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 [5–8]. 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: S → S′ a nearsemiring homomorphism if for all a, b ∈ S; 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: S → S′ is the anti-homomorphism if for all a, b ∈ S; 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 0_Γ a left S-semigroup if there exists a composition (x, γ) ↣ xγ of S × Γ satisfying: (i) (x + y)γ = xγ + yγ; (ii) (xy)γ = x(yγ); and (iii) 0_Sγ = 0_Γ, for all x, y ∈ S 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) = rφ(x), for all x, y ∈ Γ, r ∈ S and (iii) φ(0_Γ) = 0_Γ′. We refer [31–33] 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 η_A and η_B represented by η_A ∩̃ η_B, defined by η_A ∩̃ η_B = η_A∩̃B, where η_A∩̃B (x) = η_A(x) ∩ η_B(x), for all x ∈ E.

Definition 3 ([9])

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

Definition 4 ([9])

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

Definition 5 ([34])

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

for all y ∈ B and Ψ⁻¹ (η_B)(x) = η_B(Ψ(x)), for all x ∈ A. Then Ψ(η_A) is the soft image of η_A and Ψ⁻¹ (η_B) is the soft pre-image (or soft inverse image) of η_B under the napping Ψ.

Definition 6 ([34])

Let η_A be a soft set over U and α be a subset of U. Then upper α-inclusion of soft set η_A 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 x ∈ Supp(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 : R₁ → R₂ and g : A → B 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 x ∈ A.

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, y ∈ T. We denote it by (η, T) ⪝ S or η_T ⪝ S.

Corollary 1

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

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 η_T ⪝ S and η_T1 ⪝ S, then we can easily deduce the followings.

Proposition 1

• If η_T ⪝ S and η_T1 ⪝ S, then η_T ∩̃ η_T1 ⪝ S

• If η_T1 ⪝ S₁ and ϑ_T2 ⪝ S₂, then η_T1 × ϑ_T2 ⪝ S₁ × S₂.

• If η_T ⪝ S and η_T1 ⪝ S, then η_T ∧ η_T1 ⪝ S.

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, (η_T ∨ ϑ_T1)((b, 0) + (0, a)) = (η_T ∨ ϑ_T1) (b, a) = {0, a, b, d} ⊉ (η_T ∨ ϑ_T1)(b, 0) ∩ (η_T ∨ ϑ_T1)(0, a) = {0, a, b, c, d} ∩ {0, a, b, c, d} = {0, a, b, c, d}. Hence, η_T∨ ϑ_T1 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 t ∈ T, such that t=∑i=1naibi and η_T be a soft sub-nearsemiring of S, then ηT(t)=ηT(∑i=1naibi)⊇ηT(ai)∩ηT(bi) for all 1 ≤ i ≤ n.

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

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

Definition 10

Let η_T and ϑ_T be a soft sets over the nearsemiring S. Then their soft intersection sum η_T ⊕ ϑ_T is defined as

for all x ∈ T.

Definition 11

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

for all x ∈ T.

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

Definition 12

Let η_T and ϑ_T be soft set over the nearsemiring S. Then the soft union intersection (uni-int) product η_S ★ ϑ_S is defined as

Theorem 1

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

Proposition 2

Let η_T and ϑ_T be two soft sub-nearsemirings of nearsemiring S. Then f_R ⊕ h_R̃ ⊆ f_R (resp., f_R̃ ⊕ h_R = h_R).

Theorem 2

Let η_T and ϑ_T1 be soft sets over nearsemiring S and Ψ be a nearsemiring isomorphism from T to T₁. If η_T is a soft sub-nearsemiring of S, then Ψ(η_T) is also a soft sub-nearsemiring of S.

Theorem 3

Let T and T₁ be the two sub-nearsemirings of S. Let η_T and ϑ_T1 be soft sets over nearsemiring S and Ψ be a nearsemiring homomorphism from T to T₁. If ϑ_T1 is a soft sub-nearsemiring of S, then so is Ψ⁻¹(ϑ_T1).

Definition 13

Let I be an ideal of nearsemiring S (i.e., I ⊲ S) 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, y ∈ I, r ∈ S.

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

Corollary 2

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

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 η_I 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 ≤ i ≤ n.

Theorem 4

If S is a left seminearring (i.e., zero symmetric nearsemiring) and η_I be a soft right ideal (i.e., η_I ⊲_rS) and ϑ_J be a soft left ideal (i.e., ϑ_J ⊲_lS). Then, η_I ⊙ ϑ_J ⊆ η_I ∩̃ ϑ_J.

Theorem 5

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

Corollary 3

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

Remark 2

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

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 η_I be the soft set over S. Then η_I 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≠∅.

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 η_I as η_I(0) = S, η_I (a) = {0, b}, η_I (b) = {0, a, b, d}, η_I (d) = {0, a, b, d}. Clearly, η_I is a soft ideal of nearsemiring S, one can easily verify that ηI⊇α≠∅. is an ideal of S.

Theorem 7

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

Theorem 8

Let η_R and η_T be soft sets over a nearsemirings R and T, respectively. Let Ψ be a nearsemiring antihomomorphism from R to T. If η_T is a soft left ideal of T, then Ψ⁻¹(η_T) 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 η_I be a soft right ideal of S defined as η_I(0) = S, η_I (b) = {0, b, c, d} and η_I (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 η_J be a soft left ideal of S defined as η_J (0) = T, η_J (b) = {0, b, c, d} and η_J (b) = {0, b, c, d}. Let ψ : S → T be an anti-homomorphism defined by ψ(x) = x. Then, for a soft left ideal η_J of T, the inverse map ψ⁻¹ (η_J) = η_I 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 SH ⊆ H and H is called a right S-subsemigroup of S if HS ⊆ H.

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, y ∈ H and r ∈ S,

• (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., ηH∩ϑK<SrS), if it is non-null.

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) = ξ_{H × K}, where ξ_{H × K}(x, y) = η_H(x) × ϑ_K(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).

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 : S₁ → S₂ and g : H → K 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 x ∈ H.

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 : S → S be the map defined by f(x) = x. One can easily verify that f is an S-isomorphism. Let g : H → K 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) = {y ∈ S : xℜy ⇔ xy ∈ {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) = {y ∈ S : xℜy ⇔ xy ∈ {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 x ∈ H, f(η(x))= ϑ(g(x)). Hence (η, H) ≃ (ϑ, K).

Theorem 11

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

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. Right nearsemiring.

Table 2. Right nearsemiring.