧1−s≤ψR˜δx∨1−t. If ψR˜δ0∧1−s>p=ψR˜δx∨1−t, then 1−ψR˜δ0∨s<c=1−p=1−ψR˜δx∧t, and so, c∈s,t and 1−ψR˜δx≥c. Thus, R˜δp0≥R˜δpx≥1/2⇒R˜δp0≥1/2 and 1−ψR˜δ0≥c=1−p. Therefore, ψR˜δ0≤p, which is a contradiction to ψR˜δ0>p. Hence, ψR˜δ0∧1−s≤ψR˜δx∨1−t.
Next, we have to showφR˜δx∨s≥φR˜δx∗y∧φR˜δy∧t. If φR˜δx∨s<p=φR˜δx∗y∧φR˜δy∧t, then p∈s,t,φR˜δx∗y≥p and φR˜δy≥p. Thus, R˜δpx≥R˜δpx∗y∧R˜δpx=1⇒R˜δpx=1, and so, φR˜δx≥p. This is a contradiction to φR˜δx<p. Hence, φR˜δx∨s≥φR˜δx∗y∧φR˜δy∧t.
We have to showψR˜δx∧1−s≤ψR˜δx∗y∨ψR˜δy∨1−t. If ψR˜δx∧1−s>p=ψR˜δx∗y∨ψR˜δy∨1−t, then 1−ψR˜δx∨s<c=1−p=1−ψR˜δx∗y∧1−ψR˜δy∧t, and so, c∈s,t and 1−ψR˜δx∗y≥c and 1−ψR˜δy≥c. Thus, R˜δpx≥R˜δpx∗y∧R˜δpy≥1/2⇒R˜δpx≥1/2, and so, 1−ψR˜δx≥c=1−p. Therefore, ψR˜δx≤p, which is a contradiction to ψR˜δx>p. Hence, ψR˜δx∧1−s≤ψR˜δx∗y∨ψR˜δy∨1−t. Therefore, R˜δ is an IFID of X with thresholds s,t. Hence, R˜,A is an IFSID of X with thresholds s,t.


## 5. Conclusion

The main goal of the present paper is to introduce the notion ofα,β-intuitionistic fuzzy soft ideal of BCK/BCI-algebras, where αandβ are the membership values between an intuitionistic fuzzy soft point and intuitionistic fuzzy set. Moreover, intuitionistic fuzzy soft ideals with thresholds are introduced, and their related properties are investigated.We hope that this work will give a deep impact on the upcoming research in this field and other soft algebraic studies to open up new horizons of interest and innovations. In future directions, these definitions and main results can be similarly extended to some other algebraic systems such as subtraction algebras, B-algebras, MV-algebras, d-algebras, and Q-algebras.



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