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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(3): 276-286

Published online September 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.3.276

© The Korean Institute of Intelligent Systems

Soft Sub-nearsemirings, Soft Ideals and Soft -Subsemigroups of Nearsemirings

Waheed Ahmad Khan1 and Abdelghani Taouti2

1Department of Mathematics, University of Education Lahore, Attock Campus, Pakistan
2ETS-Maths and NS Engineering Division, HCT, Sharjah, United Arab Emirates

Correspondence to :

Received: March 15, 2021; Revised: July 13, 2022; Accepted: August 19, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this note, we introduce and discuss the notions of soft sub-nearsemirings, soft ideals and soft S-subsemigroups of nearsemirings. Some related properties and characterizations of these soft algebraic structures are discussed with illustrative examples. For the sake of investigations, we apply several operations of soft sets including soft intersection sum, soft product and soft uniint product. Based on these operations, we also discuss few characterizations of distributively generated nearsemirings. In due course, we investigate few relationships among these soft algebraic structures and the classical nearsemirings. We also introduce the notion of soft ideals (left and right) of nearsemirings. Firstly, we investigate these ideals by applying few operations on them. Then, we present the relationship between these soft ideals of nearsemiring with the classical ideals of nearsemirings. Moreover, we introduce the notion of soft S-homomorphism between two soft S-subsemigroups and investigate that the homomorphic image of soft S-subsemigroup is a soft S-subsemigroup. Throughout, we shift several substructures of nearsemirings towards the soft algebraic substructures of nearsemirings by utilizing different algebraic methods. Consequently, we explore a linkage among the soft set theory, classical set theory and nearsemiring theory. Mainly, our study is the interplay between soft substructures of nearsemirings and classical substructures of nearsemirings.

Keywords: Nearsemirings, Soft sub-nearsemirings, Soft S-subsemigroups, Uni-int products

Soft set theory was initiated in 1999 by Molodtsov [1] and the fuzzy set theory was initiated by Zadeh [2]. Actually, every hesitant fuzzy set introduced by Zadeh [2] can be considered a soft set on a common universe. Similarly, every soft set on a denumerable universe can be considered a fuzzy set. However, soft sets theory become an effective tool to deal with uncertainties in the given data. Numerous applications of soft sets have been explored in [3,4]. A lot of work has been done on the soft set theory and is developing rapidly. Many applications of soft sets have been explored towards data analysis, decision-making theory etc. A number of operations on soft sets have been introduced and discussed in the literature [58]. Among the others, the operation uni-int has its own importance due to its applications towards decision-making [9]. The generalized form of uni-int operation and its application in decision-making theory was presented in [10]. Moreover, Han and Geng [11] introduced some special method based on intm-intn and created a decision-making scheme. Number of algebraists introduced several soft algebraic structures such as soft groups [5], soft rings [12], soft fields [13], soft nearrings [14], soft semirings [15] and so on. Afterwards, a new view of soft intersection rings based on uni-int product has been explored in [16]. Soft subnearrings, soft ideals and soft N-subgroups of nearrings have been introduced in [17]. Notion of soft intersection h-ideals of hemirings have been introduced in [18]. Khan et al. [19] introduced the notion of soft nearsemirings. The concepts of soft intersection nearsemirings [20] and (M,N)-soft intersection nearsemirings were also introduced by Khan and Davvaz [21]. The notion of (α, β)-soft intersectional rings and ideals along with their applications have been explored in [22]. Recently, the notions of fuzzy sub-nearsemirings and fuzzy soft sub-nearsemirings [23] have been initiated by both the authors. In a sequel, we introduce and discuss the notions of soft sub-nearsemirings, soft ideals and soft S-subsemigroups of a nearsemiring S. We also provide few characterizations of distributively generated nearsemirings in the frame of the soft sets theory. Throughout our study, we interrelate several classical substructures of nearsemirings with the soft substructures of nearsemirings.

Near-semiring is the common generalization of nearrring and semiring. Basically, nearsemiring R is an algebraic structure equipped with two binary operations “+” and “・” such that (R, +) is a monoid, (R, ・) is a semigroup, and both structures are joined through a single (left or right) distributive law with 0 is the one sided absorbing element. Moreover, if 0 ∈ R such that a + 0 = 0+a = a, a ・ 0 = 0 ・ a = 0, then we say R is a zero-symmetric nearsemiring (or seminearring). Nearsemirings ascertained naturally from the mappings on monoid to itself under component-wise addition and composition of mappings. Due to having strong relationships of nearsemirings with different types of algebras, recently Chajda and Laenger [24] introduced the notion of balanced nearsemirings. Since the ideals have their own importance in algebra particularly when we study rings with two binary operations. In general, the ideals of nearsemirings are not similar to that of the standard ideals of rings (resp., nearrings, semirings) and hence many results in rings (resp., nearrings, semirings) theory have no similarities in nearsemirings using merely ideals. Hence the classified notion of ideal named S-ideal [25] has been initiated in seminearring (zero symmetric nearsemirings) theory. Some special types of prime ideals of seminearrings have been explored in [26].

On the other hands, Nearrings, semirings and nearsemirings have a number of applications in computer technology such as theory of automata, languages and machine learning. Variety of applications of semirings have been explored in [27,28]. In [28], the author explored various applications of nearrings, particularly, he extended the linear sequential machines of Eilenberg by using the theory of nearrings. Likewise, Krishna and Chatterjee [30] introduced the condition of minimality of generalized linear sequential machines at the base of nearsemirings.

This work comprises of six sections. In Section 2, we include necessary discussions about nearsemirings and useful terminologies about soft set theory. In Section 3, we introduce and discuss the notions of soft sub-nearsemirings and soft ideals of nearsemiring S. We apply a number of usual operations including soft uni-int products to these soft algebraic structures. Moreover, we investigate these soft structures by applying image, pre-image, α-inclusion mappings. We also provide few characterizations of distributively generated nearsemirings in the setting of soft set theory. In Section 4, we provide some relationships of newly established soft structures with the existing classical algebraic structures as applications. In section 5, we introduce the notions of soft S-subsemigroup and soft S-morphism between two soft S-subsemigroups of nearsemirings. Throughout our discussions, we examine the interplay of classical nearsemiring and its substructures with the soft nearsemirings and their soft substructures.

Following [31], we call a system (S, +, .) is a left(right) nearsemiring if Sis a monoid with respect to addition, semigroup with respect to multiplication and obeying left (right) distributive law. Furthermore, if there exists 0 ∈ S such that 0.a = a.0 = 0, then we call it a zero-symmetric nearsemiring. If S and S′ are two nearsemirings, then we call a mapping φ: SS′ a nearsemiring homomorphism if for all a, bS; φ(a + b) = φ(a) + φ(b), φ(ab) = φ(a)φ(b), φ(0) = 0. Similarly, if S and S′ are two nearsemirings, a mapping ψ: SS′ is the anti-homomorphism if for all a, bS; φ(a+b) = φ(a)+ φ(b); φ(ab) = φ(b)φ(a); φ(0) = 0. The kernel of semigroup homomorphism is said to be an ideal of nearsemiring R. If S is a nearsemiring, then we call a semigroup (Γ, +) having zero 0Γ a left S-semigroup if there exists a composition (x, γ) ↣ of S × Γ satisfying: (i) (x + y)γ = + ; (ii) (xy)γ = x(); and (iii) 0Sγ = 0Γ, for all x, yS and γ ∈ Γ. It is obvious that Γ is an S-semigroup with S = M(Γ), a nearsemiring consists of all mappings from Γ to itself. Also, a semigroup (S, +) of a nearsemiring (S, +, .) is also an S-semigroup. If Γ and Γ′ are two S-semigroups of a nearsemiring S. A mapping φ: Γ → Γ′ is said to be an S-morphism between S-semigroups Γ and Γ′ if it satisfies: (i) φ(x + y) = φ(x)+ φ(y), (ii) φ(rx) = (x), for all x, y ∈ Γ, rS and (iii) φ(0Γ) = 0Γ′. We refer [3133] for further discussion on nearsemirings and S-semigroups of nearsemirings. Throughout, S represents a nearsemiring, U an initial universe, E the possible parameters associated with the elements in U, ℘(U) represents the family of subsets of U, S(U) the collection of soft sets and A ≠ ∅ be a subset of E.

Definition 1 ([1])

A η be a mapping from A to ℘(U). Then the pair (η, A) is said to be soft set over the universe U.

Definition 2 ([9])

Intersection of ηA and ηB represented by ηA ∩̃ ηB, defined by ηA ∩̃ ηB = ηA∩̃B, where ηA∩̃B (x) = ηA(x) ∩ ηB(x), for all xE.

Definition 3 ([9])

The ∧-product of ηA and ηB of two soft sets over U i.e., ηAηB is the mapping ηAB : E × E → ℘(U), and ηAB(x, y) = ηA(x) ∩ ηB(y).

Definition 4 ([9])

The ∨-product of two soft sets ηA and ηB is defined by ηAB : E × E → ℘(U) where ηAB(x, y) = ηA(x) ∪ ηB(y).

Definition 5 ([34])

Let Ψ be a function from A to B and ηA, ηBS(U). Then, the soft subsets Ψ(ηA) ∈ S(U) and Ψ−1(ηB) ∈ S(U) defined as

Ψ(ηA)(y)={{ηA(x):xA,Ψ(x)=y},if yΨ(A),if yΨ(A).

for all yB and Ψ−1 (ηB)(x) = ηB(Ψ(x)), for all xA. Then Ψ(ηA) is the soft image of ηA and Ψ−1 (ηB) is the soft pre-image (or soft inverse image) of ηB under the napping Ψ.

Definition 6 ([34])

Let ηA be a soft set over U and α be a subset of U. Then upper α-inclusion of soft set ηA is defined by ηAα={xA:ηA(x)α}.

Definition 7 ([19])

Let (ζ, A) be a non-null soft set over a nearsemiring R. Then, (ζ, A) is called a soft nearsemiring over R, if ζ(x) is a sub-nearsemiring of R for all xSupp(ζ, A).

Definition 8 ([19])

Let (η, A) be the soft right nearsemiring over right nearsemiring R1 and (ζ, B) be the soft left nearsemiring over the left nearsemiring R2. Let f : R1R2 and g : AB be the two mappings. Then, the pair (f, g) is said to be a soft nearsemiring anti-homomorphism if,

• (i) f is an anti-epimorphism of nearsemirings.

• (ii) g is Onto mapping.

• (iii) f(η(x)) = ζ(g(x)) for all xA.

Now, if f is an anti-isomorphism andg is bijective then we say (f, g) a soft nearsemiring anti-isomorphism.

In this section, we initiate the notions of soft sub-nearsemirings and soft ideals of nearsemiring by using intersection operation of soft sets. For the sake of investigations, we apply several operations of soft set theory to them. We also provide few characterizations of distributively generated nearsemirings related to these soft substructures of nearsemirings.

Definition 9

Let (η, T) be a soft set over S, where T be a sub-nearsemiring of S. Then the soft set (η, T) is said to be a soft sub-nearsemiring of S if it satisfies the followings.

• (i) η(x + y) ⊇ η(x) ∩ η(y)

• (ii) η(xy) ⊇ η(x) ∩ η(y)

for all x, yT. We denote it by (η, T) ⪝ S or ηTS.

Corollary 1

It is easy to verify that if ηT (x) = S for all xT, then ηT is a soft sub-nearsemiring of S. We will abbreviate such a soft sub-nearsemiring by η.

Example 1

Let S = {0, a, b, c, d} be a (right) nearsemiring with the operations defined by the following Cayley tables.

Clearly, T = {0, a, b} is a sub-nearsemiring of nearsemiring S. For T = {0, a, b}, let η : T → ℘(S) be a set-valued function defined by η(0) = S, η(a) = {0, b, c, d}, η(b) = {0, a}. It is easy to verify that (η, T) is a soft sub-nearsemiring of S. On the other hand, if we take T1 = {0, b, c, d} a sub-nearsemiring of S, and ϑ : T1 → ℘(S) be a set-valued function defined by ϑ(0) = {0, a, b, d}, ϑ(b) = ϑ(d) = {0, b, d}, ϑ(c) = {0, a}. Then ϑ(c.c) = ϑ(b) = {0, b, d} ⊉ ϑ(c) ∩ ϑ(c) = {0, a}. Thus, (ϑ, T1) is not a soft sub-nearsemiring of S.

If ηTS and ηT1S, then we can easily deduce the followings.

Proposition 1

• If ηTS and ηT1S, then ηT ∩̃ ηT1S

• If ηT1S1 and ϑT2S2, then ηT1 × ϑT2S1 × S2.

• If ηTS and ηT1S, then ηTηT1S.

Proof

Easy to proof (1), (2) & (3) and hence omitted.

On the other hand, if ηS and ηT are two soft sub-nearsemirings of a nearsemiring R, then ηSηT need not be a soft sub-nearsemiring as we see in the below example.

Example 2

Let S = {0, a, b, c, d} be a (right) nearsemiring with the operations defined by the following Cayley tables.

Clearly, T = {0, a, b} is a sub-nearsemiring of nearsemiring S, and let η : T → ℘(S) be a set-valued function defined by η(0) = S, η(a) = {0, b, c, d}, η(b) = {0, a}. It is easy to verify that (η, T) is a soft sub-nearsemiring of S. Again by taking a sub-nearsemiring T1 = {0, a, b, d}, and a set-valued function ϑ : T1 → ℘(S) defined by ϑ(0) = S, ϑ(a) = {0, a, b, d}, ϑ(b) = ϑ(d) = {0, b, d}. Then, one can easily show that (ϑ, T1) is a soft sub-nearsemiring of S. Consider, (ηTϑT1)((b, 0) + (0, a)) = (ηTϑT1) (b, a) = {0, a, b, d} ⊉ (ηTϑT1)(b, 0) ∩ (ηTϑT1)(0, a) = {0, a, b, c, d} ∩ {0, a, b, c, d} = {0, a, b, c, d}. Hence, ηTϑT1 is not a soft sub-nearsemiring.

The following lemma is obvious.

Lemma 1

• If S is distributively generated nearsemiring and T be its sub-nearsemiring. If tT, such that t=i=1naibi and ηT be a soft sub-nearsemiring of S, then ηT(t)=ηT(i=1naibi)ηT(ai)ηT(bi) for all 1 ≤ in.

• If S is an additively commutative nearsemiring and t=i=1nai+bi such that ηT soft sub-nearsemiring of S. Then, ηT(t)=ηT(i=1nai+bi)ηT(ai)ηT(bi) for all 1 ≤ in.

Now we adjust few operations introduced in [35] by taking T a sub-nearsemiring of a nearsemiring S and ηT, ϑTS(S).

Definition 10

Let ηT and ϑT be a soft sets over the nearsemiring S. Then their soft intersection sum ηTϑT is defined as

(ηTϑT)(x)={x=y+z{ηT(y)ϑT(z)},if there exist y,zTsuch that x=y+z,,otherwise,

for all xT.

Definition 11

If ηT and ϑT are two soft sets over the nearsemiring S. Then their soft product is defined by

(ηTϑT)(x)={x=y.z{ηT(y)ϑT(z)},if there exist y,zTsuch that x=y.z,,otherwise,

for all xT.

The soft uni-int product may be define as follows.

Definition 12

Let ηT and ϑT be soft set over the nearsemiring S. Then the soft union intersection (uni-int) product ηSϑS is defined as

(ηTϑT)(x)={x=i=1naibi{ηT(ai)ϑT(bi)},if x=i=1naibiand aibi0,         for all 1in,,otherwise.

Theorem 1

Let T be a distributively generated sub-nearsemiring of nearsemiring S and ηT be the soft set over S. Then, ηT is a soft sub-nearsemiring of S iff ηT (t1 + t2) ⊇ ηT (t1) ∩ ηT (t2) and ηTηT ⊂̃ ηT, for all t1, t2T.

Proof

We assume that ηT is a soft sub-nearsemiring over S. Then ηT (t1 + t2) ⊇ ηT (t1) ∩ ηT (t2). Let t=i=1nxiyi such that (ηTηT)(t) = ∅, then nothing to prove. Let (ηTηT)(t) ≠ ∅. Then

(ηTηT)(t)=t=i=1nxiyi(ηT(xi)ηT(yi))t=i=1nxiyi(ηT(xiyi))=t=i=1nxiyi(ηT)(t)=(ηT)(t).

Hence ηTηT ⊂̃ ηT. Conversely, let ηT (t1 + t2) ⊇ ηT (t1) ∩ ηT (t2) and ηTηT ⊂̃ ηT. Then

ηT(t1t2)(ηTηT)(t1t2)=t1t2=i=1nxiyi(ηT(xi)ηT(yi))(ηT(t1)ηT(t2)).

Thus, ηT is a soft sub-nearsemiring of S.

Let T be an additive commutative sub-nearsemiring of a nearsemiring S, then we have the following.

Proposition 2

Let ηT and ϑT be two soft sub-nearsemirings of nearsemiring S. Then fRhfR (resp., fhR = hR).

Proof

Let ηT and ϑT be two soft sub-nearsemirings. Let t = x + y such that (ηTϑT)(t) ≠ ∅ and assume that ϑT(x) = ϑ(x), for all xT. Then,

(ηTϑT)(t)=t=y+z{ηT(y)ϑT(z)}t=y+z{ηT(y)ϑT˜(z)}ηT(y+z)S=(ηT)(t).

Hence ηTϑTηT. Similarly, by assuming ηT = η one can prove ηTϑTϑT.

Theorem 2

Let ηT and ϑT1 be soft sets over nearsemiring S and Ψ be a nearsemiring isomorphism from T to T1. If ηT is a soft sub-nearsemiring of S, then Ψ(ηT) is also a soft sub-nearsemiring of S.

Proof

Let t1,t2T1. Since Ψ is surjective, there exists t1, t2T such that Ψ(t1)=t1 and Ψ(t2)=t2. Then,

Ψ(ηT)(t1+t2)={η(t):tT,Ψ(t)=t1+t2}={η(t):tT,t=Ψ-1(t1+t2)}={η(t):tT,t=Ψ-1(Ψ(t1+t2))}={η(t1+t2):tiT,Ψ(ti)=ti,i=1,2}{η(t1)η(t2):tiT,Ψ(ti)=ti,i=1,2}=Ψ(ηT)(t1)Ψ(ηT)(t2).

Similarly, one can easily prove that Ψ(ηT)(t1t2)Ψ(ηT)(t1)Ψ(ηT)(t2). Hence, Ψ(ηT) is a soft sub-nearsemiring of S.

Theorem 3

Let T and T1 be the two sub-nearsemirings of S. Let ηT and ϑT1 be soft sets over nearsemiring S and Ψ be a nearsemiring homomorphism from T to T1. If ϑT1 is a soft sub-nearsemiring of S, then so is Ψ−1(ϑT1).

Proof

For each t1, t2T,

(Ψ-1(ϑT1)(t1+t2)=ϑT1(Ψ(t1+t2))=ϑT1(Ψ(t1)+Ψ(t2))ϑT1(Ψ(t1)ϑT1(Ψ(t2))=(Ψ-1(ϑT1))(t1)(Ψ-1(ϑT1))(t2).

Similarly, we can prove that (Ψ−1 (ϑT1))(t1t2) ⊇ (Ψ−1 (ϑT1)) (t1)∩ (Ψ−1 (ϑT1))(t2). Thus Ψ−1 (ϑT1) is a soft subnearsemiring of S.

Soft ideals of a nearsemirings

Definition 13

Let I be an ideal of nearsemiring S (i.e., IS) and (η, I) be a soft set over S. Then, (η, I) is said to be soft right (resp., left) ideal of S, if it satisfies the following conditions.

• (i) η(x + y) ⊇ η(x) ∩ η(y)

• (ii) η(xr) ⊇ η(x) (resp., η(rx) ⊇ η(x)) for all x, yI, rS.

We will abbreviate the soft right (left) ideal of S by ηIrS (resp., ηIlS). We call (η, I) a soft ideal if it is two sided soft ideal and will be abbreviated as ηIS.

Corollary 2

It is easy to verify that if ηI (x) = S, for all xI, then ηI is a soft (two sided) ideal of S. This type of soft ideal will be abbreviated by ηĨ.

Example 3

Let S = {0, a, b, c, d} be a (right) nearsemiring with the operations defined by the following Cayley tables, and I = {0, a, b, d} be an ideal of S.

Let η : I → ℘(S) defined by η(0) = S = {0, a, b, c, d}, η(a) = {0, b, c, d}, η(b) = η(d) = {0, b, d}. It is easy to verify that (η, I) is a soft ideal over S.

Remark 1

Every soft ideal of a nearsemiring S is a soft sub-nearsemiring of S.

Lemma 2

Let x=i=1naibi and ηI be a soft left (resp., right) ideal of S. Then, ηI(x)=ηI(i=1naibi)ηI(bi)(resp.,ηT(x)=ηI(i=1naibi)ηI(ai)), for all 1 ≤ in.

Theorem 4

If S is a left seminearring (i.e., zero symmetric nearsemiring) and ηI be a soft right ideal (i.e., ηIrS) and ϑJ be a soft left ideal (i.e., ϑJlS). Then, ηIϑJηI ∩̃ ϑJ.

Proof

Let S be a seminearring. Suppose (ηIϑJ) ≠ ∅. Since ϑJ is a soft left ideal (ϑJlS) of S, ϑJ (x) = ϑJ (yz) ⊇ ϑJ (z). Also, ηI is a soft right ideal and S is a zero-symmetric nearsemiring, then ηI (x) = ηI (yz) = ηI((0 + y)z + 0z) ⊇ ηI (y). Therefore,

(ηIϑJ)(x)=x=y.z{ηT(y)ϑT1(z)}ηI(y)(ϑJ)(z)ηI(x)ϑJ(x)=(ηIϑJ)(x).

Hence, ηIϑJηI ∩̃ ϑJ.

Theorem 5

Let I be a left ideal of a nearsemiring S, and ηI be the soft set over S. Then, ηI is a soft left ideal of S iff ηI (x + y) ⊇ ηI (x) ∩ ηI (y) and ηĨηI ⊂̃ ηI.

Proof

We assume that ηI is a soft left ideal of S. Then, ηI (x+ y) ⊇ ηI (x) ∩ ηI (y). Let z=i=1nxiyi such that (ηĨηI)(z) = ∅, then it is easy to show that ηĨηI ⊂̃ ηI. For this, let (ηĨηI)(z) ≠ ∅, then

(ηI˜ηI)(z)=z=i=1nxiyi(ηI˜(xi)ηI(yi)z=i=1nxiyi(SηI(i=1nxiyi))=z=i=1nxiyi(ηI)(z)=(ηI)(z).

Thus, ηĨηI ⊂̃ ηI. Conversely, let ηI (x + y) ⊇ ηI (x) ∩ ηI (x) and ηĨηI ⊂̃ ηI. Then

ηI(xy)(ηI˜ηT)(xy)=(xy=i=1nxiyi(ηI˜(xi)ηI(yi))(ηI˜(x)ηI(y))=SηI(y)=ηI(y).

Hence, ηT is a soft left ideal of a nearsemiring S.

Corollary 3

Let I be a right ideal of a nearsemiring S and ηI be the soft set over S. Then, ηI is a soft right ideal of S iff ηI (x + y) ⊇ ηI (x) ∩ ηI (y) and ηIηĨ ⊂̃ ηI.

Remark 2

Let I be an ideal of a nearsemiring S and ηI be the soft set over S. Then, ηI is a soft ideal of S iff ηI (x + y) ⊇ ηI (x) ∩ ηI (y) and ηIηĨ ⊂̃ ηI, ηĨηI ⊂̃ ηI.

4. Relationships between Soft Substructures and Classical Substructures of Nearsemirings

In this section, we provide few interrelationships among some soft substructures and classical substructures of nearsemirings. For this, we continue our study about the soft ideals of nearsemirings under α-inclusion, images, pre-images, nearsemiring homomorphisms and nearsemiring anti-homomorphisms.

Theorem 6

Let I be an ideal of a nearsemiring S. Let ηI be the soft set over S. Then ηI is a soft left (right) ideal of nearsemiring S iff ηIα is a left (right) ideal of S for any α ∈ ℘(S) such that ηIα.

Proof

Let ηI be the soft left ideal of S. Let a,bηIαηI(a)α and ηI (b) ⊇ α. It means ηI (a+b) ⊇ ηI (a) ∩ ηI (b) ⊇ α. Hence a+bηIα. We consider aηIα which implies ηI (a) ⊇ α and rS. Hence ηI (ra) ⊇ ηI (a) ⊇ α. Thus raηIα and hence ηIα is a left ideal of S for any α ∈ ℘(S) such that ηIα.

Conversely, suppose that ηIα is a left ideal of S such that ηIα, for any α ∈ ℘(S). Let a, a,bηIβ such that β=ηI(a)ηIβ(b) for each a,bSa+bηIβ. Thus, ηI (a + b) ⊇ β = ηI (a) ∩ ηI (b). Also, for each aηIγ such that γ = ηI (a), we get raηIβηI(ra)ηI(a). Hence ηI is a soft left ideal of S.

Example 4

Let S = {0, a, b, c, d} be a (right) nearsemiring with the operations defined by the following Cayley tables.

Let U = S, I = {0, a, b, d} be an ideal of S and α = {0, b} be a subset of S. We define the soft set ηI as ηI(0) = S, ηI (a) = {0, b}, ηI (b) = {0, a, b, d}, ηI (d) = {0, a, b, d}. Clearly, ηI is a soft ideal of nearsemiring S, one can easily verify that ηIα. is an ideal of S.

Theorem 7

Let ηR and ϑT be soft sets over nearsemirings R and T, respectively. Let Ψ be a nearsemiring antiisomorphism from R to T. If ηR is a soft left ideal of R, then Ψ(ηR) is a soft right ideal of T.

Proof

Let t1, t2T. Since Ψ is surjective, then there exist r1, r2R such that Ψ(r1) = t1, Ψ(r2) = t2. Then, we have

(Ψ(ηR))(t1+t2)={ηR(r):rR,Ψ(r)=t1+t2}={ηR(r):rR,r=Ψ-1(t1+t2)}={ηR(r):rR,r=Ψ-1(Ψ(r1+r2))=r1+r2}={ηR(r1+r2):riR,Ψ(ri)=ti,i=1,2}{ηR(r1)fR(r2):riR,Ψ(ri)=ti,i=1,2}=({ηR(r1):r1R,Ψ(r1)=t1})({ηR(r2):r2R,Ψ(r2)=t2})=(Ψ(ηR))(t1))(Ψ(ηR))(t2).

Similarly, one case easily verify that (Ψ(ηR))(t1t2) ⊇ Ψ(ηR))(t1) ∩ Ψ(ηR))(t2). Moreover,

(Ψ(ηR))(t1t2)={ηR(r):rR,Ψ(r)=t1t2}={ηR(r):rR,r=Ψ-1(t1t2)}={ηR(r):rR,r=Ψ-1(Ψ(r2r1))=r2r1}={ηR(r2r1):riR,Ψ(ri)=ti,i=1,2}{ηR(r1):r1R,Ψ(r1)=t1}=(Ψ(fR))(t1)).

Thus, Ψ(fR) is a soft right ideal of T.

Theorem 8

Let ηR and ηT be soft sets over a nearsemirings R and T, respectively. Let Ψ be a nearsemiring antihomomorphism from R to T. If ηT is a soft left ideal of T, then Ψ−1(ηT) is a soft right ideal of R.

Proof

Let r1, r2R. Then

(Ψ-1(ηT))(r1+r2)=ηT(Ψ(r1+r2)=ηT(Ψ(r1)+(Ψ(r1))ηT((Ψ(r1)(Ψ(r1))=(Ψ-1(ηT))(r1)(Ψ-1(ηT))(r2).

Similarly, one can easily verify that (Ψ−1 (ηT))(r1r2) ⊇ Ψ−1 (ηT))(r1) ∩ Ψ−1 (ηT))(r2). Also,

(Ψ-1(ηT))(r1r2)=ηT(Ψ(r1r2)=ηT(Ψ(r2)Ψ(r1))ηT(Ψ(r1))=(Ψ-1(ηT))(r1).

Hence Ψ−1 (ηT) is a soft right ideal of R.

Example 5

Let S = {0, a, b, c, d} be a nearsemiring with the operations defined by the following Cayley tables.

I = {0, b, d} be an ideal of S and ηI be a soft right ideal of S defined as ηI(0) = S, ηI (b) = {0, b, c, d} and ηI (b) = {0, b, c, d}.

Also, let T = {0, a, b, c, d} be a nearsemiring with the operations defined by the following Cayley tables.

J = {0, b, d} be an ideal of R and ηJ be a soft left ideal of S defined as ηJ (0) = T, ηJ (b) = {0, b, c, d} and ηJ (b) = {0, b, c, d}. Let ψ : ST be an anti-homomorphism defined by ψ(x) = x. Then, for a soft left ideal ηJ of T, the inverse map ψ−1 (ηJ) = ηI is a soft right ideal of S.

In this section, we introduce the notion of soft S-subsemigroup of a nearsemiring S. We shift few terms of S-morphism of classical nearsemiring theory towards soft set theory. A subsemigroup H of S is called a left S-subsemigroup of S if SHH and H is called a right S-subsemigroup of S if HSH.

Definition 14

Let H be a left (resp., right) S-subsemigroup of a nearsemiring S and (η, H) be a soft set over S. If for all x, yH and rS,

• (i) η(x + y) ⊇ η(x) ∩ η(y)

• (ii) η(rx) ⊇ η(x) (resp., η(xr) ⊇ η(x))

then the soft set (η, H) is the soft left (resp., right) S-subsemigroup of S and is denoted by (η,H)<SrS(resp.,(η,H)<SrS) or simply ηH<SlS(resp.,ηH<SrS).

Remark 3

Every soft left (resp., right) ideal of a nearsemiring S is also a soft left (resp., right) S-subsemigroup of a nearsemiring S.

Theorem 9

If ηH<SlS(resp.,ηH<SrS) and ϑK<SlS(resp.,ϑK<SrS), then ηH˜ϑK<SlS(resp.,ηHϑK<SrS), if it is non-null.

Proof

Let ηH<SlS and ϑK<SlS. Let H and K are left S-subsemigroups of S such that HK ≠ ∅, then it can easily seen that HK is a left S-subsemigroup of S. Since, ηH ∩̃ ϑK = ξH ∩̃ K, where ξH ∩̃ K (x) = η(x) ∩ ϑ(x), for all xHK. Then, for all x, yHK and for all rS,

(i)ξH˜K(x+y)=ηH(x+y)ϑK(x+y)(η(x)η(y))(ϑ(x)ϑ(y))=(η(x)ϑ(x))(η(y)ϑ(y))=ξH˜K(x)ξH˜K(y)(ii)ξH˜K(rx)=η(rx)ϑ(rx)(η(x)ϑ(x))=ξH˜K(x).

Hence, ηH˜ϑK<SlS. Similarly, one can easily prove that ηH˜ϑK<SrS.

Definition 15

Let S1 and S2 are nearsemirings and let ηH<S1lS1(resp.,ηH<S1rS1) and K<S2lS2(resp.,ϑK<S2rS2). The product of two soft left (resp., right) S-subsemigroups (η, H) and (ϑ, K) is defined as (η, H) × (ϑ, K) = ξH × K, where ξH × K(x, y) = ηH(x) × ϑK(y), for all (x, y) ∈ H × K.

Theorem 10

If ηH<S1lS1(resp.,ηH<S1rS1) and ϑK<S2lS2(resp.,ϑK<S2rS2), then ηH×ϑK<S1×S2lS1×S2(resp.,ηH×ϑK<S1×S2rS1×S2).

Proof

Let ηH<S1lS1 and ϑK<S2lS2. Since H is a left S1-subsemigroup of S1 and K is a left S2-subsemigroup of S2, it can be easily seen that H × K is a left S1 × S2-subsemigroup of S1 × S2. Since, ηH × ϑK = ξH × K, where ξH × K(x) = ηH(x) × ϑK(y) for all (x, y) ∈ H × K. Then, for all (x1, y1), (x2, y2)H × K, (r1, r2) ∈ S1× S2 we have

(i)ξH×K((x1,y1)+(x2,y2))=ξH×K(x1+x2,y1+y2)=ηH(x1+x2)×ϑK(y1+y2)(ηH(x1)ηH(x2))×(ϑK(y1)ϑK(y2))=(ηH(x1)×ϑK(y1))(ηH(x2)×ϑK(y2))=ξH×K(x1,y1)ξH×K(x2,y2)(ii)ξH×K((r1,r2)(x1,y1))=ξH×K(r1,x1,r2y1)=ηH(r1x1)×ϑK(r2y1)ηH(x1)×ϑK(y1)=ξH×K(x1,y1).

Hence, ηH×ϑK<S1×S2lS1×S2. Similarly, one can prove that ηH×ϑK<S1×S2rS1×S2.

Definition 16

Let (η, H) be a soft left (resp., right) S1-sub-semigroup of a nearsemiring S1 and (ϑ, K) be a soft left (resp., right) S2-subsemigroup of S2. Suppose f : S1S2 and g : HK be the two mappings. Then the pair (f, g) is called a soft S-homomorphism if it satisfies the following conditions.

• (i) f is an S-epimorphism

• (ii) g is a surjective mapping

• (iii) f(η(x)) = ϑ(g(x)) for all xH.

If there exists a soft S-homomorphism between (η, H) and (ϑ, K), we call (η, H) is soft S-homomorphic to (ϑ, K). Similarly, if f is an S-isomorphism of S-semigroups and g is a bijective mapping, then we call (f, g) a soft S-isomorphism, and say that (η, H) is soft S-isomorphic to (ϑ, K) i.e., (η, H) ≃ (ϑ, K).

Example 6

Let us take a nearsemiring S = {0, a, b, c, d} given in the Example2. And let H = {0, a, b, d} be a S-subsemigroup of an S-semigroup (S, +). We also assume that H = K and S1 = S2 = S. Then, let f : SS be the map defined by f(x) = x. One can easily verify that f is an S-isomorphism. Let g : HK be the map defined by g(x) = x, which is clearly a bijective mapping. Let η : H → ℘(R) be the map defined by η(x) = {yS : xyxy ∈ {0, a}}. Then, η(0) = η(a) = S, η(b) = η(d) = {0, a}. By doing a simple calculations, one can prove that (η, H) is a soft S-subsemigroup of S. Similarly, we define the map ϑ : K → ℘(S) by ϑ(x) = {yS : xyxy ∈ {0, a}}. Then, ϑ(0) = ϑ(a) = S, ϑ(b) = ϑ(d) = {0, a}. Evidently, (ϑ, K) is a soft S-subsemigroup of S. Finally, consider f(η(0)) = S, and ϑ(g(0)) = ϑ(0) = S, so f(η(0)) = ϑ(g(0)). Again, consider f(η(a)) = f(S) = S and ϑ(g(a)) = ϑ({0, a}) = S, so f(η(a)) = ϑ(g(a)). Similarly, we can show that for all xH, f(η(x))= ϑ(g(x)). Hence (η, H) ≃ (ϑ, K).

Theorem 11

Let H, K be the two S-subsemigroups of S. Let ηH and ϑK be the soft sets over S and Ψ be an S-isomorphism from H to K. If ηH is a soft S-subsemigroup of S, then so is Ψ(ηH).

Proof

Let k1, k2K. Since Ψ is surjective, there exists h1, h2H such that Ψ(h1) = k1 and Ψ(h2) = k2. Thus

Ψ(ηH)(k1+k2)={ηH(h):hH,Ψ(h)=(k1+k2)}={ηH(h):hH,h=Ψ-1(k1+k2}={ηH(h):hH,h=Ψ-1(Ψ(h1+h2))=h1+h2}={ηH(h1+h2):kiK,Ψ(hi)=ki,i=1,2}{η(h1)η(h2):hiH,Ψ(hi)=ki,i=1,2}=({η(h1):k1K,Ψ(h1)=k1})({η(h2):k2K,Ψ(h2)=k2})=(Ψ(ηH))(h1)(Ψ(ηH))(h2).

Now, let rS and kK. Since Ψ is surjective, there exists h′ ∈ H such that Ψ(h′) = k. Then,

Ψ(ηH)(rk)={ηH(h):hH,Ψ(h)=rk}={ηH(h):hH,h=Ψ-1(rk)}={ηH(h):hH,h=Ψ-1(rΨ(h))}={ηH(h):hH,h=Ψ-1(Ψ(rh))=rh}={ηH(rh):hH,Ψ(h)=k}{ηH(h):hH,Ψ(h)=k}=(Ψ(ηH))(k).

Hence Ψ(ηH) is a soft S-subsemigroup of S.

In this study, we have introduced soft sub-nearsemirings, soft ideals of nearsemirings and soft S-subsemigroups of nearsemirings by using the definition of soft sets introduced by Molodtsov [1]. We have applied a number of soft set operations to our newly established soft algebraic structures. Moreover, we investigated these soft algebraic structures through soft nearsemiring homomorphism and soft nearsemiring anti-homorphism. Throughout our study, we provide bunch of illustrative examples. In addition, we have provided few relationships among these soft substructures and classical substructures of nearsemirings. Through these relationships, we established the connection among the soft set theory, classical set theory and nearsemiring theory. To extend this study, one can discuss some other classical algebraic structures such as balanced nearsemirings, affine near-semirings etc on the same pattern. Classical nearsemiring is used in the theory of automata and other computer languages thus we expect that the work presented in this manuscript would be useful in the theory of fuzzy (soft) automata.

Table. 1.

Table 1. Right nearsemiring.

+0abcd.0abcd
00abcd000000
aaabdda0aaaa
bbbbddb0abbd
ccddcdc0abbd
ddddddd0abbd

Table. 2.

Table 2. Right nearsemiring.

 + 0 a b c d . 0 a b c d 0 0 a b c d 0 0 0 0 0 0 a a a b d d a 0 a a a a b b b b d d b 0 a b b d c c d d c d c 0 a b b d d d d d d d d 0 a b b d

Table. 3.

Table 3. Right nearsemiring.

 + 0 a b c d . 0 a b c d 0 0 a b c d 0 0 0 0 0 0 a a a b d d a 0 a a a a b b b b d d b 0 a b b b c c d d c d c 0 a b b b d d d d d d d 0 a d d d

Table. 4.

Table 4. Right nearsemiring.

 + 0 a b c d . 0 a b c d 0 0 a b c d 0 0 0 0 0 0 a a a b d d a 0 a a a a b b b b d d b 0 b b b b c c d d d d c 0 a a a a d d d d d d d 0 a a a a

Table. 5.

Table 5. Right nearsemiring.

 + 0 a b c d . 0 a b c d 0 0 a b c d 0 0 0 0 0 0 a a a b d d a 0 a a a a b b b b d d b 0 b b b d c c d d c d c 0 c c c c d d d d d d d 0 d d d d

Table. 6.

Table 6. Right nearsemiring.

 + 0 a b c d . 0 a b c d 0 0 a b c d 0 0 0 0 0 0 a a a b d d a 0 a a a a b b b b d d b 0 a b b b c c d d c d c 0 a b b b d d d d d d d 0 a d d d

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Waheed Ahmad Khan is working as an assistant professor in the Department of Mathematics at the University of Education, Attock Campus, Lahore, Pakistan. He has been working on research related to fuzzy set theory, soft set theory, and commutative algebra.

Abdelghani Taouti is working as a lecturer at the ETS-Maths and NS Engineering Division, HCT, University City P.O. Box 7947, Sharjah, United Arab Emirates. He has been working on research related to fuzzy set theory and soft set theory.

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(3): 276-286

Published online September 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.3.276

Soft Sub-nearsemirings, Soft Ideals and Soft -Subsemigroups of Nearsemirings

Waheed Ahmad Khan1 and Abdelghani Taouti2

1Department of Mathematics, University of Education Lahore, Attock Campus, Pakistan
2ETS-Maths and NS Engineering Division, HCT, Sharjah, United Arab Emirates

Received: March 15, 2021; Revised: July 13, 2022; Accepted: August 19, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this note, we introduce and discuss the notions of soft sub-nearsemirings, soft ideals and soft S-subsemigroups of nearsemirings. Some related properties and characterizations of these soft algebraic structures are discussed with illustrative examples. For the sake of investigations, we apply several operations of soft sets including soft intersection sum, soft product and soft uniint product. Based on these operations, we also discuss few characterizations of distributively generated nearsemirings. In due course, we investigate few relationships among these soft algebraic structures and the classical nearsemirings. We also introduce the notion of soft ideals (left and right) of nearsemirings. Firstly, we investigate these ideals by applying few operations on them. Then, we present the relationship between these soft ideals of nearsemiring with the classical ideals of nearsemirings. Moreover, we introduce the notion of soft S-homomorphism between two soft S-subsemigroups and investigate that the homomorphic image of soft S-subsemigroup is a soft S-subsemigroup. Throughout, we shift several substructures of nearsemirings towards the soft algebraic substructures of nearsemirings by utilizing different algebraic methods. Consequently, we explore a linkage among the soft set theory, classical set theory and nearsemiring theory. Mainly, our study is the interplay between soft substructures of nearsemirings and classical substructures of nearsemirings.

Keywords: Nearsemirings, Soft sub-nearsemirings, Soft S-subsemigroups, Uni-int products

1. Introduction

Soft set theory was initiated in 1999 by Molodtsov [1] and the fuzzy set theory was initiated by Zadeh [2]. Actually, every hesitant fuzzy set introduced by Zadeh [2] can be considered a soft set on a common universe. Similarly, every soft set on a denumerable universe can be considered a fuzzy set. However, soft sets theory become an effective tool to deal with uncertainties in the given data. Numerous applications of soft sets have been explored in [3,4]. A lot of work has been done on the soft set theory and is developing rapidly. Many applications of soft sets have been explored towards data analysis, decision-making theory etc. A number of operations on soft sets have been introduced and discussed in the literature [58]. Among the others, the operation uni-int has its own importance due to its applications towards decision-making [9]. The generalized form of uni-int operation and its application in decision-making theory was presented in [10]. Moreover, Han and Geng [11] introduced some special method based on intm-intn and created a decision-making scheme. Number of algebraists introduced several soft algebraic structures such as soft groups [5], soft rings [12], soft fields [13], soft nearrings [14], soft semirings [15] and so on. Afterwards, a new view of soft intersection rings based on uni-int product has been explored in [16]. Soft subnearrings, soft ideals and soft N-subgroups of nearrings have been introduced in [17]. Notion of soft intersection h-ideals of hemirings have been introduced in [18]. Khan et al. [19] introduced the notion of soft nearsemirings. The concepts of soft intersection nearsemirings [20] and (M,N)-soft intersection nearsemirings were also introduced by Khan and Davvaz [21]. The notion of (α, β)-soft intersectional rings and ideals along with their applications have been explored in [22]. Recently, the notions of fuzzy sub-nearsemirings and fuzzy soft sub-nearsemirings [23] have been initiated by both the authors. In a sequel, we introduce and discuss the notions of soft sub-nearsemirings, soft ideals and soft S-subsemigroups of a nearsemiring S. We also provide few characterizations of distributively generated nearsemirings in the frame of the soft sets theory. Throughout our study, we interrelate several classical substructures of nearsemirings with the soft substructures of nearsemirings.

Near-semiring is the common generalization of nearrring and semiring. Basically, nearsemiring R is an algebraic structure equipped with two binary operations “+” and “・” such that (R, +) is a monoid, (R, ・) is a semigroup, and both structures are joined through a single (left or right) distributive law with 0 is the one sided absorbing element. Moreover, if 0 ∈ R such that a + 0 = 0+a = a, a ・ 0 = 0 ・ a = 0, then we say R is a zero-symmetric nearsemiring (or seminearring). Nearsemirings ascertained naturally from the mappings on monoid to itself under component-wise addition and composition of mappings. Due to having strong relationships of nearsemirings with different types of algebras, recently Chajda and Laenger [24] introduced the notion of balanced nearsemirings. Since the ideals have their own importance in algebra particularly when we study rings with two binary operations. In general, the ideals of nearsemirings are not similar to that of the standard ideals of rings (resp., nearrings, semirings) and hence many results in rings (resp., nearrings, semirings) theory have no similarities in nearsemirings using merely ideals. Hence the classified notion of ideal named S-ideal [25] has been initiated in seminearring (zero symmetric nearsemirings) theory. Some special types of prime ideals of seminearrings have been explored in [26].

On the other hands, Nearrings, semirings and nearsemirings have a number of applications in computer technology such as theory of automata, languages and machine learning. Variety of applications of semirings have been explored in [27,28]. In [28], the author explored various applications of nearrings, particularly, he extended the linear sequential machines of Eilenberg by using the theory of nearrings. Likewise, Krishna and Chatterjee [30] introduced the condition of minimality of generalized linear sequential machines at the base of nearsemirings.

This work comprises of six sections. In Section 2, we include necessary discussions about nearsemirings and useful terminologies about soft set theory. In Section 3, we introduce and discuss the notions of soft sub-nearsemirings and soft ideals of nearsemiring S. We apply a number of usual operations including soft uni-int products to these soft algebraic structures. Moreover, we investigate these soft structures by applying image, pre-image, α-inclusion mappings. We also provide few characterizations of distributively generated nearsemirings in the setting of soft set theory. In Section 4, we provide some relationships of newly established soft structures with the existing classical algebraic structures as applications. In section 5, we introduce the notions of soft S-subsemigroup and soft S-morphism between two soft S-subsemigroups of nearsemirings. Throughout our discussions, we examine the interplay of classical nearsemiring and its substructures with the soft nearsemirings and their soft substructures.

2. Preliminaries

Following [31], we call a system (S, +, .) is a left(right) nearsemiring if Sis a monoid with respect to addition, semigroup with respect to multiplication and obeying left (right) distributive law. Furthermore, if there exists 0 ∈ S such that 0.a = a.0 = 0, then we call it a zero-symmetric nearsemiring. If S and S′ are two nearsemirings, then we call a mapping φ: SS′ a nearsemiring homomorphism if for all a, bS; φ(a + b) = φ(a) + φ(b), φ(ab) = φ(a)φ(b), φ(0) = 0. Similarly, if S and S′ are two nearsemirings, a mapping ψ: SS′ is the anti-homomorphism if for all a, bS; φ(a+b) = φ(a)+ φ(b); φ(ab) = φ(b)φ(a); φ(0) = 0. The kernel of semigroup homomorphism is said to be an ideal of nearsemiring R. If S is a nearsemiring, then we call a semigroup (Γ, +) having zero 0Γ a left S-semigroup if there exists a composition (x, γ) ↣ of S × Γ satisfying: (i) (x + y)γ = + ; (ii) (xy)γ = x(); and (iii) 0Sγ = 0Γ, for all x, yS and γ ∈ Γ. It is obvious that Γ is an S-semigroup with S = M(Γ), a nearsemiring consists of all mappings from Γ to itself. Also, a semigroup (S, +) of a nearsemiring (S, +, .) is also an S-semigroup. If Γ and Γ′ are two S-semigroups of a nearsemiring S. A mapping φ: Γ → Γ′ is said to be an S-morphism between S-semigroups Γ and Γ′ if it satisfies: (i) φ(x + y) = φ(x)+ φ(y), (ii) φ(rx) = (x), for all x, y ∈ Γ, rS and (iii) φ(0Γ) = 0Γ′. We refer [3133] for further discussion on nearsemirings and S-semigroups of nearsemirings. Throughout, S represents a nearsemiring, U an initial universe, E the possible parameters associated with the elements in U, ℘(U) represents the family of subsets of U, S(U) the collection of soft sets and A ≠ ∅ be a subset of E.

Definition 1 ([1])

A η be a mapping from A to ℘(U). Then the pair (η, A) is said to be soft set over the universe U.

Definition 2 ([9])

Intersection of ηA and ηB represented by ηA ∩̃ ηB, defined by ηA ∩̃ ηB = ηA∩̃B, where ηA∩̃B (x) = ηA(x) ∩ ηB(x), for all xE.

Definition 3 ([9])

The ∧-product of ηA and ηB of two soft sets over U i.e., ηAηB is the mapping ηAB : E × E → ℘(U), and ηAB(x, y) = ηA(x) ∩ ηB(y).

Definition 4 ([9])

The ∨-product of two soft sets ηA and ηB is defined by ηAB : E × E → ℘(U) where ηAB(x, y) = ηA(x) ∪ ηB(y).

Definition 5 ([34])

Let Ψ be a function from A to B and ηA, ηBS(U). Then, the soft subsets Ψ(ηA) ∈ S(U) and Ψ−1(ηB) ∈ S(U) defined as

$Ψ(ηA)(y)={∪{ηA(x):x∈A, Ψ(x)=y},if y∈Ψ(A),∅if y∉Ψ(A).$

for all yB and Ψ−1 (ηB)(x) = ηB(Ψ(x)), for all xA. Then Ψ(ηA) is the soft image of ηA and Ψ−1 (ηB) is the soft pre-image (or soft inverse image) of ηB under the napping Ψ.

Definition 6 ([34])

Let ηA be a soft set over U and α be a subset of U. Then upper α-inclusion of soft set ηA is defined by $ηA⊇α={x∈A:ηA(x)⊇α}$.

Definition 7 ([19])

Let (ζ, A) be a non-null soft set over a nearsemiring R. Then, (ζ, A) is called a soft nearsemiring over R, if ζ(x) is a sub-nearsemiring of R for all xSupp(ζ, A).

Definition 8 ([19])

Let (η, A) be the soft right nearsemiring over right nearsemiring R1 and (ζ, B) be the soft left nearsemiring over the left nearsemiring R2. Let f : R1R2 and g : AB be the two mappings. Then, the pair (f, g) is said to be a soft nearsemiring anti-homomorphism if,

• (i) f is an anti-epimorphism of nearsemirings.

• (ii) g is Onto mapping.

• (iii) f(η(x)) = ζ(g(x)) for all xA.

Now, if f is an anti-isomorphism andg is bijective then we say (f, g) a soft nearsemiring anti-isomorphism.

3. Soft Sub-nearsemirings and Soft Ideals

In this section, we initiate the notions of soft sub-nearsemirings and soft ideals of nearsemiring by using intersection operation of soft sets. For the sake of investigations, we apply several operations of soft set theory to them. We also provide few characterizations of distributively generated nearsemirings related to these soft substructures of nearsemirings.

Definition 9

Let (η, T) be a soft set over S, where T be a sub-nearsemiring of S. Then the soft set (η, T) is said to be a soft sub-nearsemiring of S if it satisfies the followings.

• (i) η(x + y) ⊇ η(x) ∩ η(y)

• (ii) η(xy) ⊇ η(x) ∩ η(y)

for all x, yT. We denote it by (η, T) ⪝ S or ηTS.

Corollary 1

It is easy to verify that if ηT (x) = S for all xT, then ηT is a soft sub-nearsemiring of S. We will abbreviate such a soft sub-nearsemiring by η.

Example 1

Let S = {0, a, b, c, d} be a (right) nearsemiring with the operations defined by the following Cayley tables.

Clearly, T = {0, a, b} is a sub-nearsemiring of nearsemiring S. For T = {0, a, b}, let η : T → ℘(S) be a set-valued function defined by η(0) = S, η(a) = {0, b, c, d}, η(b) = {0, a}. It is easy to verify that (η, T) is a soft sub-nearsemiring of S. On the other hand, if we take T1 = {0, b, c, d} a sub-nearsemiring of S, and ϑ : T1 → ℘(S) be a set-valued function defined by ϑ(0) = {0, a, b, d}, ϑ(b) = ϑ(d) = {0, b, d}, ϑ(c) = {0, a}. Then ϑ(c.c) = ϑ(b) = {0, b, d} ⊉ ϑ(c) ∩ ϑ(c) = {0, a}. Thus, (ϑ, T1) is not a soft sub-nearsemiring of S.

If ηTS and ηT1S, then we can easily deduce the followings.

Proposition 1

• If ηTS and ηT1S, then ηT ∩̃ ηT1S

• If ηT1S1 and ϑT2S2, then ηT1 × ϑT2S1 × S2.

• If ηTS and ηT1S, then ηTηT1S.

Proof

Easy to proof (1), (2) & (3) and hence omitted.

On the other hand, if ηS and ηT are two soft sub-nearsemirings of a nearsemiring R, then ηSηT need not be a soft sub-nearsemiring as we see in the below example.

Example 2

Let S = {0, a, b, c, d} be a (right) nearsemiring with the operations defined by the following Cayley tables.

Clearly, T = {0, a, b} is a sub-nearsemiring of nearsemiring S, and let η : T → ℘(S) be a set-valued function defined by η(0) = S, η(a) = {0, b, c, d}, η(b) = {0, a}. It is easy to verify that (η, T) is a soft sub-nearsemiring of S. Again by taking a sub-nearsemiring T1 = {0, a, b, d}, and a set-valued function ϑ : T1 → ℘(S) defined by ϑ(0) = S, ϑ(a) = {0, a, b, d}, ϑ(b) = ϑ(d) = {0, b, d}. Then, one can easily show that (ϑ, T1) is a soft sub-nearsemiring of S. Consider, (ηTϑT1)((b, 0) + (0, a)) = (ηTϑT1) (b, a) = {0, a, b, d} ⊉ (ηTϑT1)(b, 0) ∩ (ηTϑT1)(0, a) = {0, a, b, c, d} ∩ {0, a, b, c, d} = {0, a, b, c, d}. Hence, ηTϑT1 is not a soft sub-nearsemiring.

The following lemma is obvious.

Lemma 1

• If S is distributively generated nearsemiring and T be its sub-nearsemiring. If tT, such that $t=∑i=1naibi$ and ηT be a soft sub-nearsemiring of S, then $ηT(t)=ηT(∑i=1naibi)⊇ηT(ai)∩ηT(bi)$ for all 1 ≤ in.

• If S is an additively commutative nearsemiring and $t=∑i=1nai+bi$ such that ηT soft sub-nearsemiring of S. Then, $ηT(t)=ηT(∑i=1nai+bi)⊇ηT(ai)∩ηT(bi)$ for all 1 ≤ in.

Now we adjust few operations introduced in [35] by taking T a sub-nearsemiring of a nearsemiring S and ηT, ϑTS(S).

Definition 10

Let ηT and ϑT be a soft sets over the nearsemiring S. Then their soft intersection sum ηTϑT is defined as

$(ηT⊕ϑT)(x)={∪x=y+z{ηT(y)∩ϑT(z)},if there exist y,z∈T such that x=y+z,∅,otherwise,$

for all xT.

Definition 11

If ηT and ϑT are two soft sets over the nearsemiring S. Then their soft product is defined by

$(ηT⊙ϑT)(x)={∪x=y.z{ηT(y)∩ϑT(z)},if there exist y,z∈T such that x=y.z,∅,otherwise,$

for all xT.

The soft uni-int product may be define as follows.

Definition 12

Let ηT and ϑT be soft set over the nearsemiring S. Then the soft union intersection (uni-int) product ηSϑS is defined as

$(ηT★ϑT)(x)={∪x=∑i=1naibi{ηT(ai)∩ϑT(bi)},if x=∑i=1naibi and aibi≠0, for all 1≤i≤n,∅,otherwise.$

Theorem 1

Let T be a distributively generated sub-nearsemiring of nearsemiring S and ηT be the soft set over S. Then, ηT is a soft sub-nearsemiring of S iff ηT (t1 + t2) ⊇ ηT (t1) ∩ ηT (t2) and ηTηT ⊂̃ ηT, for all t1, t2T.

Proof

We assume that ηT is a soft sub-nearsemiring over S. Then ηT (t1 + t2) ⊇ ηT (t1) ∩ ηT (t2). Let $t=∑i=1nxiyi$ such that (ηTηT)(t) = ∅, then nothing to prove. Let (ηTηT)(t) ≠ ∅. Then

$(ηT★ηT)(t)=∪t=∑i=1nxiyi(ηT(xi)∩ηT(yi))⊆∪t=∑i=1nxiyi(ηT(xiyi))=∪t=∑i=1nxiyi(ηT)(t)=(ηT)(t).$

Hence ηTηT ⊂̃ ηT. Conversely, let ηT (t1 + t2) ⊇ ηT (t1) ∩ ηT (t2) and ηTηT ⊂̃ ηT. Then

$ηT(t1t2)⊇(ηT★ηT)(t1t2)=∪t1t2=∑i=1nxiyi(ηT(xi)∩ηT(yi))⊇(ηT(t1)∩ηT(t2)).$

Thus, ηT is a soft sub-nearsemiring of S.

Let T be an additive commutative sub-nearsemiring of a nearsemiring S, then we have the following.

Proposition 2

Let ηT and ϑT be two soft sub-nearsemirings of nearsemiring S. Then fRhfR (resp., fhR = hR).

Proof

Let ηT and ϑT be two soft sub-nearsemirings. Let t = x + y such that (ηTϑT)(t) ≠ ∅ and assume that ϑT(x) = ϑ(x), for all xT. Then,

$(ηT⊕ϑT)(t)=∪t=y+z{ηT(y)∩ϑT(z)}⊆∪t=y+z{ηT(y)∩ϑT˜(z)}⊆ηT(y+z)∩S=(ηT)(t).$

Hence ηTϑTηT. Similarly, by assuming ηT = η one can prove ηTϑTϑT.

Theorem 2

Let ηT and ϑT1 be soft sets over nearsemiring S and Ψ be a nearsemiring isomorphism from T to T1. If ηT is a soft sub-nearsemiring of S, then Ψ(ηT) is also a soft sub-nearsemiring of S.

Proof

Let $t1′,t2′∈T1$. Since Ψ is surjective, there exists t1, t2T such that $Ψ(t1)=t1′$ and $Ψ(t2)=t2′$. Then,

$Ψ(ηT)(t1′+t2′)=∪{η(t):t∈T, Ψ(t)=t1′+t2′}=∪{η(t):t∈T, t=Ψ-1(t1′+t2′)}=∪{η(t):t∈T, t=Ψ-1(Ψ(t1+t2))}=∪{η(t1+t2):ti∈T, Ψ(ti)=ti′, i=1, 2}⊇{η(t1)∩η(t2):ti∈T, Ψ(ti)=ti′, i=1, 2}=Ψ(ηT)(t1′)∩ Ψ(ηT)(t2′).$

Similarly, one can easily prove that $Ψ(ηT)(t1′t2′)⊇Ψ(ηT)(t1′)∩Ψ(ηT)(t2′)$. Hence, Ψ(ηT) is a soft sub-nearsemiring of S.

Theorem 3

Let T and T1 be the two sub-nearsemirings of S. Let ηT and ϑT1 be soft sets over nearsemiring S and Ψ be a nearsemiring homomorphism from T to T1. If ϑT1 is a soft sub-nearsemiring of S, then so is Ψ−1(ϑT1).

Proof

For each t1, t2T,

$(Ψ-1(ϑT1)(t1+t2)=ϑT1(Ψ(t1+t2))=ϑT1(Ψ(t1)+Ψ(t2))⊇ϑT1(Ψ(t1)∩ϑT1(Ψ(t2))=(Ψ-1(ϑT1))(t1)∩(Ψ-1(ϑT1))(t2).$

Similarly, we can prove that (Ψ−1 (ϑT1))(t1t2) ⊇ (Ψ−1 (ϑT1)) (t1)∩ (Ψ−1 (ϑT1))(t2). Thus Ψ−1 (ϑT1) is a soft subnearsemiring of S.

Soft ideals of a nearsemirings

Definition 13

Let I be an ideal of nearsemiring S (i.e., IS) and (η, I) be a soft set over S. Then, (η, I) is said to be soft right (resp., left) ideal of S, if it satisfies the following conditions.

• (i) η(x + y) ⊇ η(x) ∩ η(y)

• (ii) η(xr) ⊇ η(x) (resp., η(rx) ⊇ η(x)) for all x, yI, rS.

We will abbreviate the soft right (left) ideal of S by ηIrS (resp., ηIlS). We call (η, I) a soft ideal if it is two sided soft ideal and will be abbreviated as ηIS.

Corollary 2

It is easy to verify that if ηI (x) = S, for all xI, then ηI is a soft (two sided) ideal of S. This type of soft ideal will be abbreviated by ηĨ.

Example 3

Let S = {0, a, b, c, d} be a (right) nearsemiring with the operations defined by the following Cayley tables, and I = {0, a, b, d} be an ideal of S.

Let η : I → ℘(S) defined by η(0) = S = {0, a, b, c, d}, η(a) = {0, b, c, d}, η(b) = η(d) = {0, b, d}. It is easy to verify that (η, I) is a soft ideal over S.

Remark 1

Every soft ideal of a nearsemiring S is a soft sub-nearsemiring of S.

Lemma 2

Let $x=∑i=1naibi$ and ηI be a soft left (resp., right) ideal of S. Then, $ηI(x)=ηI(∑i=1naibi)⊇ηI(bi) (resp., ηT(x)=ηI(∑i=1naibi)⊇ηI(ai))$, for all 1 ≤ in.

Theorem 4

If S is a left seminearring (i.e., zero symmetric nearsemiring) and ηI be a soft right ideal (i.e., ηIrS) and ϑJ be a soft left ideal (i.e., ϑJlS). Then, ηIϑJηI ∩̃ ϑJ.

Proof

Let S be a seminearring. Suppose (ηIϑJ) ≠ ∅. Since ϑJ is a soft left ideal (ϑJlS) of S, ϑJ (x) = ϑJ (yz) ⊇ ϑJ (z). Also, ηI is a soft right ideal and S is a zero-symmetric nearsemiring, then ηI (x) = ηI (yz) = ηI((0 + y)z + 0z) ⊇ ηI (y). Therefore,

$(ηI⊙ϑJ)(x)=∪x=y.z{ηT(y)∩ϑT1(z)}⊆ηI(y)∩(ϑJ)(z)⊆ηI(x)∩ϑJ(x)=(ηI∩ϑJ)(x).$

Hence, ηIϑJηI ∩̃ ϑJ.

Theorem 5

Let I be a left ideal of a nearsemiring S, and ηI be the soft set over S. Then, ηI is a soft left ideal of S iff ηI (x + y) ⊇ ηI (x) ∩ ηI (y) and ηĨηI ⊂̃ ηI.

Proof

We assume that ηI is a soft left ideal of S. Then, ηI (x+ y) ⊇ ηI (x) ∩ ηI (y). Let $z=∑i=1nxiyi$ such that (ηĨηI)(z) = ∅, then it is easy to show that ηĨηI ⊂̃ ηI. For this, let (ηĨηI)(z) ≠ ∅, then

$(ηI˜★ηI)(z)=∪z=∑i=1nxiyi(ηI˜(xi)∩ηI(yi)⊆∪z=∑i=1nxiyi(S∩ηI(∑i=1nxiyi))=∪z=∑i=1nxiyi(ηI)(z)=(ηI)(z).$

Thus, ηĨηI ⊂̃ ηI. Conversely, let ηI (x + y) ⊇ ηI (x) ∩ ηI (x) and ηĨηI ⊂̃ ηI. Then

$ηI(xy)⊇(ηI˜★ηT)(xy)=(∪xy=∑i=1nxiyi(ηI˜(xi)∩ηI(yi))⊇(ηI˜(x)∩ηI(y))=S∩ηI(y)=ηI(y).$

Hence, ηT is a soft left ideal of a nearsemiring S.

Corollary 3

Let I be a right ideal of a nearsemiring S and ηI be the soft set over S. Then, ηI is a soft right ideal of S iff ηI (x + y) ⊇ ηI (x) ∩ ηI (y) and ηIηĨ ⊂̃ ηI.

Remark 2

Let I be an ideal of a nearsemiring S and ηI be the soft set over S. Then, ηI is a soft ideal of S iff ηI (x + y) ⊇ ηI (x) ∩ ηI (y) and ηIηĨ ⊂̃ ηI, ηĨηI ⊂̃ ηI.

4. Relationships between Soft Substructures and Classical Substructures of Nearsemirings

In this section, we provide few interrelationships among some soft substructures and classical substructures of nearsemirings. For this, we continue our study about the soft ideals of nearsemirings under α-inclusion, images, pre-images, nearsemiring homomorphisms and nearsemiring anti-homomorphisms.

Theorem 6

Let I be an ideal of a nearsemiring S. Let ηI be the soft set over S. Then ηI is a soft left (right) ideal of nearsemiring S iff $ηI⊇α$ is a left (right) ideal of S for any α ∈ ℘(S) such that $ηI⊇α≠∅$.

Proof

Let ηI be the soft left ideal of S. Let $a,b∈ηI⊇α⇒ηI(a)⊇α$ and ηI (b) ⊇ α. It means ηI (a+b) ⊇ ηI (a) ∩ ηI (b) ⊇ α. Hence $a+b∈ηI⊇α$. We consider $a∈ηI⊇α$ which implies ηI (a) ⊇ α and rS. Hence ηI (ra) ⊇ ηI (a) ⊇ α. Thus $ra∈ηI⊇α$ and hence $ηI⊇α$ is a left ideal of S for any α ∈ ℘(S) such that $ηI⊇α≠∅$.

Conversely, suppose that $ηI⊇α$ is a left ideal of S such that $ηI⊇α≠∅$, for any α ∈ ℘(S). Let a, $a, b∈ηI⊇β$ such that $β=ηI(a)∩ηI⊇β(b)$ for each $a, b∈S⇒a+b∈ηI⊇β$. Thus, ηI (a + b) ⊇ β = ηI (a) ∩ ηI (b). Also, for each $a∈ηI⊇γ$ such that γ = ηI (a), we get $ra∈ηI⊇β⇒ηI(ra)⊇ηI(a)$. Hence ηI is a soft left ideal of S.

Example 4

Let S = {0, a, b, c, d} be a (right) nearsemiring with the operations defined by the following Cayley tables.

Let U = S, I = {0, a, b, d} be an ideal of S and α = {0, b} be a subset of S. We define the soft set ηI as ηI(0) = S, ηI (a) = {0, b}, ηI (b) = {0, a, b, d}, ηI (d) = {0, a, b, d}. Clearly, ηI is a soft ideal of nearsemiring S, one can easily verify that $ηI⊇α≠∅$. is an ideal of S.

Theorem 7

Let ηR and ϑT be soft sets over nearsemirings R and T, respectively. Let Ψ be a nearsemiring antiisomorphism from R to T. If ηR is a soft left ideal of R, then Ψ(ηR) is a soft right ideal of T.

Proof

Let t1, t2T. Since Ψ is surjective, then there exist r1, r2R such that Ψ(r1) = t1, Ψ(r2) = t2. Then, we have

$(Ψ(ηR))(t1+t2)=∪{ηR(r):r∈R, Ψ(r)=t1+t2}=∪{ηR(r):r∈R, r=Ψ-1(t1+t2)}=∪{ηR(r):r∈R, r=Ψ-1(Ψ(r1+r2))=r1+r2}=∪{ηR(r1+r2):ri∈R, Ψ(ri)=ti, i=1, 2}⊇∪{ηR(r1)∩fR(r2):ri∈R, Ψ(ri)=ti, i=1, 2}=(∪{ηR(r1):r1∈R, Ψ(r1)=t1})∩(∪{ηR(r2):r2∈R, Ψ(r2)=t2})=(Ψ(ηR))(t1))∩ (Ψ(ηR))(t2).$

Similarly, one case easily verify that (Ψ(ηR))(t1t2) ⊇ Ψ(ηR))(t1) ∩ Ψ(ηR))(t2). Moreover,

$(Ψ(ηR))(t1t2)=∪{ηR(r):r∈R, Ψ(r)=t1t2}=∪{ηR(r):r∈R, r=Ψ-1(t1t2)}=∪{ηR(r):r∈R, r=Ψ-1(Ψ(r2r1))=r2r1}=∪{ηR(r2r1):ri∈R, Ψ(ri)=ti, i=1, 2}⊇∪{ηR(r1):r1∈R, Ψ(r1)=t1}=(Ψ(fR))(t1)).$

Thus, Ψ(fR) is a soft right ideal of T.

Theorem 8

Let ηR and ηT be soft sets over a nearsemirings R and T, respectively. Let Ψ be a nearsemiring antihomomorphism from R to T. If ηT is a soft left ideal of T, then Ψ−1(ηT) is a soft right ideal of R.

Proof

Let r1, r2R. Then

$(Ψ-1(ηT))(r1+r2)=ηT(Ψ(r1+r2)=ηT(Ψ(r1)+(Ψ(r1))⊇ηT((Ψ(r1)∩(Ψ(r1))=(Ψ-1(ηT))(r1)∩(Ψ-1(ηT))(r2).$

Similarly, one can easily verify that (Ψ−1 (ηT))(r1r2) ⊇ Ψ−1 (ηT))(r1) ∩ Ψ−1 (ηT))(r2). Also,

$(Ψ-1(ηT))(r1r2)=ηT(Ψ(r1r2)=ηT(Ψ(r2)Ψ(r1))⊇ηT(Ψ(r1))=(Ψ-1(ηT))(r1).$

Hence Ψ−1 (ηT) is a soft right ideal of R.

Example 5

Let S = {0, a, b, c, d} be a nearsemiring with the operations defined by the following Cayley tables.

I = {0, b, d} be an ideal of S and ηI be a soft right ideal of S defined as ηI(0) = S, ηI (b) = {0, b, c, d} and ηI (b) = {0, b, c, d}.

Also, let T = {0, a, b, c, d} be a nearsemiring with the operations defined by the following Cayley tables.

J = {0, b, d} be an ideal of R and ηJ be a soft left ideal of S defined as ηJ (0) = T, ηJ (b) = {0, b, c, d} and ηJ (b) = {0, b, c, d}. Let ψ : ST be an anti-homomorphism defined by ψ(x) = x. Then, for a soft left ideal ηJ of T, the inverse map ψ−1 (ηJ) = ηI is a soft right ideal of S.

5. Soft S-Subsemigroup of Nearsemirings and Soft S-Morphisms

In this section, we introduce the notion of soft S-subsemigroup of a nearsemiring S. We shift few terms of S-morphism of classical nearsemiring theory towards soft set theory. A subsemigroup H of S is called a left S-subsemigroup of S if SHH and H is called a right S-subsemigroup of S if HSH.

Definition 14

Let H be a left (resp., right) S-subsemigroup of a nearsemiring S and (η, H) be a soft set over S. If for all x, yH and rS,

• (i) η(x + y) ⊇ η(x) ∩ η(y)

• (ii) η(rx) ⊇ η(x) (resp., η(xr) ⊇ η(x))

then the soft set (η, H) is the soft left (resp., right) S-subsemigroup of S and is denoted by $(η, H) or simply $ηH.

Remark 3

Every soft left (resp., right) ideal of a nearsemiring S is also a soft left (resp., right) S-subsemigroup of a nearsemiring S.

Theorem 9

If $ηH and $ϑK, then $ηH∩˜ϑK, if it is non-null.

Proof

Let $ηH and $ϑK. Let H and K are left S-subsemigroups of S such that HK ≠ ∅, then it can easily seen that HK is a left S-subsemigroup of S. Since, ηH ∩̃ ϑK = ξH ∩̃ K, where ξH ∩̃ K (x) = η(x) ∩ ϑ(x), for all xHK. Then, for all x, yHK and for all rS,

$(i) ξH∩˜K(x+y) =ηH(x+y)∩ϑK(x+y) ⊇(η(x)∩η(y))∩(ϑ(x)∩ϑ(y)) =(η(x)∩ϑ(x))∩(η(y)∩ϑ(y)) =ξH∩˜K(x)∩ξH∩˜K(y)(ii)ξH∩˜K(rx) =η(rx)∩ϑ(rx) ⊇(η(x)∩ϑ(x))= ξH∩˜K(x).$

Hence, $ηH∩˜ϑK. Similarly, one can easily prove that $ηH∩˜ϑK.

Definition 15

Let S1 and S2 are nearsemirings and let $ηH and $⊇K. The product of two soft left (resp., right) S-subsemigroups (η, H) and (ϑ, K) is defined as (η, H) × (ϑ, K) = ξH × K, where ξH × K(x, y) = ηH(x) × ϑK(y), for all (x, y) ∈ H × K.

Theorem 10

If $ηH and $ϑK, then $ηH×ϑK.

Proof

Let $ηH and $ϑK. Since H is a left S1-subsemigroup of S1 and K is a left S2-subsemigroup of S2, it can be easily seen that H × K is a left S1 × S2-subsemigroup of S1 × S2. Since, ηH × ϑK = ξH × K, where ξH × K(x) = ηH(x) × ϑK(y) for all (x, y) ∈ H × K. Then, for all (x1, y1), (x2, y2)H × K, (r1, r2) ∈ S1× S2 we have

$(i) ξH×K((x1, y1)+(x2, y2)) =ξH×K(x1+x2,y1+y2) =ηH(x1+x2)×ϑK(y1+y2) ⊇(ηH(x1)∩ηH(x2))×(ϑK(y1)∩ϑK(y2)) =(ηH(x1)×ϑK(y1))∩(ηH(x2)×ϑK(y2)) =ξH×K(x1, y1)∩ξH×K(x2, y2) (ii) ξH×K((r1, r2)(x1, y1)) = ξH×K(r1,x1,r2y1) =ηH(r1x1)×ϑK(r2y1) ⊇ηH(x1)×ϑK(y1) = ξH×K(x1, y1).$

Hence, $ηH×ϑK. Similarly, one can prove that $ηH×ϑK.

Definition 16

Let (η, H) be a soft left (resp., right) S1-sub-semigroup of a nearsemiring S1 and (ϑ, K) be a soft left (resp., right) S2-subsemigroup of S2. Suppose f : S1S2 and g : HK be the two mappings. Then the pair (f, g) is called a soft S-homomorphism if it satisfies the following conditions.

• (i) f is an S-epimorphism

• (ii) g is a surjective mapping

• (iii) f(η(x)) = ϑ(g(x)) for all xH.

If there exists a soft S-homomorphism between (η, H) and (ϑ, K), we call (η, H) is soft S-homomorphic to (ϑ, K). Similarly, if f is an S-isomorphism of S-semigroups and g is a bijective mapping, then we call (f, g) a soft S-isomorphism, and say that (η, H) is soft S-isomorphic to (ϑ, K) i.e., (η, H) ≃ (ϑ, K).

Example 6

Let us take a nearsemiring S = {0, a, b, c, d} given in the Example2. And let H = {0, a, b, d} be a S-subsemigroup of an S-semigroup (S, +). We also assume that H = K and S1 = S2 = S. Then, let f : SS be the map defined by f(x) = x. One can easily verify that f is an S-isomorphism. Let g : HK be the map defined by g(x) = x, which is clearly a bijective mapping. Let η : H → ℘(R) be the map defined by η(x) = {yS : xyxy ∈ {0, a}}. Then, η(0) = η(a) = S, η(b) = η(d) = {0, a}. By doing a simple calculations, one can prove that (η, H) is a soft S-subsemigroup of S. Similarly, we define the map ϑ : K → ℘(S) by ϑ(x) = {yS : xyxy ∈ {0, a}}. Then, ϑ(0) = ϑ(a) = S, ϑ(b) = ϑ(d) = {0, a}. Evidently, (ϑ, K) is a soft S-subsemigroup of S. Finally, consider f(η(0)) = S, and ϑ(g(0)) = ϑ(0) = S, so f(η(0)) = ϑ(g(0)). Again, consider f(η(a)) = f(S) = S and ϑ(g(a)) = ϑ({0, a}) = S, so f(η(a)) = ϑ(g(a)). Similarly, we can show that for all xH, f(η(x))= ϑ(g(x)). Hence (η, H) ≃ (ϑ, K).

Theorem 11

Let H, K be the two S-subsemigroups of S. Let ηH and ϑK be the soft sets over S and Ψ be an S-isomorphism from H to K. If ηH is a soft S-subsemigroup of S, then so is Ψ(ηH).

Proof

Let k1, k2K. Since Ψ is surjective, there exists h1, h2H such that Ψ(h1) = k1 and Ψ(h2) = k2. Thus

$Ψ(ηH)(k1+k2)=∪{ηH(h):h∈H, Ψ(h)=(k1+k2)}=∪{ηH(h):h∈H, h=Ψ-1(k1+k2}=∪{ηH(h):h∈H, h=Ψ-1(Ψ(h1+h2))=h1+h2}=∪{ηH(h1+h2):ki∈K, Ψ(hi)=ki, i=1, 2}⊇∪{η(h1)∩η(h2):hi∈H, Ψ(hi)=ki, i=1, 2}=(∪{η(h1):k1∈K, Ψ(h1)=k1})∩(∪{η(h2):k2∈K, Ψ(h2)=k2})=(Ψ(ηH))(h1)∩(Ψ(ηH))(h2).$

Now, let rS and kK. Since Ψ is surjective, there exists h′ ∈ H such that Ψ(h′) = k. Then,

$Ψ(ηH)(rk)=∪{ηH(h):h∈H, Ψ(h)=rk}=∪{ηH(h):h∈H, h=Ψ-1(rk)}=∪{ηH(h):h∈H, h=Ψ-1(rΨ(h′))}=∪{ηH(h):h∈H, h=Ψ-1(Ψ(rh′))=rh′}=∪{ηH(rh′):h′∈H, Ψ(h′)=k}⊇∪{ηH(h′):h′∈H, Ψ(h′)=k}=(Ψ(ηH))(k).$

Hence Ψ(ηH) is a soft S-subsemigroup of S.

6. Conclusion

In this study, we have introduced soft sub-nearsemirings, soft ideals of nearsemirings and soft S-subsemigroups of nearsemirings by using the definition of soft sets introduced by Molodtsov [1]. We have applied a number of soft set operations to our newly established soft algebraic structures. Moreover, we investigated these soft algebraic structures through soft nearsemiring homomorphism and soft nearsemiring anti-homorphism. Throughout our study, we provide bunch of illustrative examples. In addition, we have provided few relationships among these soft substructures and classical substructures of nearsemirings. Through these relationships, we established the connection among the soft set theory, classical set theory and nearsemiring theory. To extend this study, one can discuss some other classical algebraic structures such as balanced nearsemirings, affine near-semirings etc on the same pattern. Classical nearsemiring is used in the theory of automata and other computer languages thus we expect that the work presented in this manuscript would be useful in the theory of fuzzy (soft) automata.

Right nearsemiring.

+0abcd.0abcd
00abcd000000
aaabdda0aaaa
bbbbddb0abbd
ccddcdc0abbd
ddddddd0abbd

Right nearsemiring.

 + 0 a b c d . 0 a b c d 0 0 a b c d 0 0 0 0 0 0 a a a b d d a 0 a a a a b b b b d d b 0 a b b d c c d d c d c 0 a b b d d d d d d d d 0 a b b d

Right nearsemiring.

 + 0 a b c d . 0 a b c d 0 0 a b c d 0 0 0 0 0 0 a a a b d d a 0 a a a a b b b b d d b 0 a b b b c c d d c d c 0 a b b b d d d d d d d 0 a d d d

Right nearsemiring.

 + 0 a b c d . 0 a b c d 0 0 a b c d 0 0 0 0 0 0 a a a b d d a 0 a a a a b b b b d d b 0 b b b b c c d d d d c 0 a a a a d d d d d d d 0 a a a a

Right nearsemiring.

 + 0 a b c d . 0 a b c d 0 0 a b c d 0 0 0 0 0 0 a a a b d d a 0 a a a a b b b b d d b 0 b b b d c c d d c d c 0 c c c c d d d d d d d 0 d d d d

Right nearsemiring.

 + 0 a b c d . 0 a b c d 0 0 a b c d 0 0 0 0 0 0 a a a b d d a 0 a a a a b b b b d d b 0 a b b b c c d d c d c 0 a b b b d d d d d d d 0 a d d d

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