Applications of Soft Sets in 𝐾-Algebras

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 to K-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.

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 of 𝐾-algebras.

The notion of a 𝐾-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. A 𝐾-algebra 𝒦 is called abelian if and only if 𝑥⊙(𝑒⊙𝑦)=𝑦⊙(𝑒⊙𝑥) for all 𝑥, 𝑦∈𝑆.
If a 𝐾-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 subset 𝐻 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 pair 𝐹_𝐴=(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 𝐹_𝐴. It is easy to represent a soft set 𝐹_𝐴 by a set of ordered pairs as follows: 𝐹_𝐴=(𝐹,𝐴)={(𝑥,𝐹(𝑥))∣𝑥∈𝐴}. (1)
It is clear to see that a soft set is a parameterized family of subsets of the set 𝑈. Hereafter unless stated otherwise, we always identify a soft set 𝐹_𝐴=(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. Let 𝐹_𝐴 and 𝐺_𝐵 be two soft sets over a common universe 𝑈. 𝐹_𝐴 is a said to be a soft subset of 𝐺_𝐵, denoted by 𝐹_𝐴⊂𝐺_𝐵, if 𝐹(𝑥)⊆𝐺(𝑥) for all 𝑥∈𝐸.

Definition 5. The restricted intersection of a nonempty family soft sets {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} over a common universe 𝑈 is defined as the soft set 𝐻_𝐵=⊓_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖, where 𝐵=⋂_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=⋂_𝑥∈Λ𝐹_𝑖(𝑥) for all 𝑥∈𝐵.

Definition 6. The extended intersection of a nonempty family soft sets {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} over a common universe 𝑈 is defined as the soft set 𝐻_𝐵=⋂_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖, where 𝐵=⋃_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=⋂_𝑥∈Λ𝐹_𝑖(𝑥), Λ(𝑥)={𝑖∣𝑖∈𝐴_𝑖} for all 𝑥∈𝐵.

Definition 7. The restricted union of a nonempty family soft sets {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} over a common universe 𝑈 is defined as the soft set 𝐻_𝐵=⋃_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖, where 𝐵=⋂_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=⋃_𝑥∈Λ𝐹_𝑖(𝑥), Λ(𝑥)={𝑖∣𝑖∈𝐴_𝑖} for all 𝑥∈𝐵.

Definition 8. The ∧-intersection of a nonempty family soft sets {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} over a common universe 𝑈 is defined as the soft set 𝐻_𝐵=⋀_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖, where 𝐵=∏_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=⋂_𝑥∈Λ𝐹_𝑖(𝑥), Λ(𝑥)={𝑖∣𝑖∈𝐴_𝑖} for all 𝑥∈𝐵.

Definition 9. The ∨-union of a nonempty family soft sets {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} over a common universe 𝑈 is defined as the soft set 𝐻_𝐵=⋁_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖, where 𝐵=∏_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=⋃_𝑥∈Λ𝐹_𝑖(𝑥), Λ(𝑥)={𝑖∣𝑖∈𝐴_𝑖} for all 𝑥∈𝐵.

Definition 10. The Cartesian product of the nonempty family soft sets {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} over a common universe 𝑈 is defined as the soft set 𝐻_𝐵=∏_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖, where 𝐵=∏_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=∏_𝑥∈Λ𝐹_𝑖(𝑥), Λ(𝑥)={𝑖∣𝑖∈𝐴_𝑖} for all 𝑥∈𝐵.

Definition 11. For a soft set 𝐹_𝐴, the set Supp 𝐹_𝐴={𝑥∈𝐴∣𝐹(𝑥)≠∅} is called the support of the soft set 𝐹_𝐴, and the soft set 𝐹_𝐴 is called nonnull if Supp 𝐹_𝐴≠∅.
We refer the readers to the papers [22–24] for further information regarding the application of soft set theory.

3. Applications of Soft Sets in 𝐾-Algebras

If 𝒦 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 𝐹_𝐴 is then a soft set over 𝒦.

Definition 12 (see [19]). Let 𝐹_𝐴 be a soft set over 𝒦. Then, 𝐹_𝐴 is called a soft 𝐾-algebra over 𝒦 if 𝐹(𝑥) is a 𝐾-subalgebra of a 𝐾-algebra 𝒦 for all 𝑥∈𝐴.

Definition 13. Let 𝐹_𝐴 be a nonnull soft set over 𝒦. Then, 𝐹_𝐴 is called a soft 𝐾-algebra over 𝒦 if 𝐹(𝑥) is a 𝐾-subalgebra of 𝒦 for all 𝑥∈ Supp 𝐹_𝐴.

Example 14. Consider the 𝐾-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: ⊙𝑒𝑥𝑦𝑧𝑎𝑏𝑒𝑥𝑒𝑥𝑦𝑧𝑏𝑎𝑦𝑥𝑒𝑎𝑏𝑧𝑦𝑧𝑦𝑏𝑒𝑎𝑥𝑧𝑎𝑧𝑎𝑏𝑒𝑦𝑥𝑏𝑎𝑧𝑥𝑦𝑒𝑏𝑏𝑦𝑧𝑥𝑎𝑒 (2) Let 𝐹_𝐴 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 𝐹_𝐴. Therefore, 𝐹_𝐴 is a soft 𝒦-algebra over 𝒦.

Example 15. Consider the 𝐾-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: ⊙𝑒𝑎𝑢𝑣𝑏𝑥𝑦𝑧𝑒𝑎𝑒𝑣𝑢𝑎𝑏𝑥𝑦𝑧𝑢𝑎𝑒𝑣𝑢𝑥𝑦𝑧𝑏𝑣𝑢𝑎𝑒𝑣𝑦𝑧𝑏𝑥𝑏𝑣𝑢𝑎𝑒𝑧𝑏𝑥𝑦𝑥𝑏𝑥𝑦𝑧𝑒𝑣𝑢𝑎𝑦𝑥𝑦𝑧𝑏𝑎𝑒𝑣𝑢𝑧𝑦𝑧𝑏𝑥𝑢𝑎𝑒𝑣𝑧𝑏𝑥𝑦𝑣𝑢𝑎𝑒 (3) Let 𝐹_𝐴 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, 𝐹_𝐴 is a soft 𝐾-algebra over 𝒦.

Lemma 16. Let 𝐹_𝐴 be a soft 𝐾-algebra over 𝒦, then(i)if 𝑥∈𝐹(𝑥)⇒𝑥⁻¹∈𝐹(𝑥) for all 𝑥∈𝐴,(ii)if 𝑎⊙𝑏∈𝐹(𝑥)⇒𝑏⊙𝑎∈𝐹(𝑥) for all 𝑎,𝑏∈𝐴.

Proposition 17. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a nonempty family of soft 𝐾-algebras over 𝒦. Then, the bi-intersection ⊓_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over 𝒦 if it is nonnull.

Proof. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a nonempty family of soft 𝐾-algebras over 𝒦. By Definition 5, we can write ⊓_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖=𝐻_𝐵, where 𝐵=⋂_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=⋂_𝑖∈Λ𝐹_𝑖(𝑥) for all 𝑥∈𝐵. Let 𝑥∈Supp𝐻_𝐵. Then, ⋂_𝑖∈Λ𝐹_𝑖(𝑥)≠∅, and so we have 𝐹_𝑖(𝑥)≠∅ for all 𝑖∈Λ. Since {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} is a nonempty family of soft 𝐾-algebras over 𝒦, it follows that 𝐹_𝑖(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∈Λ𝐹_𝑖(𝑥) is a 𝐾-subalgebra of 𝒦 for all 𝑥∈Supp𝐻_𝐵. Hence, 𝐻_𝐵=⊓_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over 𝒦.

Proposition 18. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a nonempty family of soft 𝐾-algebras over 𝒦. Then, the extended intersection ⋂_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over 𝒦.

Proof. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a nonempty family of soft 𝐾-algebras over 𝒦. By Definition 6, we can write ⋂_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖=𝐻_𝐵, where 𝐵=⋂_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=⋂_𝑖∈Λ𝐹_𝑖(𝑥) for all 𝑥∈𝐵. Let 𝑥∈Supp𝐻_𝐵. Then, ⋂_𝑖∈Λ𝐹_𝑖(𝑥)≠∅, and so we have 𝐹_𝑖(𝑥)≠∅ for all 𝑖∈Λ. Since {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} is a nonempty family of soft 𝐾-algebras over 𝒦, it follows that 𝐹_𝑖(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∈Λ𝐹_𝑖(𝑥) is a 𝐾-subalgebra of 𝒦 for all 𝑥∈Supp𝐻_𝐵. Hence, 𝐻_𝐵=⋂_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over 𝒦.

Proposition 19. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a nonempty family of soft 𝐾-algebras over 𝒦. If 𝐹_𝑖(𝑥_𝑖)⊆𝐹_𝑗(𝑥_𝑗) or 𝐹_𝑗(𝑥_𝑗)⊆𝐹_𝑖(𝑥_𝑖) for all 𝑖, 𝑗∈Λ, 𝑥_𝑖∈𝐴_𝑖, then the restricted union ⋃(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over 𝑋.

Proof. Suppose that {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} is a nonempty family of soft 𝐾-algebras over 𝒦. By Definition 7, we can write ⋃_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖=𝐻_𝐵, where 𝐵=⋂_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=⋂_𝑖∈Λ𝐹_𝑖(𝑥) for all 𝑥∈𝐵. Let 𝑥∈Supp𝐻_𝐵. Since Supp𝐻_𝐵=⋃_𝑖∈ΛSupp(𝐹_𝑖)_𝐴𝑖≠∅, 𝐹_𝑖0(𝑥)≠∅ for some 𝑖₀∈Λ. By assumption, ⋃_𝑖∈Λ𝐹_𝑖(𝑥) is a 𝐾-subalgebra of 𝒦 for all 𝑥∈Supp𝐻_𝐵. Hence, restricted union ⋃_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over 𝒦.

Proposition 20. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a nonempty family of soft 𝐾-algebras over 𝒦. Then, the ∧-intersection ⋀_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over 𝒦 if it is nonnull.

Proof. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a nonempty family of soft 𝐾-algebras over 𝒦. By Definition 11, we can write ⋀_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖=𝐻_𝐵, where 𝐵=∏_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=⋂_𝑖∈Λ𝐹_𝑖(𝑥) for all 𝑥=(𝑥_𝑖)_𝑖∈∧∈𝐵. Suppose that the soft set 𝐻_𝐵 is nonempty. If 𝑥=(𝑥_𝑖)_𝑖∈∧∈Supp𝐻_𝐵, 𝐻(𝑥)=⋂_𝑖∈Λ𝐹_𝑖(𝑥)≠∅. Since {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} is a nonempty family of soft 𝐾-algebras over 𝒦, nonempty set 𝐹_𝑖(𝑥) is a 𝐾-subalgebra of 𝒦 for all 𝑖∈Λ. It follows that 𝐻(𝑥)=⋂_𝑖∈Λ𝐹_𝑖(𝑥) is a 𝐾-subalgebra of 𝒦 for all 𝑥=(𝑥_𝑖)_𝑖∈∧∈Supp𝐻_𝐵. Hence ∧-intersection ⋀_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over 𝒦.

Proposition 21. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a nonempty family of soft 𝐾-algebras over 𝒦. If 𝐹_𝑖(𝑥_𝑖)⊆𝐹_𝑗(𝑥_𝑗) or 𝐹_𝑗(𝑥_𝑗)⊆𝐹_𝑖(𝑥_𝑖) for all 𝑖,𝑗∈Λ, 𝑥_𝑖∈𝐴_𝑖, then the ∨-union ⋁(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over 𝒦.

Proof. Assume that {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} is a nonempty family of soft 𝐾-algebras over 𝒦. By Definition 9, we can write ⋁_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖=𝐻_𝐵, where 𝐵=∏_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=⋃_𝑖∈Λ𝐹_𝑖(𝑥) for all 𝑥=(𝑥_𝑖)_𝑖∈∧∈𝐵. Let 𝑥=(𝑥_𝑖)_𝑖∈∧∈Supp𝐻_𝐵. Then, 𝐻(𝑥)=⋃_𝑖∈Λ𝐹_𝑖(𝑥)≠∅, so we have 𝐹_𝑖0(𝑥_𝑖0)≠∅ for some 𝑖_𝑖0∈Λ. By assumption, ⋃_𝑖∈Λ𝐹_𝑖(𝑥) is a 𝐾-subalgebra of 𝒦 for all 𝑥=(𝑥_𝑖)_𝑖∈∧∈Supp𝐻_𝐵. Hence, ∨-union ⋁(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over 𝒦.

Proposition 22. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a nonempty family of soft 𝐾-algebras over 𝒦. Then, the cartesian product ∏_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over ∏_𝑖∈Λ𝒦_𝑖.

Proof. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a nonempty family of soft 𝐾-algebras over 𝒦. By Definition 10, we can write ∏_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖=𝐻_𝐵, where 𝐵=∏_𝑖∈Λ𝐴_𝑖 and 𝐻(𝑥)=∏_𝑖∈Λ𝐹_𝑖(𝑥) for all 𝑥=(𝑥_𝑖)_𝑖∈∧∈𝐵. Suppose that the soft set 𝐻_𝐵 is nonnull. If 𝑥=(𝑥_𝑖)_𝑖∈∧∈Supp𝐻_𝐵, 𝐻(𝑥)=∏_𝑖∈Λ𝐹_𝑖(𝑥)≠∅. Since {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} is a nonempty family of soft 𝐾-algebras over 𝒦, the nonempty set 𝐹_𝑖(𝑥) is a 𝐾-subalgebra of 𝒦 for all 𝑖∈Λ. It follows that 𝐻(𝑥)=∏_𝑖∈Λ𝐹_𝑖(𝑥) is a 𝐾-subalgebra of 𝒦 for all 𝑥=(𝑥_𝑖)_𝑖∈∧∈Supp𝐻_𝐵. Hence, ∧-intersection ∏_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft 𝐾-algebra over ∏_𝑖∈Λ𝒦_𝑖.

Definition 23. Let 𝐹_𝐴 be a soft 𝐾-algebra over 𝒦. (i)𝐹_𝐴 is called the trivial soft 𝐾-algebra over 𝒦 if 𝐹(𝑥)={𝑒} for all 𝑥∈𝐴. (ii)𝐹_𝐴 is called the whole soft 𝐾-algebra over 𝒦 if 𝐹(𝑥)=𝒦 for all 𝑥∈𝐴.

Definition 24. Let 𝐹_𝐴 be a soft set over a 𝐾-algebra 𝒦. Then, the inverse of 𝐹_𝐴 is denoted by 𝐹_𝐴⁻¹ and is defined as follows 𝐹_𝐴⁻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. Let 𝐹_𝐴 and 𝐺_𝐵 be any two soft sets over 𝒦. Then (𝐹_𝐴∘̂𝐺_𝐵)⁻¹=𝐺_𝐵⁻¹∘̂𝐹_𝐴⁻¹.

Theorem 26. If 𝐹_𝐴 is a soft 𝐾-algebra over 𝒦, then 𝐹_𝐴⁻¹=𝐹_𝐴.

The converse of above theorem is not true in general, and it can be seen in the following example.

Example 27. Consider the 𝐾-algebra 𝐾=(𝐺,⋅,⊙,𝑒) on the dihedral group 𝐺={𝑒,𝑎,𝑢,𝑣,𝑏,𝑥,𝑦,𝑧} which is given in Example 15. Let 𝐹_𝐴 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 𝐹_𝐴⁻¹=𝐹_𝐴, but 𝐹_𝐴 is not soft 𝐾-algebra over 𝒦 because 𝐹(𝑎)={𝑒,𝑎,𝑢} is not a 𝐾-subalgebra of 𝐾.

Definition 28. A soft 𝐾-algebra 𝐹_𝐴 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. Let 𝐹_𝐴 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, 𝐹_𝐴 is an Abelian soft 𝐾-algebra over 𝐾.

Definition 30. Let 𝐹_𝐴 be a soft 𝐾-algebra over 𝒦 and 𝐻_𝐵 a soft 𝐾-subalgebra of 𝐹_𝐴. Then, we say that 𝐻_𝐵 is an Abelian soft 𝐾-subalgebra of 𝐹_𝐴 if 𝐻(𝑥) is an Abelian 𝐾-subalgebra of 𝐹(𝑥) for all 𝑥∈𝐵.

Example 31. Let 𝐹_𝐴 be a soft 𝐾-algebra over 𝒦 which is given in Example 14, and let 𝐻_𝐵 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, 𝐻_𝐵 is an Abelian soft 𝐾-subalgebra of 𝐹_𝐴.

Theorem 32. Let 𝐹_𝐴 be an Abelian soft 𝐾-algebra over 𝒦 and 𝐺_𝐵 be a soft 𝐾-algebras over 𝒦. Then their restricted intersection 𝐹_𝐴⊓𝐺_𝐵 is an Abelian soft 𝐾-algebra over 𝒦 for all 𝑐∈𝐴∩𝐵.

Definition 33. Let 𝒦₁, 𝒦₂ be two 𝐾-algebras and 𝜑∶𝒦₁→𝒦₂ a mapping of 𝐾-algebras. If 𝐹_𝐴 and 𝐺_𝐵 are soft sets over 𝒦₁ and 𝒦₂, respectively, then 𝜑(𝐹_𝐴) 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(𝐺_𝐵) 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. Let 𝐹_𝐴 and 𝐻_𝐵 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, if 𝜑 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 𝐹_𝐴 is soft isomorphic to 𝐻_𝐵. Notation, 𝐹_𝐴≃𝐻_𝐵.

Example 35. Let 𝑆={𝑒,𝑎,𝑏,𝑐} be a Klein four-group. Consider a 𝐾-algebra on 𝐺 with the following Cayley table: ⊙𝑒𝑎𝑏𝑐𝑒𝑎𝑒𝑎𝑏𝑐𝑏𝑎𝑒𝑐𝑏𝑐𝑏𝑐𝑒𝑎𝑐𝑏𝑎𝑒 (4)
This is an improper 𝐾-algebra on Klein four-group since it is an elementary Abelian 2-group, that is, 𝑥⊙𝑦=𝑥⋅𝑦⁻¹=𝑥⋅𝑦. Let 𝐹_𝐴 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∈𝐴_𝑥} where 𝐴_𝑥={𝑒,𝑥,𝑥⁻¹}. Then, 𝐹_𝐴 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⇔𝑥^𝑛=𝑟,𝑛∈ℕ}. Then 𝐺_𝐴 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 the 𝐾-algebra 𝐾=(𝑆₃,⋅,⊙,𝑒) which is given in Example 14. Let 𝐹_𝐴 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∈𝐴_𝑥}, where 𝐴_𝑥={𝑒,𝑥,𝑥⁻¹}. Then, 𝐹_𝐴 is a soft 𝐾-algebra over 𝒦. Let 𝐺_𝐴 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⇔𝑥^𝑛=𝑟,𝑛∈ℕ}. Then, 𝐺_𝐴 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. Let 𝑓∶𝒦₁→𝒦₂ be an onto homomorphism of 𝐾-algebras and 𝐹_𝐴, 𝐺_𝐵 two soft 𝐾-algebras over 𝒦₁ and 𝒦₂, respectively. (i)The soft function (𝜑,𝐼_𝐴) from 𝐹_𝐴 to (𝐻,𝐴) is a soft homomorphism from 𝒦₁ to 𝒦₂, where 𝐼_𝐴∶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)If 𝜑∶𝒦₁→𝒦₂ is an isomorphism, then the soft function (𝑓⁻¹,𝐼_𝐵) from 𝐺_𝐵 to 𝑆_𝐵 is a soft homomorphism from 𝒦₂ to 𝒦₁, where 𝐼_𝐵∶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. Let 𝒦₁, 𝒦₂, and 𝒦₃ be 𝐾-algebras and 𝐹_𝐴, 𝐺_𝐵 and 𝐻_𝐶 soft 𝐾-algebras over 𝒦₁, 𝒦₂, and 𝒦₃, respectively. Let the soft function (𝜑,𝜙) from 𝐹_𝐴 to 𝐺_𝐵 be a soft homomorphism from 𝒦₁ to 𝒦₂, and the soft function (𝑓^′,𝑔^′) from 𝐺_𝐵 to 𝐻_𝐶 a soft homomorphism from 𝒦₂ to 𝒦₃. Then, the soft function (𝜑^′∘𝜑,𝜙^′∘𝜙) from 𝐹_𝐴 to 𝐻_𝐶 is a soft homomorphism from 𝒦₁ to 𝒦₃.

Theorem 39. Let 𝒦₁ and 𝒦₂ be 𝐾-algebras and 𝐹_𝐴, 𝐺_𝐵 soft sets over 𝒦₁ and 𝒦₂, respectively. If 𝐹_𝐴 is a soft 𝐾-algebra over 𝒦₁ and 𝐹_𝐴≃𝐺_𝐵, then 𝐺_𝐵 is a soft 𝐾-algebra over 𝒦₂.

Definition 40. Let 𝐹_𝐴 and 𝐻_𝐵 be two soft 𝐾-algebras over 𝒦₁ and 𝒦₂, respectively. Then the Cartesian product of soft 𝐾-algebras 𝐹_𝐴 and 𝐻_𝐵 is denoted by 𝐹_𝐴×𝐻_𝐵=(𝑈,𝐴×𝐵) and 𝑈 is defined as 𝑈(𝑎,𝑏)=𝐹(𝑎)×𝐻(𝑏) for all (𝑎,𝑏)∈𝐴×𝐵.

Theorem 41. Let 𝐹_𝐴 and 𝐻_𝐵 be two soft 𝐾-algebras over 𝒦₁ and 𝒦₂, respectively. Then, (1) the Cartesian product 𝐹_𝐴×𝐻_𝐵 is a soft 𝐾-algebra over 𝒦₁×𝒦₂, and (2) 𝐹_𝐴×𝐻_𝐵 is soft isomorphic to 𝐻_𝐵×𝐹_𝐴.

Proof. We will prove second part. We show that (𝜑,𝜙)∶𝐹_𝐴×𝐻_𝐵→𝐻_𝐵×𝐹_𝐴 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 that 𝜑∶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 that 𝜙∶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 𝜑(𝑈((𝐹(𝑎)(𝑏)(𝑎)(𝑏=)(𝑏)(𝑎(𝑏)(𝑎)()𝑔()𝑎,𝑏))=𝜑×𝐻))=𝜑({(𝑟,𝑠∶𝑟∈𝐹,𝑠∈𝐻)}){(𝑠,𝑟∶𝑠∈𝐻,𝑟∈𝐹)}=𝐻×𝐹=𝑊𝑏,𝑎=𝑊𝑎,𝑏 (5)for all (𝑎,𝑏)∈𝐴×𝐵. This implies that (𝜑,𝜙)∶𝐹_𝐴×𝐻_𝐵→𝐻_𝐵×𝐹_𝐴 is a soft isomorphism. Hence, 𝐹_𝐴×𝐻_𝐵≃𝐻_𝐵×𝐹_𝐴.

We now introduce the concept of soft intersection soft 𝐾-algebras and investigate some of their properties

Definition 42. Let 𝒦=𝐸 be a 𝐾-algebra and let 𝐴 be a subset of 𝒦. Let 𝐹_𝐴 be a soft set over 𝑈. Then, 𝐹_𝐴 is called a soft intersection 𝐾-subalgebra over 𝑈 if it satisfies the following condition: 𝐹𝑥⊙𝑦⊇𝐹(𝑥)∩𝐹𝑦 (6) for all 𝑥, 𝑦∈𝐴.

Example 43. Assume that 𝑈=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: ⊙𝑒𝑎𝑎²𝑒𝑒𝑎²𝑎𝑎𝑎𝑒𝑎²𝑎²𝑎²𝑎𝑒 (7) Let 𝐹_𝐴 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 𝐹_𝐴 is a soft intersection 𝐾-subalgebra over 𝑈.
From now on, we will always assume that 𝒦=𝐸 unless otherwise specified.

Proposition 44. Let 𝒦 be a 𝐾-algebra and let 𝐴 and 𝐵 be 𝐾-subalgebras of 𝒦. If 𝐹_𝐴 and 𝐺_𝐵 are soft intersection 𝐾-subalgebras over 𝑈. Then, 𝐹_𝐴∧𝐺_𝐵 is a soft intersection 𝐾-subalgebra over 𝑈, where 𝐹_𝐴∧𝐺_𝐵 is defined by 𝐹_𝐴∧𝐺_𝐵(𝑥,𝑦)=𝐹(𝑥)∩𝐺(𝑦) for all (𝑥,𝑦)∈𝐴×𝐵.

Proof. Let (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝐴×𝐵. Then, 𝐹_𝐴∧𝐺_𝐵𝑥₁,𝑦₁⊙𝑥₂,𝑦₂=𝐹_𝐴∧𝐺_𝐵𝑥₁⊙𝑥₂,𝑦₁⊙𝑦₂=𝐹𝑥₁⊙𝑥₂∩𝐺𝑦₁⊙𝑦₂⊇𝐹𝑥₁∩𝐹𝑥₂∩𝐺𝑦₁∩𝐺𝑦₂=𝐹𝑥₁∩𝐺𝑦₁∩𝐹𝑥₂∩𝐺𝑦₂=𝐹_𝐴∧𝐺_𝐵𝑥₁,𝑦₁∩𝐹_𝐴∧𝐺_𝐵𝑥₂,𝑦₂. (8) Hence, 𝐹_𝐴∧𝐺_𝐵 is a soft intersection 𝐾-subalgebra over 𝑈.

Theorem 45. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a family of soft intersection 𝐾-subalgebras (resp., soft intersection ideals) over 𝑈. Then, ⋀_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft intersection 𝐾-subalgebra over 𝑈.

Proposition 46. Let 𝒦 be a 𝐾-algebra and let 𝐴 be a 𝐾-subalgebra of 𝒦. If 𝐹_𝐴 and 𝐺_𝐴 are soft intersection 𝐾-subalgebras over 𝑈. Then, 𝐹_𝐴∩𝐺_𝐴 is a soft intersection 𝐾-algebra over 𝑈, where 𝐹_𝐴∩𝐺_𝐴 is defined by 𝐹_𝐴∩𝐺_𝐴(𝑥)=𝐹(𝑥)∩𝐺(𝑥) for all 𝑥∈𝐴.

Proof. Let 𝑥,𝑦∈𝐴. Then, 𝐹_𝐴∩𝐺_𝐴⊇𝐹(𝑥)𝑦∩𝐺(𝑥)𝑦=(𝐹(𝑥)(𝑥∩𝐹𝑦𝑦=𝐹𝑥⊙𝑦=𝐹𝑥⊙𝑦∩𝐺𝑥⊙𝑦∩𝐹∩𝐺∩𝐺))∩𝐺_𝐴∩𝐺_𝐴(𝑥)∩𝐹_𝐴∩𝐺_𝐴𝑦. (9) Hence, 𝐹_𝐴∩𝐺_𝐴 is a soft intersection 𝐾-subalgebras over 𝑈.

Theorem 47. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a family of soft intersection 𝐾-subalgebras over 𝑈. Then, ⋂_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft intersection 𝐾-subalgebra over 𝑈.

Proposition 48. Let 𝒦 be a 𝐾-algebra and let 𝐴 and 𝐵, be 𝐾-subalgebras of 𝒦. If 𝐹_𝐴 and 𝐺_𝐵 are soft intersection 𝐾-subalgebras over 𝑈. Then 𝐹_𝐴×𝐺_𝐵, is a soft intersection 𝐾-algebra over 𝑈, where 𝐹_𝐴×𝐺_𝐵 is defined by 𝐹_𝐴×𝐺_𝐵(𝑥,𝑦)=𝐹(𝑥)×𝐺(𝑦) for all (𝑥,𝑦)∈𝐴×𝐵.

Proof. Let (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝐴×𝐵. Then 𝐹_𝐴×𝐺_𝐵𝑥₁,𝑦₁⊙𝑥₂,𝑦₂=𝐹_𝐴×𝐺_𝐵𝑥₁⊙𝑥₂,𝑦₁⊙𝑦₂=𝐹𝑥₁⊙𝑥₂×𝐺𝑦₁⊙𝑦₂⊇𝐹𝑥₁∩𝐹𝑥₂×𝐺𝑦₁∩𝐺𝑦₂=𝐹𝑥₁×𝐺𝑦₁∩𝐹𝑥₂×𝐺𝑦₂=𝐹_𝐴×𝐺_𝐵𝑥₁,𝑦₁∩𝐹_𝐴×𝐺_𝐵𝑥₂,𝑦₂. (10)
Hence, 𝐹_𝐴×𝐺_𝐵 is a soft intersection 𝐾-subalgebras over 𝑈.

Theorem 49. Let {(𝐹_𝑖)_𝐴𝑖∣𝑖∈Λ} be a family of soft intersection 𝐾-subalgebras over 𝑈. Then, ∏_𝑖∈Λ(𝐹_𝑖)_𝐴𝑖 is a soft intersection 𝐾-subalgebra over 𝑈.

Proposition 50. Let 𝒦 be a 𝐾-algebra and let 𝐴, 𝐵 and 𝐶 be 𝐾-subalgebras of 𝒦. If 𝐹_𝐴, 𝐺_𝐵 and 𝐹_𝐶 are soft intersection 𝐾-subalgebras over 𝑈, 𝐹_𝐴≤𝐺_𝐵, and 𝐹_𝐶≤𝐺_𝐵, then 𝐹_𝐴∩𝐹_𝐶≤𝐺_𝐵 over 𝑈.

Proof. Straightforward.

Definition 51 (see [25]). Let 𝐹_𝐴 and 𝐺_𝐵 be two soft sets over the common on universe 𝑈 and let 𝜑 be a function from 𝐴 to 𝐵. Then, soft image of 𝐹_𝐴 under 𝜑 denoted by 𝜑(𝐹_𝐴) is a soft set over 𝑈 by 𝜑(𝐹𝑦=⋃𝐹(𝑥)(𝑥))∶𝑥∈𝐴,𝜑=𝑦if𝜑⁻¹𝑦∅≠∅,otherwise (11) for all 𝑦∈𝐵, and soft preimage (or soft inverse image) of 𝐺_𝐵 under 𝜑 denoted by 𝜑⁻¹(𝐺_𝐵) is a soft set over 𝑈 by 𝜑⁻¹(𝐺)(𝑥)=𝐺(𝜑(𝑥)) for all 𝑥∈𝐴.

Proposition 52. Let 𝒦 be a 𝐾-algebra and let 𝐴 be ideal of 𝒦. If 𝐹_𝐴 is a soft intersection 𝐾-subalgebra over 𝑈, then 𝐴_𝐹={𝑥∈𝐴∶𝐹(𝑥)=𝐹(𝑒)} is a soft intersection 𝐾-subalgebra over 𝑈.

Theorem 53. Let 𝒦 be a 𝐾-algebra and 𝐴 and 𝐵 ideals of 𝒦. Let 𝜑 be a 𝒦-homomorphism from 𝐴 to 𝐵. If 𝐺_𝐵 is soft intersection 𝐾-subalgebra over 𝑈, then 𝜑⁻¹(𝐺_𝐵) is a soft intersection 𝐾-subalgebra over 𝑈.

Proof. Let 𝑥, 𝑦, 𝑧∈𝐴 and 𝑚∈𝐾. Then, 𝜑⁻¹𝐺_𝐵𝑥⊙𝑦=𝐺_𝐵𝜑𝑥⊙𝑦=𝐺_𝐵𝜑(𝑥)𝑦𝜑(𝑥𝜑𝑦⊙𝜑⊇𝐺)∩𝐺=𝜑⁻¹(𝐺)(𝑥)∩𝜑⁻¹(𝐺)𝑦. (12) Hence, 𝜑⁻¹(𝐺_𝐵) is a soft intersection 𝐾-subalgebra over 𝑈.

Theorem 54. Let 𝒦 be a 𝐾-algebra and 𝐴 and 𝐵 ideals of 𝒦, and let 𝜑 be a 𝒦-isomorphism from 𝐴 to 𝐵. If 𝐹_𝐴 is a soft intersection 𝐾-subalgebra over 𝑈, then 𝜑(𝐹_𝐴) is a soft intersection 𝐾-subalgebra over 𝑈.

Proof. Since 𝜑 is surjective, there exist 𝑥, 𝑦∈𝐴 such that 𝑎=𝜑(𝑥) and 𝑏=𝜑(𝑦) for all 𝑎, 𝑏∈𝐵. Then, 𝜑𝐹_𝐴𝐹(𝑧)(𝑧)𝐹𝜑(𝑥)𝑦𝐹(𝑥)𝑦𝜑(𝑥)𝑦=∪𝐹(𝑥)(𝑥)∩∪𝐹𝑦𝑦𝐹𝑥⊙𝑦=∪∶𝑧∈𝐴,𝜑=𝑎⊙𝑏=∪𝑥⊙𝑦∶𝑥,𝑦∈𝐴,=𝑎,𝜑=𝑏⊇∪∩𝐹∶𝑥,𝑦∈𝐴,=𝑎,𝜑=𝑏∶𝑥∈𝐴,𝜑=𝑎∶𝑦∈𝐴,𝜑=𝑏=𝜑_𝐴(𝑎)∩𝜑𝐹_𝐴(𝑏). (13) Hence, 𝜑(𝐹_𝐴) 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 to 𝐾-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.

Acknowledgment

The authors are highly thankful to the referees for their valuable comments and suggestions.