# Applications of Soft Sets inK-Algebras

Authors: N. O. Alshehri; M. Akram; R. S. Al-ghamdi
Journal: Advances in Fuzzy Systems
 (2013)

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## Abstract

In 1999, Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty and vagueness. In this paper, we apply the concept of soft sets toK-algebras and investigate some properties of Abelian soft K-algebras. We also introduce the concept of soft intersection K-algebras and investigate some of their properties.


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## Body

## 1. Introduction

Most of the problems in engineering, medical science, economics, environments, and so forth, have various uncertainties. The problems in system identification involve characteristics which are essentially nonprobabilistic in nature. In response to this situation, Zadeh [1] 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 [2]. Molodtsov [3] initiated the concept of soft set theory as a new mathematical tool for dealing with uncertainties. In soft set theory, the problem of setting the membership function does not arise, which makes the theory easily applied to many different fields including game theory, operations research, Riemann integration, and Perron integration. At present, work on soft set theory is progressing rapidly. After Molodtsov’s work, some operations and application of soft sets were studied by many researchers including Ali et al. [4], Aktaş and Çağman [5], Chen et al. [6], and Maji et al. [7]. Maji et al. [7] gave first practical application of soft sets in decision making problems. To address decision making problems based on fuzzy soft sets, Feng et al. introduced the concept of soft level sets of fuzzy soft sets and initiated an adjustable decision making scheme using fuzzy soft sets [8]. It is interesting to see that soft sets are closely related to many other soft computing models such as rough sets and fuzzy sets. Feng et al. [9] first considered the combination of soft sets, fuzzy sets, and rough sets. Using soft sets as the granulation structures, Feng et al. [10] defined soft approximation spaces, soft rough approximations, and soft rough sets, which are generalizations of Pawlak’s rough set model based on soft sets. It has been proven that in some cases Feng’s soft rough set model could provide better approximations than classical rough sets. The algebraic structure of soft set theories has been studied increasingly in recent years. Aktaş and Çağman [5] defined the notion of soft groups. Feng et al. [11] initiated the study of soft semirings, and soft rings were defined by Acar et al. [12]. Jun [13] introduced soft BCK/BCI-algebras, and Kazancı et al. [14] introduced soft BCH-algebras. Along this direction, we apply soft set theory to K-algebras and investigate some of their properties. We introduce the notion of Abelian soft K-algebras and investigate some of their properties. We also introduce the concept of soft intersection soft K-algebras and investigate some of their properties.


## 2. Review of the Literature

In this section, we recall some basic concepts that are necessary for subsequent discussion ofK-algebras.The notion of aK-algebra was first introduced by Dar and Akram [15] in 2003 and published in 2005. A K-algebra is an algebra built on a group by adjoining an induced binary operation ⊙ on group which is attached to an abstract K-algebra. This system is, in general noncommutative and nonassociative with a right identity e, if group is noncommutative.Definition 1 (see [15]).
Let(S,·,e) be a group in which each nonidentity element is not of order 2. Then, a K-algebra is a structure 𝒦=(S,·,⊙,e) on a group S, in which induced binary operation ⊙:S×S→S is defined by ⊙(x,y)=x⊙y=x·y-1 and satisfies the following axioms: (K1)
(x⊙y)⊙(x⊙z)=(x⊙((e⊙z)⊙(e⊙y)))⊙x,(K2)
x⊙(x⊙y)=(x⊙(e⊙y))⊙x,(K3)
(x⊙x)=e,(K4)
(x⊙e)=x,(K5)
(e⊙x)=x-1 for all x, y, z∈S.Definition 2.
AK-algebra 𝒦 is called abelian if and only if x⊙(e⊙y)=y⊙(e⊙x) for all x, y∈S.
If aK-algebra 𝒦 is abelian, then the axioms (K1) and (K2) can be written as (K1¯)
(x⊙y)⊙(x⊙z)=z⊙y,(K2¯)
x⊙(x⊙y)=y. In what follows, we denote a K-algebra by 𝒦 unless otherwise specified.
A nonempty subsetH of a K-algebra 𝒦 is called a subalgebra [15] of the K-algebra 𝒦 if a⊙b∈H for all a, b∈H. Note that every subalgebra of a K-algebra 𝒦 contains the identity e of the group (G,·,e). Naturally, the mapping f:𝒦1→𝒦2 of K-algebras is called a homomorphism [16] if f(x⊙y)=f(x)⊙f(y) for all x,y∈𝒦1. We refer the readers to the book [17] and research papers [16–20] for further information regarding K-algebras.Soft Set Theory. In 1999, Molodtsov [3] initiated soft set theory as a new approach for modelling uncertainties. Later on, Maji et al. [21] expanded this theory to fuzzy soft set theory. Based on the idea of parametrization, a soft set gives a series of approximate descriptions of a complicate object from various different aspects. Each approximate description has two parts, namely, predicate and approximate value set. A soft set can be determined by a set-valued mapping assigning to each parameter exactly one crisp subset of the universe. More specifically, we can define the notion of soft set in the following way. Let U be an initial universe and E a set of parameters. Let P(U) denote the power set of U, and let A be a nonempty subset of E.Definition 3.
A pairFA=(F,A) is called a soft set over U, where A⊆E and F:A→P(U) is a set-valued mapping, called the approximate function of the soft set FA. It is easy to represent a soft set FA by a set of ordered pairs as follows:
(1)FA=(F,A)={(x,F(x))∣x∈A}.
It is clear to see that a soft set is a parameterized family of subsets of the setU. Hereafter unless stated otherwise, we always identify a soft set FA=(F,A) with the corresponding soft set (F,E) with F(e)=∅ for all e∈E∖A. In this fashion, we can implicitly view any soft set as a soft set with the whole parameter set E.Definition 4.
LetFA and GB be two soft sets over a common universe U. FA is a said to be a soft subset of GB, denoted by FA⊂~GB, if F(x)⊆G(x) for all x∈E.Definition 5.
The restricted intersection of a nonempty family soft sets{(Fi)Ai∣i∈Λ} over a common universe U is defined as the soft set HB=⊓~i∈Λ(Fi)Ai, where B=⋂i∈Λ‍Ai and H(x)=⋂x∈Λ‍Fi(x) for all x∈B.Definition 6.
The extended intersection of a nonempty family soft sets{(Fi)Ai∣i∈Λ} over a common universe U is defined as the soft set HB=⋂‍~i∈Λ(Fi)Ai, where B=⋃i∈Λ‍Ai and H(x)=⋂x∈Λ‍Fi(x), Λ(x)={i∣i∈Ai} for all x∈B.Definition 7.
The restricted union of a nonempty family soft sets{(Fi)Ai∣i∈Λ} over a common universe U is defined as the soft set HB=⋃‍~i∈Λ(Fi)Ai, where B=⋂i∈Λ‍Ai and H(x)=⋃x∈Λ‍Fi(x), Λ(x)={i∣i∈Ai} for all x∈B.Definition 8.
The∧-intersection of a nonempty family soft sets {(Fi)Ai∣i∈Λ} over a common universe U is defined as the soft set HB=⋀~i∈Λ(Fi)Ai, where B=∏i∈Λ‍Ai and H(x)=⋂x∈Λ‍Fi(x), Λ(x)={i∣i∈Ai} for all x∈B.Definition 9.
The∨-union of a nonempty family soft sets {(Fi)Ai∣i∈Λ} over a common universe U is defined as the soft set HB=⋁~i∈Λ(Fi)Ai, where B=∏i∈Λ‍Ai and H(x)=⋃x∈Λ‍Fi(x), Λ(x)={i∣i∈Ai} for all x∈B.Definition 10.
The Cartesian product of the nonempty family soft sets{(Fi)Ai∣i∈Λ} over a common universe U is defined as the soft set HB=∏‍~i∈Λ(Fi)Ai, where B=∏i∈Λ‍Ai and H(x)=∏x∈Λ‍Fi(x), Λ(x)={i∣i∈Ai} for all x∈B.Definition 11.
For a soft setFA, the set Supp FA={x∈A∣F(x)≠∅} is called the support of the soft set FA, and the soft set FA is called nonnull if Supp FA≠∅.
We refer the readers to the papers [22–24] for further information regarding the application of soft set theory.


## 3. Applications of Soft Sets inK-Algebras

If𝒦 is a K-algebra and A a nonempty set, a set-valued function F:A→𝒫(𝒦) can be defined by F(x)={y∈𝒦∣xRy}, x∈A, where R is an arbitrary binary relation from A to 𝒦; that is, R is a subset of A×𝒦 unless otherwise specified. The pair FA is then a soft set over 𝒦.Definition 12 (see [19]).
LetFA be a soft set over 𝒦. Then, FA is called a softK-algebra over 𝒦 if F(x) is a K-subalgebra of a K-algebra 𝒦 for all x∈A.Definition 13.
LetFA be a nonnull soft set over 𝒦. Then, FA is called a softK-algebra over 𝒦 if F(x) is a K-subalgebra of 𝒦 for all x∈ Supp FA.Example 14.
Consider theK-algebra 𝒦=(S3,·,⊙,e) on the symmetric group S3={e,a,b,x,y,z}, where e=(1), a=(123), b=(132), x=(12), y=(13), z=(23), and ⊙ are given by the following Cayley table:
(2)⊙exyzabeexyzbaxxeabzyyybeaxzzzabeyxaazxyebbbyzxae
Let FA be a soft set over 𝒦, where A=𝒦 and F:A→P(𝒦) is a set-valued function defined by F(e)={e}, F(a)=F(b)={e,a,b}, F(x)={e,x}, F(y)={e,y}, and F(z)={e,z} being K-subalgebras of 𝒦 for all x∈ Supp FA. Therefore, FA is a soft 𝒦-algebra over 𝒦.Example 15.
Consider theK-algebra 𝒦=(G,·,⊙,e) on the Dihedral group G={e,a,u,v,b,x,y,z}, where u=a2, v=a3, x=ab, y=a2b, z=a3b, and ⊙ are given by the following Cayley table:
(3)⊙eauvbxyzeevuabxyzaaevuxyzbuuaevyzbxvvuaezbxybbxyzevuaxxyzbaevuyyzbxuaevzzbxyvuae
Let FA be a soft set over 𝒦, where A=𝒦 and F:A→P(𝒦) is set-valued function defined by F(e)={e}, F(a)=F(v)={e,a,u,v}, F(u)={e,u}, F(x)={e,x}, F(y)={e,y}, and F(z)={e,z} being K-subalgebras of 𝒦. Therefore, FA is a soft K-algebra over 𝒦.Lemma 16.
LetFA be a soft K-algebra over 𝒦, then(i)
ifx∈F(x)⇒x-1∈F(x) for all x∈A,(ii)
ifa⊙b∈F(x)⇒b⊙a∈F(x) for all a,b∈A.Proposition 17.
Let{(Fi)Ai∣i∈Λ} be a nonempty family of soft K-algebras over 𝒦. Then, the bi-intersection ⊓~i∈Λ(Fi)Ai is a soft K-algebra over 𝒦 if it is nonnull.Proof.
Let{(Fi)Ai∣i∈Λ} be a nonempty family of soft K-algebras over 𝒦. By Definition 5, we can write ⊓~i∈Λ(Fi)Ai=HB, where B=⋂i∈Λ‍Ai and H(x)=⋂i∈Λ‍Fi(x) for all x∈B. Let x∈SuppHB. Then, ⋂i∈Λ‍Fi(x)≠∅, and so we have Fi(x)≠∅ for all i∈Λ. Since {(Fi)Ai∣i∈Λ} is a nonempty family of soft K-algebras over 𝒦, it follows that Fi(x) is a K-subalgebra of X for all i∈Λ, and its intersection is also a K-subalgebra of 𝒦; that is, H(x)=⋂i∈Λ‍Fi(x) is a K-subalgebra of 𝒦 for all x∈SuppHB. Hence, HB=⊓~i∈Λ(Fi)Ai is a soft K-algebra over 𝒦.Proposition 18.
Let{(Fi)Ai∣i∈Λ} be a nonempty family of soft K-algebras over 𝒦. Then, the extended intersection ⋂~i∈Λ(Fi)Ai is a soft K-algebra over 𝒦.Proof.
Let{(Fi)Ai∣i∈Λ} be a nonempty family of soft K-algebras over 𝒦. By Definition 6, we can write ⋂~i∈Λ(Fi)Ai=HB, where B=⋂i∈Λ‍Ai and H(x)=⋂i∈Λ‍Fi(x) for all x∈B. Let x∈SuppHB. Then, ⋂i∈Λ‍Fi(x)≠∅, and so we have Fi(x)≠∅ for all i∈Λ. Since {(Fi)Ai∣i∈Λ} is a nonempty family of soft K-algebras over 𝒦, it follows that Fi(x) is a K-subalgebra of X for all i∈Λ, and its intersection is also a K-subalgebra of 𝒦; that is, H(x)=⋂i∈Λ‍Fi(x) is a K-subalgebra of 𝒦 for all x∈SuppHB. Hence, HB=⋂~i∈Λ(Fi)Ai is a soft K-algebra over 𝒦.Proposition 19.
Let{(Fi)Ai∣i∈Λ} be a nonempty family of soft K-algebras over 𝒦. If Fi(xi)⊆Fj(xj) or Fj(xj)⊆Fi(xi) for all i, j∈Λ, xi∈Ai, then the restricted union ⋃‍~(Fi)Ai is a soft K-algebra over X.Proof.
Suppose that{(Fi)Ai∣i∈Λ} is a nonempty family of soft K-algebras over 𝒦. By Definition 7, we can write ⋃~i∈Λ(Fi)Ai=HB, where B=⋂i∈Λ‍Ai and H(x)=⋂i∈Λ‍Fi(x) for all x∈B. Let x∈SuppHB. Since SuppHB=⋃i∈Λ‍Supp(Fi)Ai≠∅, Fi0(x)≠∅ for some i0∈Λ. By assumption, ⋃i∈Λ‍Fi(x) is a K-subalgebra of 𝒦 for all x∈SuppHB. Hence, restricted union ⋃~i∈Λ(Fi)Ai is a soft K-algebra over 𝒦.Proposition 20.
Let{(Fi)Ai∣i∈Λ} be a nonempty family of soft K-algebras over 𝒦. Then, the ∧-intersection ⋀~i∈Λ(Fi)Ai is a soft K-algebra over 𝒦 if it is nonnull.Proof.
Let{(Fi)Ai∣i∈Λ} be a nonempty family of soft K-algebras over 𝒦. By Definition 11, we can write ⋀~i∈Λ(Fi)Ai=HB, where B=∏i∈Λ‍Ai and H(x)=⋂i∈Λ‍Fi(x) for all x=(xi)i∈∧∈B. Suppose that the soft set HB is nonempty. If x=(xi)i∈∧∈SuppHB, H(x)=⋂i∈Λ‍Fi(x)≠∅. Since {(Fi)Ai∣i∈Λ} is a nonempty family of soft K-algebras over 𝒦, nonempty set Fi(x) is a K-subalgebra of 𝒦 for all i∈Λ. It follows that H(x)=⋂i∈Λ‍Fi(x) is a K-subalgebra of 𝒦 for all x=(xi)i∈∧∈SuppHB. Hence ∧-intersection ⋀~i∈Λ(Fi)Ai is a soft K-algebra over 𝒦.Proposition 21.
Let{(Fi)Ai∣i∈Λ} be a nonempty family of soft K-algebras over 𝒦. If Fi(xi)⊆Fj(xj) or Fj(xj)⊆Fi(xi) for all i,j∈Λ, xi∈Ai, then the ∨-union ⋁~(Fi)Ai is a soft K-algebra over 𝒦.Proof.
Assume that{(Fi)Ai∣i∈Λ} is a nonempty family of soft K-algebras over 𝒦. By Definition 9, we can write ⋁~i∈Λ(Fi)Ai=HB, where B=∏i∈Λ‍Ai and H(x)=⋃i∈Λ‍Fi(x) for all x=(xi)i∈∧∈B. Let x=(xi)i∈∧∈SuppHB. Then, H(x)=⋃i∈Λ‍Fi(x)≠∅, so we have Fi0(xi0)≠∅ for some ii0∈Λ. By assumption, ⋃i∈Λ‍Fi(x) is a K-subalgebra of 𝒦 for all x=(xi)i∈∧∈SuppHB. Hence, ∨-union ⋁~(Fi)Ai is a soft K-algebra over 𝒦.Proposition 22.
Let{(Fi)Ai∣i∈Λ} be a nonempty family of soft K-algebras over 𝒦. Then, the cartesian product ∏~i∈Λ(Fi)Ai is a soft K-algebra over ∏i∈Λ‍𝒦i.Proof.
Let{(Fi)Ai∣i∈Λ} be a nonempty family of soft K-algebras over 𝒦. By Definition 10, we can write ∏~i∈Λ(Fi)Ai=HB, where B=∏i∈Λ‍Ai and H(x)=∏i∈Λ‍Fi(x) for all x=(xi)i∈∧∈B. Suppose that the soft set HB is nonnull. If x=(xi)i∈∧∈SuppHB, H(x)=∏i∈Λ‍Fi(x)≠∅. Since {(Fi)Ai∣i∈Λ} is a nonempty family of soft K-algebras over 𝒦, the nonempty set Fi(x) is a K-subalgebra of 𝒦 for all i∈Λ. It follows that H(x)=∏i∈Λ‍Fi(x) is a K-subalgebra of 𝒦 for all x=(xi)i∈∧∈SuppHB. Hence, ∧-intersection ∏~i∈Λ(Fi)Ai is a soft K-algebra over ∏i∈Λ‍𝒦i.Definition 23.
LetFA be a soft K-algebra over 𝒦. (i)
FA is called the trivial softK-algebra over 𝒦 if F(x)={e} for all x∈A.(ii)
FA is called the whole softK-algebra over 𝒦 if F(x)=𝒦 for all x∈A.Definition 24.
LetFA be a soft set over a K-algebra 𝒦. Then, the inverse of FA is denoted by FA-1 and is defined as follows FA-1={(F(a))-1:a∈A}, where (F(a))-1 is called the inverse of F(a) and is defined as (F(a))-1={x-1:x∈F(a)}Theorem 25.
LetFA and GB be any two soft sets over 𝒦. Then (FA∘^GB)-1=GB-1∘^FA-1.Theorem 26.
IfFA is a soft K-algebra over 𝒦, then FA-1=FA.The converse of above theorem is not true in general, and it can be seen in the following example.Example 27.
Consider theK-algebra K=(G,·,⊙,e) on the dihedral group G={e,a,u,v,b,x,y,z} which is given in Example 15. Let FA be a soft set over 𝒦, where A=𝒦 and F:A→P(𝒦) is a set-valued function defined by (F(e))-1={e}, (F(a))-1={e,a,u}, (F(u))-1={e,u}, (F(v))-1={e,v,u}, (F(b))-1={e,b), (F(x))-1={e,x}, (F(y))-1={e,y}, and (F(z))-1={e,z}. Therefore, we find that F(α)=(F(α))-1 for all α∈A. Hence FA-1=FA, but FA is not soft K-algebra over 𝒦 because F(a)={e,a,u} is not a K-subalgebra of K.Definition 28.
A softK-algebra FA over 𝒦 is said to be Abelian soft K-algebra over 𝒦 if each F(α) is an Abelian K-subalgebra of 𝒦 for all α∈A.Example 29.
LetFA be a soft K-algebra over 𝒦, which is given in Example 14. Then, it is easy to verify that each F(x) is an Abelian K-subalgebra of K for all x∈A. Hence, FA is an Abelian soft K-algebra over K.Definition 30.
LetFA be a soft K-algebra over 𝒦 and HB a soft K-subalgebra of FA. Then, we say that HB is an Abelian soft K-subalgebra of FA if H(x) is an Abelian K-subalgebra of F(x) for all x∈B.Example 31.
LetFA be a soft K-algebra over 𝒦 which is given in Example 14, and let HB be a soft set over 𝒦, where B=A3 and H:B→P(K) is the set-valued function defined by H(e)={e}, H(a)={e,a,b}, and H(b)={e,a,b} being Abelian K-subalgebras of F(e), F(a), and F(b), respectively. Hence, HB is an Abelian soft K-subalgebra of FA.Theorem 32.
LetFA be an Abelian soft K-algebra over 𝒦 and GB be a soft K-algebras over 𝒦. Then their restricted intersection FA⊓~GB is an Abelian soft K-algebra over 𝒦 for all c∈A∩B.Definition 33.
Let𝒦1, 𝒦2 be two K-algebras and φ:𝒦1→𝒦2 a mapping of K-algebras. If FA and GB are soft sets over 𝒦1 and 𝒦2, respectively, then φ(FA) is a soft set over 𝒦2 where φ(F):E→P(𝒦2) is defined by φ(F)(x)=φ(F(x)) for all x∈E and φ-1(GB) is a soft set over 𝒦1 where φ-1(G):E→P(𝒦1) is defined by φ-1(G)(y)=φ-1(G(y)) for all y∈E.Definition 34.
LetFA and HB be two soft sets over K-algebras 𝒦1 and 𝒦2, respectively, and let φ:𝒦1→𝒦2 and ϕ:A→B be two functions. Then, we say that (φ,ϕ) is a soft homomorphism, if the following conditions are satisfied: (i)
φ is a homomorphism from 𝒦1 onto 𝒦2;(ii)
ϕ is mapping from A onto B;(iii)
φ(F(x))=H(ϕ(x)).
In this definition, ifφ is an isomorphism 𝒦1 to 𝒦2 and ϕ is a one-to-one mapping from A on to B, then we say that (φ,ϕ) is a soft isomorphism and that FA is soft isomorphic to HB. Notation, FA≃HB.Example 35.
LetS={e,a,b,c} be a Klein four-group. Consider a K-algebra on G with the following Cayley table:
(4)⊙eabceeabcaaecbbbceaccbae
This is an improperK-algebra on Klein four-group since it is an elementary Abelian 2-group, that is, x⊙y=x·y-1=x·y. Let FA be a soft set over 𝒦, where A=𝒦 and F:A→P(𝒦); the set-valued function defined by F(x)={r∈𝒦:xRr⇔x⊙r∈Ax} where Ax={e,x,x-1}. Then, FA is a soft 𝒦-algebra over 𝒦. Let (G,A) be a soft set over 𝒦, where A=𝒦 and G:A→P(𝒦) is the set-valued function defined by G(x)={r∈𝒦:xRr⇔xn=r,n∈ℕ}. Then GA is a soft 𝒦-algebra over 𝒦. Let φ:𝒦→𝒦 be the mapping defined by φ(x)=x. It is clear that φis a 𝒦-homomorphism. Consider the mapping ϕ:A→A given by ϕ(x)=x3. Then, one can easily verify that φ(F(x))=G(ϕ(x)) for all x∈A. Hence, (φ,ϕ) is a soft homomorphism from 𝒦 to 𝒦.Example 36.
Consider theK-algebra K=(S3,·,⊙,e) which is given in Example 14. Let FA be a soft set over 𝒦, where A=A3⊂K and F:A→P(𝒦) is the set-valued function defined by F(x)={r∈𝒦:xRr⇔x⊙r∈Ax}, where Ax={e,x,x-1}. Then, FA is a soft K-algebra over 𝒦. Let GA be a soft set over 𝒦, where A=A3 and G:A→P(𝒦) is the set-valued function defined by G(x)={r∈𝒦:xRr⇔xn=r,n∈ℕ}. Then, GA is a soft K-algebra over 𝒦. Let f:→𝒦 be the mapping defined by φ(x)=x4. It is clear that φ is a K-homomorphism. Consider the mapping ϕ:A→A given by ϕ(x)=x. Then, one can easily verify that φ(F(x))=G(ϕ(x)) for all x∈A. Hence, (φ,ϕ) is a soft homomorphism from 𝒦 to 𝒦.
We state the following propositions without their proofs.Proposition 37.
Letf:𝒦1→𝒦2 be an onto homomorphism of K-algebras and FA, GB two soft K-algebras over 𝒦1 and 𝒦2, respectively. (i)
The soft function(φ,IA) from FA to (H,A) is a soft homomorphism from 𝒦1 to 𝒦2, where IA:A→A is the identity mapping and the set-valued function H:A→P(𝒦2) is defined by H(x)=φ(F(x)) for all x∈A.(ii)
Ifφ:𝒦1→𝒦2 is an isomorphism, then the soft function (f-1,IB) from GB to SB is a soft homomorphism from 𝒦2 to 𝒦1, where IB:B→B is the identity mapping and the set-valued function S:B→P(𝒦1) is defined by S(x)=φ-1(G(x)) for all x∈B.Proposition 38.
Let𝒦1, 𝒦2, and 𝒦3 be K-algebras and FA, GB and HC soft K-algebras over 𝒦1, 𝒦2, and 𝒦3, respectively. Let the soft function (φ,ϕ) from FA to GB be a soft homomorphism from 𝒦1 to 𝒦2, and the soft function (f′,g′) from GB to HC a soft homomorphism from 𝒦2 to 𝒦3. Then, the soft function (φ′∘φ,ϕ′∘ϕ) from FA to HC is a soft homomorphism from 𝒦1 to 𝒦3.Theorem 39.
Let𝒦1 and 𝒦2 be K-algebras and FA, GB soft sets over 𝒦1 and 𝒦2, respectively. If FA is a soft K-algebra over 𝒦1 and FA≃GB, then GB is a soft K-algebra over 𝒦2.Definition 40.
LetFA and HB be two soft K-algebras over 𝒦1 and 𝒦2, respectively. Then the Cartesian product of soft K-algebras FA and HB is denoted by FA×~HB=(U,A×B) and U is defined as U(a,b)=F(a)×H(b) for all (a,b)∈A×B.Theorem 41.
LetFA and HB be two soft K-algebras over 𝒦1 and 𝒦2, respectively. Then, (1) the Cartesian product FA×~HB is a soft K-algebra over 𝒦1×𝒦2, and (2) FA×~HB is soft isomorphic to HB×~FA.Proof.
We will prove second part. We show that(φ,ϕ):FA×~HB→HB×~FA is a soft isomorphism, that is, (φ,ϕ):(U,A×B)→(W,B×A) is a soft isomorphism where W(b,a) is defined as W(b,a)=H(b)×F(a). We prove the following three conditions:(i)
we show thatφ:𝒦1×𝒦2→𝒦2×𝒦1 is an isomorphism. Let φbe a function defined by φ(r,s)=(s,r). Then obviously φ is an isomorphism.(ii)
We now show thatϕ:A×B→B×A is a bijective mapping. The mapping ϕ is defined by ϕ(a,b)=(b,a) then obviously ϕ is a bijective mapping.(iii)
Consider(5)φ(U(a,b))=φ(F(a)×H(b))=φ({(r,s):r∈F(a),s∈H(b)})={(s,r):s∈H(b),r∈F(a)}=H(b)×F(a)=W(b,a)=W(g(a,b))for all (a,b)∈A×B. This implies that (φ,ϕ):FA×~HB→HB×~FA is a soft isomorphism. Hence, FA×~HB≃HB×~FA.We now introduce the concept of soft intersection softK-algebras and investigate some of their propertiesDefinition 42.
Let𝒦=E be a K-algebra and let A be a subset of 𝒦. Let FA be a soft set over U. Then, FA is called a soft intersectionK-subalgebra over U if it satisfies the following condition:
(6)F(x⊙y)⊇F(x)∩F(y)
for all x, y∈A.Example 43.
Assume thatU=ℤ is the universal set. Let A=G={e,a,a2} be the cyclic group of order 3. Then, (G,·,⊙,e) is a K-algebra 𝒦, and ⊙ is given by the following Cayley table:
(7)⊙eaa2eea2aaaea2a2a2ae
Let FA be a soft set over U. Then, F(e)=ℤ and F(a)=F(a2)={-2,-1,0,1,2}. It is easy to see that FA is a soft intersection K-subalgebra over U.
From now on, we will always assume that𝒦=E unless otherwise specified.Proposition 44.
Let𝒦 be a K-algebra and let A and B be K-subalgebras of 𝒦. If FA and GB are soft intersection K-subalgebras over U. Then, FA∧~GB is a soft intersection K-subalgebra over U, where FA∧~GB is defined by FA∧~GB(x,y)=F(x)∩G(y) for all (x,y)∈A×B.Proof.
Let(x1,y1),(x2,y2)∈A×B. Then,
(8)(FA∧~GB)((x1,y1)⊙(x2,y2))=(FA∧~GB)((x1⊙x2,y1⊙y2))=F(x1⊙x2)∩G(y1⊙y2)⊇(F(x1)∩F(x2))∩(G(y1)∩G(y2))=(F(x1)∩G(y1))∩(F(x2)∩G(y2))=(FA∧~GB)(x1,y1)∩(FA∧~GB)(x2,y2).
Hence, FA∧~GB is a soft intersection K-subalgebra over U.Theorem 45.
Let{(Fi)Ai∣i∈Λ} be a family of soft intersection K-subalgebras (resp., soft intersection ideals) over U. Then, ⋀~i∈Λ(Fi)Ai is a soft intersection K-subalgebra over U.Proposition 46.
Let𝒦 be a K-algebra and let A be a K-subalgebra of 𝒦. If FA and GA are soft intersection K-subalgebras over U. Then, FA∩~GA is a soft intersection K-algebra over U, where FA∩~GA is defined by FA∩~GA(x)=F(x)∩G(x) for all x∈A.Proof.
Letx,y∈A. Then,
(9)(FA∩~GA)(x⊙y)=F(x⊙y)∩G(x⊙y)⊇(F(x)∩F(y))∩(G(x)∩G(y))=(F(x)∩G(x))∩(F(y)∩G(y))=(FA∩~GA)(x)∩(FA∩~GA)(y).
Hence, FA∩~GA is a soft intersection K-subalgebras over U.Theorem 47.
Let{(Fi)Ai∣i∈Λ} be a family of soft intersection K-subalgebras over U. Then, ⋂‍~i∈Λ(Fi)Ai is a soft intersection K-subalgebra over U.Proposition 48.
Let𝒦 be a K-algebra and let A and B, be K-subalgebras of 𝒦. If FA and GB are soft intersection K-subalgebras over U. Then FA×~GB, is a soft intersection K-algebra over U, where FA×~GB is defined by FA×~GB(x,y)=F(x)×G(y) for all (x,y)∈A×B.Proof.
Let(x1,y1),(x2,y2)∈A×B. Then
(10)(FA×~GB)((x1,y1)⊙(x2,y2))=(FA×~GB)((x1⊙x2,y1⊙y2))=F(x1⊙x2)×G(y1⊙y2)⊇(F(x1)∩F(x2))×(G(y1)∩G(y2))=(F(x1)×G(y1))∩(F(x2)×G(y2))=(FA×~GB)(x1,y1)∩(FA×~GB)(x2,y2).
Hence,FA×~GB is a soft intersection K-subalgebras over U.Theorem 49.
Let{(Fi)Ai∣i∈Λ} be a family of soft intersection K-subalgebras over U. Then, ∏~i∈Λ(Fi)Ai is a soft intersection K-subalgebra over U.Proposition 50.
Let𝒦 be a K-algebra and let A, B and C be K-subalgebras of 𝒦. If FA, GB and FC are soft intersection K-subalgebras over U, FA≤~GB, and FC≤~GB, then FA∩~FC≤~GB over U.Proof.
Straightforward.Definition 51 (see [25]).
LetFA and GB be two soft sets over the common on universe U and let φ be a function from A to B. Then, soft image of FA under φ denoted by φ(FA) is a soft set over U by
(11)φ(F)(y)={⋃‍{F(x):x∈A,φ(x)=y}ifφ-1(y)≠∅,∅otherwise
for all y∈B, and soft preimage (or soft inverse image) of GB under φ denoted by φ-1(GB) is a soft set over U by φ-1(G)(x)=G(φ(x)) for all x∈A.Proposition 52.
Let𝒦 be a K-algebra and let A be ideal of 𝒦. If FA is a soft intersection K-subalgebra over U, then AF={x∈A:F(x)=F(e)} is a soft intersection K-subalgebra over U.Theorem 53.
Let𝒦 be a K-algebra and A and B ideals of 𝒦. Let φ be a 𝒦-homomorphism from A to B. If GB is soft intersection K-subalgebra over U, then φ-1(GB) is a soft intersection K-subalgebra over U.Proof.
Letx, y, z∈A and m∈K. Then,
(12)φ-1(GB)(x⊙y)=GB(φ(x⊙y))=GB(φ(x)⊙φ(y))⊇G(φ(x))∩G(φ(y))=φ-1(G)(x)∩φ-1(G)(y).
Hence, φ-1(GB) is a soft intersection K-subalgebra over U.Theorem 54.
Let𝒦 be a K-algebra and A and B ideals of 𝒦, and let φ be a 𝒦-isomorphism from A to B. If FA is a soft intersection K-subalgebra over U, then φ(FA) is a soft intersection K-subalgebra over U.Proof.
Sinceφ is surjective, there exist x, y∈A such that a=φ(x) and b=φ(y) for all a, b∈B. Then,
(13)φ(FA)(x⊙y)=∪{F(z):z∈A,φ(z)=a⊙b}=∪{F(x⊙y):x,y∈A,φ(x)=a,φ(y)=b}⊇∪{F(x)∩F(y):x,y∈A,φ(x)=a,φ(y)=b}=(∪{F(x):x∈A,φ(x)=a})∩(∪{F(y):y∈A,φ(y)=b})=φ(FA)(a)∩φ(FA)(b).
Hence, φ(FA) is a soft intersection K-subalgebra over U.


## 4. Conclusions

Presently, science and technology are featured with complex processes and phenomena for which complete information is not always available. For such cases, mathematical models are developed to handle various types of systems containing elements of uncertainty. A large number of these models are based on an extension of the ordinary set theory, namely, soft sets. In 1999, Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty and vagueness, and many researchers have created some models to solve problems in decision making and medical diagnosis. We have applied the concept of soft set theory toK-algebras and have investigated some of their properties. The natural extension of this research work is connected with the study of (i) fuzzy soft intersection K-algebras and (ii) roughness in K-algebras.



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