# Applications of Soft Sets inBE-Algebras

Authors: Young Bae Jun; Sun Shin Ahn
Journal: Algebra
 (2013)

---

## Abstract

The notion of intersectional soft subalgebras of a BE-algebra is introduced, and related properties are investigated. Characterization of an intersectional soft subalgebra is discussed. The problem of classifying intersectional soft subalgebras by their inclusive subalgebras will be solved.


---

## Body

## 1. Introduction

In 1966, Imai and Iséki [1] and Iséki [2] introduced two classes of abstract algebras: BCK-algebras and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. Ma et al. studied (∈,∈∨q)-tyle (interval-valued) fuzzy ideals in BCI-algebras and soft R0-algebras (see [3–5]). As a generalization of a BCK-algebra, H. S. Kim and Y. H. Kim [6] introduced the notion of a BE-algebra and investigated several properties. In [7], Ahn and So introduced the notion of ideals in BE-algebras. They gave several descriptions of ideals in BE-algebras. Song et al. [8] considered the fuzzification of ideals in BE-algebras. They introduced the notion of fuzzy ideals in BE-algebras and investigated related properties. They gave characterizations of a fuzzy ideal in BE-algebras.Various problems in system identification involve characteristics which are essentially nonprobabilistic in nature [9]. In response to this situation, Zadeh [10] introduced fuzzy set theory as an alternative to probability theory. Uncertainty is an attribute of information. In order to suggest a more general framework, the approach to uncertainty is outlined by Zadeh [11]. To solve complicated problem in economics, engineering, and environment, we cannot successfully use classical methods because of various uncertainties typical for those problems. There are three theories: theory of probability, theory of fuzzy sets, and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. Uncertainties cannot be handled using traditional mathematical tools but may be dealt with using a wide range of existing theories such as probability theory, theory of (intuitionistic) fuzzy sets, theory of vague sets, theory of interval mathematics, and theory of rough sets. However, all of these theories have their own difficulties which are pointed out in [12]. Maji et al. [13] and Molodtsov [12] suggested that one reason for these difficulties may be due to the inadequacy of the parametrization tool of the theory. To overcome these difficulties, Molodtsov [12] introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties that have troubled the usual theoretical approaches. Molodtsov pointed out several directions for the applications of soft sets. At present, works on the soft set theory are progressing rapidly. Maji et al. [13] described the application of soft set theory to a decision making problem. Maji et al. [14] also studied several operations on the theory of soft sets. Chen et al. [15] presented a new definition of soft set parametrization reduction and compared this definition to the related concept of attributes reduction in rough set theory. The algebraic structure of set theories dealing with uncertainties has been studied by some authors. Çağman et al. [16] introduced fuzzy parameterized (FP) soft sets and their related properties. They proposed a decision making method based on FP-soft set theory and provided an example which shows that the method can be successfully applied to the problems that contain uncertainties. Feng [17] considered the application of soft rough approximations in multicriteria group decision making problems. Aktaş and Çağman [18] studied the basic concepts of soft set theory and compared soft sets to fuzzy and rough sets, providing examples to clarify their differences. They also discussed the notion of soft groups. After that, many algebraic properties of soft sets are studied (see [19–29]).In this paper, we introduce the notion of int-soft subalgebras of aBE-algebra and investigate their properties. We consider characterization of an int-soft subalgebra, and solve the problem of classifying int-soft subalgebras by their inclusive subalgebras.


## 2. Preliminaries

LetK(τ) be the class of all algebras of type τ=(2,0). By aBE-algebra we mean a system (X;*,1)∈K(τ) in which the following axioms hold (see [6]):
(1)(∀x∈X)(x*x=1),(2)(∀x∈X)(x*1=1),(3)(∀x∈X)(1*x=x),(4)(∀x,y,z∈X)(x*(y*z)=y*(x*z)).(exchange)A relation “≤” on a BE-algebra X is defined by
(5)(∀x,y∈X)(x≤y⟺x*y=1).ABE-algebra (X;*,1) is said to be transitive (see [7]) if it satisfies
(6)(∀x,y,z∈X)(y*z≤(x*y)*(x*z)).ABE-algebra (X;*,1) is said to be self distributive (see [6]) if it satisfies
(7)(∀x,y,z∈X)(x*(y*z)=(x*y)*(x*z)).Note that every self distributiveBE-algebra is transitive, but the converse is not true in general (see [7]).A mappingμ:X→Y of BE-algebras is called a homomorphism if μ(x*y)=μ(x)*μ(y) for all x,y∈X.A soft set theory is introduced by Molodtsov [12], and Çağman and Enginoğlu [30] provided new definitions and various results on soft set theory.In what follows, letU be an initial universe set, and let E be a set of parameters. Let 𝒫(U) denote the power set of U and A,B,C,…⊆E.Definition 1 (see [12, 13]).
A soft set(f,A) over U is defined to be the set of ordered pairs
(8)(f,A):={(x,f(x)):x∈E,f(x)∈𝒫(U)},
where f:E→𝒫(U) such that f(x)=∅ if x∉A.The functionf is called an approximate function of the soft set (f,A).In what follows, denote byS(U) the set of all soft sets over U by Çağman and Enginoğlu [30].For any soft sets(f,X) and (g,X) over U, we call (f,X) a soft subset of (g,X), denoted by (f,X)⊆~(g,X) if f(x)⊆g(x) for all x∈X. The soft{unionintersection} of (f,X) and (g,X) is defined to be a soft set {(f,X)∪~(g,X)=(f∪~g,X)(f,X)∩~(g,X)=(f∩~g,X)}, where {(f∪~g)(x)=f(x)∪g(x)(f∩~g)(x)=f(x)∩g(x)} for all x∈X.Definition 2 (see [31, 32]).
Assume thatE has a binary operation ↪. For any nonempty subset A of E, a soft set (f,A) over U is said to be intersectional over U if it satisfies
(9)(∀x,y∈A)(x↪y∈A⟹f(x)∩f(y)⊆f(x↪y)).For a soft set(f,A) over U and a subset γ of U, the γ-inclusive set of (f,A), denoted by iA(f;γ), is defined to be the set
(10)iA(f;γ):={x∈A∣γ⊆f(x)}.


## 3. Intersectional Soft Subalgebras

In what follows, we takeE=X as a set of parameters, which is a BE-algebra under the operation “*” unless otherwise specified.Definition 3.
A soft set(f,X) over U is called an intersectional soft subalgebra (briefly, int-soft subalgebra) over U if it satisfies
(11)(∀x,y∈X)(f(x*y)⊇f(x)∩f(y)).Example 4.
LetE=X be the set of parameters where X={1,a,b,c,d} is a BE-algebra with the following Cayley table:
(12)*1abcd11abcda11bcdb1a1ccc11b11d11111
Let (f,X) be a soft set over U defined as follows:
(13)f:X⟶𝒫(U),x↦{γ3ifx=1,γ1ifx∈{a,c,d},γ2ifx=b,
where γ1,γ2, and γ3 are subsets of U with γ1⊊γ2⊊γ3. It is easy to check that (f,X) is an int-soft subalgebra over U.Example 5.
LetE=X be the set of parameters, and let U=X be the initial universe set, where X={1,a,b,c,d,0} is a BE-algebra [7] with the following Cayley table:
(14)*1abcd011abcd0a11accdb111cccc1ab1abd11a11a0111111
Let {γn∣n=1,2,3,4} be a class of subsets of U which is a poset under the following Hasse diagram:
Let(f,X) be a soft set over U defined as follows:
(15)f:X⟶𝒫(U),x↦{γ4ifx=1,γ1ifx∈{a,c,d},γ2ifx=b,γ3ifx=0.
It is easy to check that (f,X) is an int-soft subalgebra over U.Theorem 6.
A soft set(f,X) over U is an int-soft subalgebra over U if and only if iX(f;γ) is a subalgebra of X for all γ∈𝒫(U).The subalgebraiX(f;γ) in Theorem 6 is called the inclusive subalgebra of X.Proof.
Assume that(f,X) is an int-soft subalgebra over U. Let γ∈𝒫(U) and x,y∈iX(f;γ). Then, γ⊆f(x) and γ⊆f(y). It follows from (11) that
(16)f(x*y)⊇f(x)∩f(y)⊇γ,
that is, x*y∈iX(f;γ). Thus, iX(f;γ) is a subalgebra of X.
Conversely, suppose thatiX(f;γ) is a subalgebra of X for all γ∈𝒫(U). Let x,y∈X be such that f(x)=γx and f(y)=γy. Take γ=γx∩γy. Then, x,y∈iX(f;γ), and so x*y∈iX(f;γ) by assumption. Hence,
(17)f(x*y)⊇γ=γx∩γy=f(x)∩f(y).
Therefore, (f,X) is an int-soft subalgebra over U.Lemma 7.
Every int-soft subalgebra(f,X) over U satisfies the following inclusion:
(18)(∀x∈X)(f(x)⊆f(1)).Proof.
Using (1) and (11), we have
(19)f(1)=f(x*x)⊇f(x)∩f(x)=f(x)
for all x∈X.Proposition 8.
For any int-soft subalgebra(f,X) over U, if a fixed element x∈X satisfies f(x)=f(1), then
(20)(∀y∈X)(f(y)⊆f(x*y)).Proof.
Assume that a fixed elementx∈X satisfies f(x)=f(1). Then,
(21)f(y)=f(1)∩f(y)=f(x)∩f(y)⊆f(x*y)
for all y∈X.Proposition 9.
Let(f,X) be an int-soft subalgebra over U. If a fixed element x∈X satisfies the following condition:
(22)(∀y∈X)(f(y)⊆f(y*x)),
then f(x)=f(1).Proof.
Takingy=1 in (22) implies that f(1)⊆f(1*x)=f(x) by (3). It follows from Lemma 7 that f(x)=f(1).For anyBE-algebras X and Y, let μ:X→Y be a function and (f,X), and let (g,Y) be soft sets over U.(1) The soft set(23)μ-1(g,Y)={(x,μ-1(g)(x)):x∈X,μ-1(g)(x)∈𝒫(U)},
where μ-1(g)(x)=g(μ(x)), is called the soft preimage of (g,Y) under μ.(2) The soft set(24)μ(f,X)={(y,μ(f)(y)):y∈Y,μ(f)(y)∈𝒫(U)},
where
(25)μ(f)(y)={⋃x∈μ-1(y)f(x)ifμ-1(y)≠∅,∅otherwise,
is called the soft image of (f,X) under μ.Proposition 10.
For anyBE-algebras X and Y, let μ:X→Y be a function. Then,
(26)(∀(f,X)∈S(U))((f,X)⊆~μ-1(μ(f,X))).Proof.
Note thatμ-1(μ(x))≠∅ for all x∈X. Hence,
(27)f(x)⊆⋃a∈μ-1(μ(x))‍f(a)=μ(f)(μ(x))=μ-1(μ(f))(x)
for all x∈X, and therefore (26) is valid.Theorem 11.
Letμ:X→Y be a homomorphism of BE-algebras and (g,Y)∈S(U). If (g,Y) is an int-soft subalgebra over U, then the soft preimage μ-1(g,Y) of (g,Y) under μ is also an int-soft subalgebra over U.Proof.
For anyx1,x2∈X, we have
(28)μ-1(g)(x1)∩μ-1(g)(x2)=g(μ(x1))∩g(μ(x2))⊆g(μ(x1)*μ(x2))=g(μ(x1*x2))=μ-1(g)(x1*x2).
Hence, μ-1(g,Y) is also an int-soft subalgebra over U.Theorem 12.
Letμ:X→Y be a homomorphism of BE-algebras and (f,X)∈S(U). If (f,X) is an int-soft subalgebra over U and μ is injective, then the soft image μ(f,X) of (f,X) under μ is also an int-soft subalgebra over U.Proof.
Lety1,y2∈Y. If at least one of μ-1(y1) and μ-1(y2) is empty, then the inclusion
(29)μ(f)(y1)∩μ(f)(y2)⊆μ(f)(y1*y2)
is clear. Assume that μ-1(y1)≠∅ and μ-1(y2)≠∅. Since μ is injective, we have
(30)μ(f)(y1)∩μ(f)(y2)=(⋃x1∈μ-1(y1)‍f(x1))∩(⋃x2∈μ-1(y2)‍f(x2))=⋃x1∈μ-1(y1)x2∈μ-1(y2)(f(x1)∩f(x2))⊆⋃x1∈μ-1(y1)x2∈μ-1(y2)(f(x1*x2))=⋃x∈μ-1(y1*y2)‍f(x)=μ(f)(y1*y2).
Therefore, μ(f,X) is an int-soft subalgebra over U.Theorem 13.
Let(f,X)∈S(U) and define a soft set (f*,X) over U by
(31)f*:X⟶𝒫(U),x↦{f(x)ifx∈iX(f;γ),δotherwise,
where γ is any subset of U and δ is a subset of U satisfying δ⊊⋂x∉iX(f;γ)f(x). If (f,X) is an int-soft subalgebra over U, then so is (f*,X).Proof.
If(f,X) is an int-soft subalgebra over U, then iX(f;γ) is a subalgebra of X for all γ⊆U by Theorem 6. Let x,y∈X. If x,y∈iX(f;γ), then x*y∈iX(f;γ). Hence,
(32)f*(x*y)=f(x*y)⊇f(x)∩f(y)=f*(x)∩f*(y).
If x∉iX(f;γ) or y∉iX(f;γ), then f*(x)=δ or f*(y)=δ. Thus,
(33)f*(x*y)⊇δ=f*(x)∩f*(y).
Therefore, (f*,X) is an int-soft subalgebra over U.Theorem 14.
If(f,X) and (g,X) are int-soft subalgebras over U, then the soft intersection (f,X)∩~(g,X) of (f,X) and (g,X) is an int-soft subalgebra over U.Proof.
Letx,y∈X. Then,
(34)(f∩~g)(x*y)=f(x*y)∩g(x*y)⊇(f(x)∩f(y))∩(g(x)∩g(y))=(f(x)∩g(x))∩(f(y)∩g(y))=(f∩~g)(x)∩(f∩~g)(y).
Hence, (f,X)∩~(g,X) is an int-soft subalgebra over U.The following example shows that the soft union of int-soft subalgebras overU may not be an int-soft subalgebra over U.Example 15.
LetE=X be the set of parameters where X={1,a,b,c} is a BE-algebra [7] with the following Cayley table:
(35)*1abc11abca11aab111ac11a1
Let (f,X) and (g,X) be soft sets over U defined, respectively, as follows:
(36)f:X⟶𝒫(U),x↦{γ5ifx=1,γ2ifx=a,γ1ifx=b,γ3ifx=c,,g:X⟶𝒫(U),x↦{γ4ifx=1,γ2ifx=a,γ3ifx=b,γ1ifx=c,,
where γ1,γ2,γ3,γ4, and γ5 are subsets of U with γ1⊊γ2⊊γ3⊊γ4⊊γ5. It is easy to check that (f,X) and (g,X) are int-soft subalgebras over U. But (f,X)∪~(g,X)=(f∪~g,X) is not an int-soft subalgebra over U, since (f∪~g)(c*b)=(f∪~g)(a)=γ2⊉γ3=(f∪~g)(c)∩(f∪~g)(b).Theorem 16.
Let(f,X) be an int-soft subalgebra over U. Let γ1 and γ2 be subsets of U such that γ1⊊γ2. If the γ1-inclusive set of (f,X) is equal to the γ2-inclusive set of (f,X), then there is no x∈X such that γ1⊆f(x)⊊γ2.Proof.
Straightforward.The converse of Theorem16 is not true in general as seen in the following example.Example 17.
LetE=X be the set of parameters, and let U=X be the initial universe set where X={1,a,b,c,d} is a BE-algebra as in Example 4. Consider a soft set (f,X) over U which is given by
(37)f:X⟶𝒫(U),x↦{Xifx=1,{1,a}ifx∈{a,c,d},{1,a,c}ifx=b.
Then, (f,X) is an int-soft subalgebra over U. The γ-inclusive sets of (f,X) are described as follows:
(38)iX(f;γ)={Xifγ∈{∅,{1},{a},{1,a}}{1,b}ifγ∈{{c},{a,c},{1,a,c}},{1}ifγ∈𝒫(U)∖{∅,{1},{a},{1,a},{c},{a,c},{1,a,c}}.
If we take γ1={1,c} and γ2={1,b,c}, then γ1⊊γ2 and there is no x∈X such that γ1⊆f(x)⊊γ2. But iX(f;γ1)={1,b}≠{1}=iX(f;γ2).Theorem 18.
Let(f,X) be an int-soft subalgebra over U. Let γ1 and γ2 be subsets of U such that γ1⊊γ2 and {γ1,γ2,f(x)} are totally ordered by set inclusion for all x∈X. If there is no x∈X such that γ1⊆f(x)⊊γ2, then the γ1-inclusive set of (f,X) is equal to the γ2-inclusive set of (f,X).Proof.
Sinceγ1⊊γ2, we have iX(f;γ2)⊆iX(f;γ1). If x∈iX(f;γ1), then γ1⊆f(x). Since {γ1,γ2,f(x)∣x∈X} is totally ordered by inclusion and there is no x∈X such that γ1⊆f(x)⊊γ2, it follows that γ2⊆f(x), that is, x∈iX(f;γ2). Therefore, the γ1-inclusive set of (f,X) is equal to the γ2-inclusive set of (f,X).Theorem 19.
Let(f,X) be a soft set over U in which Im(f) is totally ordered by set inclusion. For each subset γ of Im(f), if the γ-inclusive set of (f,X) is a subalgebra of X, then (f,X) is an int-soft subalgebra over U.Proof.
Letx,y∈X be such that f(x)=γ1 and f(y)=γ2. Then, either γ1⊆γ2 or γ2⊆γ1. We may assume that γ1⊆γ2 without loss of generality. Then, x∈iX(f;γ1),y∈iX(f;γ2), and iX(f;γ2)⊆iX(f;γ1). Since iX(f;γ1) is a subalgebra of X, it follows that x*y∈iX(f;γ1) so that
(39)f(x*y)⊇γ1=γ1∩γ2=f(x)∩f(y).
Therefore, (f,X) is an int-soft subalgebra over U.We have the following question.Question. Let (f,X) be an int-soft subalgebra over U. Does any subalgebra can be represented as a γ-inclusive set of (f,X)?The following example shows that the answer to the question above is false.Example 20.
LetE=X be the set of parameters, and let U=X be the initial universe set where X={1,a,b,c} is a BE-algebra as in Example 15. Consider a soft set (f,X) over U which is given by
(40)f:X⟶𝒫(U),x↦{{1,c}ifx=1,{c}ifx∈{a,b,c}.
Then, (f,X) is an int-soft subalgebra over U. The γ-inclusive sets of (f,X) are described as follows:
(41)iX(f;γ)={Xifγ∈{∅,{c}},{1}ifγ∈{{1},{1,c}},∅otherwise.
The subalgebra {1,b} cannot be a γ-inclusive set iX(f;γ), since there is no γ⊆U such that iX(f;γ)={1,b}.However, we have the following theorem.Theorem 21.
Every subalgebra of aBE-algebra can be represented as a γ-inclusive set of an int-soft subalgebra.Proof.
LetA be a subalgebra of a BE-algebra X. For a subset γ of U, define a soft set (f,X) over U by
(42)f:X⟶𝒫(U),x↦{γifx∈A,∅ifx∉A.
Obviously, A=iX(f;γ). We now prove that (f,X) is an int-soft subalgebra over U. Let x,y∈X. If x,y∈A, then x*y∈A because A is a subalgebra of X. Hence, f(x)=f(y)=f(x*y)=γ, and so f(x)∩f(y)⊆f(x*y). If x∈A and y∉A, then f(x)=γ and f(y)=∅ which imply that
(43)f(x)∩f(y)=γ∩∅=∅⊆f(x*y).
Similarly, if x∉A and y∈A, then f(x)∩f(y)⊆f(x*y). Obviously, if x∉A and y∉A, then f(x)∩f(y)⊆f(x*y). Therefore, (f,X) is an int-soft subalgebra over U.Note that ifE=X is a finite BE-algebra, then the number of subalgebras of X is finite whereas the number of γ-inclusive sets of an int-soft subalgebra over U appears to be infinite. But, since every γ-inclusive set is indeed a subalgebra of X, not all these γ-inclusive sets are distinct. The next theorem characterizes this aspect.Theorem 22.
Let(f,X) be an int-soft subalgebra over U and let γ1⊊γ2⊆U such that {γ1,γ2,f(x)} is a chain for all x∈X. Two γ-inclusive sets iX(f;γ1) and iX(f;γ2) are equal if and only if there is no x∈X such that γ1⊊f(x)⊊γ2.Proof.
Letγ1 and γ2 be subsets of U such that iX(f;γ1)=iX(f;γ2). Assume that there exists x∈X such that γ1⊊f(x)⊊γ2. Then, iX(f;γ2) is a proper subset of iX(f;γ1), which contradicts the hypothesis.
Conversely, suppose that there is nox∈X such that γ1⊊f(x)⊊γ2. Obviously, iX(f;γ2)⊆iX(f;γ1). If x∈iX(f;γ1), then γ1⊆f(x). It follows from the assumption that γ2⊆f(x), that is, x∈iX(f;γ2). Therefore, iX(f;γ1)=iX(f;γ2).Remark 23.
As a consequence of Theorem22, if E=X is a finite BE-algebra, then the γ-inclusive sets of an int-soft subalgebra (f,X) over U form a chain. But f(x)⊆f(1) for all x∈X. Therefore, iX(f;γ0), where γ0=f(1), is the smallest inclusive subalgebra but not always iX(f;γ0)={1} as seen in the following example, and so we have the chain
(44)iX(f;γ0)⊊iX(f;γ1)⊊⋯⊊iX(f;γr)=X,
where γ0⊋γ1⊋⋯⊋γr.Example 24.
LetA be a subalgebra of a BE-algebra X such that A≠{1}. Let (f;X) be the int-soft subalgebra over U which is given in the proof of Theorem 21. Then, Im(f)={∅,γ}. Further, the γ-inclusive sets of (f,X) are iX(f;∅)=X and iX(f;γ)=A. Thus, we have f(1)=γ but iX(f;γ)=A≠{1}.Corollary 25.
LetE=X be a finite BE-algebra, and let (f,X) be an int-soft subalgebra over U. If Im(f)={γ1,γ2,…,γn}, then the family of γi-inclusive sets iX(f;γi),1≤i≤n, constitutes all the γ-inclusive sets of (f,X).Proof.
Letγ⊆U and γ∉Im(f). If γi⊊γ⊊γj, where γi,γj∈Im(f), then iX(f;γi)=iX(f;γj)=iX(f;γ) by Theorem 22. If γ⊊γr, where γr is the least element (under the set inclusion) of Im(f), then iX(f;γr)=X=iX(f;γ). Assume that γ⊋γk, where γk is the greatest element (under the set inclusion) of Im(f). If there is x∈X such that f(x)=γx⊆U and γk⊊γx⊊γ, then γx∈Im(f). It  is a contradiction. It follows from Theorem 22 that iX(f;γk)=iX(f;γ). Thus, for any γ⊆U, the inclusive subalgebra is one of {iX(f;γi)∣γi∈Im(f)}.The following example shows that two int-soft subalgebras overU may have an identical family of γ-inclusive sets but the int-soft subalgebras over U may not be equal.Example 26.
LetE=X be the set of parameters, and let U=X be the initial universe set where X={1,a,b,c} is a BE-algebra as in Example 15. Consider a soft set (f,X) over U which is given by
(45)f:X⟶𝒫(U),x↦{γ1ifx=1,γ2ifx=a,γ3ifx∈{b,c},
where γ1⊋γ2⊋γ3 are subsets of U. It is easy to verify that (f;X) is an int-soft subalgebra over U. The γ-inclusive sets of (f;X) are iX(f;γ1)={1},iX(f;γ2)={1,a} and iX(f;γ3)=X. Now let δ1,δ2, and δ3 be subsets of U such that δ1⊋δ2⊋δ3 and δi≠γj for i=1,2,3 and j=1,2,3. Define a soft set (g;X) over U as follows:
(46)g:X⟶𝒫(U),x↦{δ1ifx=1,δ2ifx=a,δ3ifx∈{b,c}.
Then, (g,X) is an int-soft subalgebra over U, and the γ-inclusive sets of (g;X) are iX(g;δ1)={1},iX(g;δ2)={1,a} and iX(g;δ3)=X. Hence, the two int-soft subalgebras (f,X) and (g,X) over U have an identical family of γ-inclusive sets. However, it is clear that (f,X) is not equal to (g,X).Lemma 27.
LetE=X be a finite BE-algebra, and let (f,X) be an int-soft subalgebra over U. If γi and γj are elements of Im(f) such that iX(f;γi)=iX(f;γj), then γi=γj.Proof.
Straightforward.Theorem 28.
LetE=X be a finite BE-algebra, and let (f,X) and (g,X) be two int-soft subalgebras over U having the identical family of γ-inclusive sets. If Im(f)={γ0,γ1,…,γr} and Im(g)={δ0,δ1,…,δk}, where
(47)γ0⊋γ1⊋⋯⊋γr,δ0⊋δ1⊋⋯⊋δk,
then we have
(1)r=k,
(2)iX(f;γi)=iX(g;δi),0≤i≤r,
(3)(∀x∈X)(f(x)=γi⇒g(x)=δi,0≤i≤r).Proof.
Corollary25 implies that the only γ-inclusive sets of (f,X) and (g,X) are the two families iX(f;γi) and iX(g;δi). Since (f,X) and (g,X) have the same family of γ-inclusive sets, we have r=k which proves (1).
(2) Using (1) and Remark23, we have two chains of γ-inclusive sets:
(48)iX(f;γ0)⊊iX(f;γ1)⊊⋯⊊iX(f;γr)=X,iX(g;δ0)⊊iX(g;δ1)⊊⋯⊊iX(g;δr)=X.
Clearly, we have
(49)(∀γi,γj∈Im(f))(γi⊋γj⟹iX(f;γi)⊊iX(f;γj)),(50)(∀δi,δj∈Im(g))(δi⊋δj⟹iX(g;δi)⊊iX(g;δj)).
Since two families of γ-inclusive sets are identical, it is clear that iX(f;γ0)=iX(g;δ0). By hypothesis, iX(f;γ1)=iX(g;δj) for some j>0. Assume that iX(f;γ1)≠iX(g;δ1). Then, iX(f;γ1)=iX(g;δj) for some j>1, and iX(g;δ1)=iX(f;γi) for some γi⊊γ1. Thus, by (49) and (50), we have iX(g;δj)=iX(f;γ1)⊊iX(f;γi) and iX(f;γi)=iX(g;δ1)⊊iX(g;δj). This is a contradiction, and so iX(f;γ1)=iX(g;δ1). By mathematical induction on i,0≤i≤r, we finally obtain iX(f;γi)=iX(g;δi),0≤i≤r.
(3) Letx∈X be such that f(x)=γi and g(x)=δj, where 0≤i≤r and 0≤j≤r. It is sufficient to show that δj=δi. Now, x∈iX(f;γi)=iX(g;δi) implies that g(x)=δj⊋δi. This gives from (50) that iX(g;δj)⊊iX(g;δi). Since x∈iX(g;δj), it follows from (2) that x∈iX(f;γj) and so f(x)=γi⊋γj. Hence, iX(f;γi)⊊iX(f;γj) by (49). Using (2), we have iX(g;δi)=iX(f;γi)⊊iX(f;γj)=iX(g;δj). Thus, iX(g;δi)=iX(g;δj), and so δj=δi. This completes the proof.Theorem 29.
LetE=X be a BE-algebra. Given any chain of subalgebras
(51)A0⊊A1⊊A2⊊⋯⊊Ar=X,
there exists an int-soft subalgebra over U whose γ-inclusive sets are exactly the subalgebras of this chain.Proof.
Consider a class{γi∣i=0,1,2,…,r} of subsets of U such that
(52)γ0⊋γ1⊋⋯⊋γr.
Define a soft set (f,X):E→𝒫(U) by f(A0)=γ0 and f(Ai-Ai-1)=γi,0<i≤r. We will prove that (f,X) is an intersectional soft BE-algebra over U. Note that if x∈Ai, then γi⊆f(x). If x∈Ai, then either x∈Ai-Aj or x∈Aj for i>j. Thus, if x∈Ai-Aj, then f(x)=γi. If x∈Aj, then γi⊊γj⊆f(x). Let x,y∈X. We distinguish two cases as follows:
Case  1. Let x,y∈Ai-Ai-1. Then, f(x)=γi=f(y). Since Ai is a subalgebra, we have x*y∈Ai, and so either x*y∈Ai-Ai-1 or x*y∈Ai-1. In any case we know that
(53)f(x)∩f(y)=γi⊆f(x*y).Case  2. For i>j, let x∈Ai-Ai-1 and y∈Aj-Aj-1. Then, f(x)=γi,f(y)=γj and x*y∈Ai. It follows that
(54)f(x)∩f(y)=γi∩γj=γi⊆f(x*y).
Hence, we conclude that (f,X) is an intersectional soft BE-algebra over U. From the definition of (f,X), we have Im(f)={γ0,γ1,…,γr}. Thus, the γ-inclusive sets of X are given by the chain of subalgebras
(55)iX(f;γ0)⊊iX(f;γ1)⊊⋯⊊iX(f;γr)=X.
Now, iX(f;γ0)={x∈X∣γ0⊆f(x)}=A0. Finally, we prove that iX(f;γi)=Ai for 0<i≤r. Clearly Ai⊆iX(f;γi). If x∈iX(f;γi), then γi⊆f(x), and so x∉Aj for j>i. Hence, f(x)∈{γ1,γ2,…,γr}, and thus x∈Ak for some k≤i. Since Ak⊆Ai, we have x∈Ai, and so iX(f;γi)=Ai for 0≤i≤r. This completes the proof.Theorem29 is illustrated as an example.Example 30.
LetU=ℤ be the initial universe set, and let E=X be the set of parameters where X={1,a,b,c,d,0} is a BE-algebra as in Example 5. Consider subalgebras A1={1},A2={1,a},A3={1,a,c,d}, and A4=X. Then, A1⊊A2⊊A3⊊A4. Define a soft set (f,X) over U by
(56)f:X⟶𝒫(U),x↦{ℤifx=1,2ℤifx=a,4ℤifx∈{c,d},8ℤifx∈{b,0}.
Then, (f,X) is an int-soft subalgebra over U with iX(f;ℤ)={1}=A1, iX(f;2ℤ)={1,a}=A2, iX(f;4ℤ)={1,a,c,d}=A3, and iX(f;8ℤ)=X=A4.Theorem 31.
Let(f,X) be a soft set over U, and let γ be a subset of U. Define a soft set (f*,X) over U by
(57)f*:X⟶𝒫(U),x↦{f(x)ifx∈iX(f;γ),∅otherwise.
If (f,X) is an int-soft subalgebra over U, then so is (f*,X).Proof.
Letx,y∈X. If x,y∈iX(f;γ), then x*y∈iX(f;γ), and so
(58)f*(x*y)=f(x*y)⊇f(x)∩f(y)=f*(x)∩f*(y).
If x∉iX(f;γ) or y∉iX(f;γ), then f*(x)=∅ or f*(y)=∅. Hence,
(59)f*(x)∩f*(y)=∅⊆f*(x*y).
Therefore, (f*,X) is an int-soft subalgebra over U.


## 4. Conclusion

Using the notion of int-soft sets, we have introduced the concept of int-soft subalgebras inBE-algebras and investigated related properties. We have considered characterization of an int-soft subalgebra and solved the problem of classifying int-soft subalgebras by their inclusive subalgebras. We have shown that(1)
every soft image of an int-soft subalgebra is also an int-soft subalgebra;(2)
the soft intersection of int-soft subalgebras is an int-soft subalgebra.We have made a new int-soft subalgebra from the old one. Work is ongoing. Some important issues for future work are as follows:(1)
to develop strategies for obtaining more valuable results,(2)
to apply these notions and results for studying related notions in other (soft) algebraic structures,(3)
to study the soft set application in ideal and filter theory ofBE-algebras.



---

*Source: 368962-2013-04-10.xml*