Soft intersection hyperstructures: Application in Krasner hyperrings

Abstract

In this paper first, we define the notions of left and right soft int-hypergroups and derive some of their basic properties. Second, we study these concepts in the context of complete hypergroups and polygroups. Then, we introduce the notions of left and right soft int-additive hyperrings and soft int-hyperideal. In special case we study these notions for the class of Krasner hyperrings. Finally, a characterization of soft int-hyperideal for the class of Krasner hyperfields is investigated.

Keywords

left(right) soft int-hypergroup soft int-polygroup left(right) soft int-additive hyperring soft int-hyperideal

1 Introduction

The theory of hyperstructures was introduced in 1934 by Marty [21] at the 8th Congress of Scandinavian Mathematicians. This theory has been developed by Corsini, Leoreanu, Mittas, Stratigopoulos, Vougiouklis, Davvaz [8–10 , 32] and by various authors [11 , 31]. Basic definitions and propositions about the hyperstructures are found in [9 , 13]. Krasner [18] has studied the notion of hyperfields, hyperrings and then some other researchers did, for example see [11 , 27]. The axioms of hyperrings are the same with rings. There are different types of hyperrings. If the addition + is a hyperoperation and the multiplication is a binary operation, then the hyperring is called additive hyperring. A special case of this type is the Krasner hyperring [18]. Rota [28] introduced a multiplicative hyperring, where + is a binary operation and the multiplication is a hyperoperation. De Salvo [15] studied hyperrings in which the additions and the multiplications were hyperoperations. Surveys of hyperrings theory and its applications can be found in the book of Davvaz and Leoreanu-Fotea [13]. In particular, the relationships between the fuzzy sets, soft sets and hyperrings have been considered by many researchers, for example see [1–6 , 12] and also [16 , 35].

N. Ça $\overset{ˇ}{g}$ man et al. [6] studied soft int-groups, which are different from the definition of soft groups in [2], then F. Çitak and N. Ça $\overset{ˇ}{g}$ man introduced and studied the notion of soft int-ring [7]. In this paper we introduce and study algebraic applications of the notions of left and right soft int-hypergroups, soft int-additive hyperrings, soft int-hyperideal. A characterization of soft int-hyperideal for the class of Krasner hyperfields is investigated. These new concepts are as a bridge among soft sets theory, set theory and hyperstructure theory show the effect of soft sets on hyperstructures in the sense of intersection of sets.

2 Preliminaries

We recall some basic notions of hyperstructure theory. Let H be a non-empty set and $P^{*} (H)$ be the set of all non-empty subsets of H . Let · be ahyperoperation on H, that is, · is a function from H × H into $P^{*} (H) .$ If (a, b) ∈ H × H, its image under · in $P^{*} (H)$ is denoted by a · b or sometimes for simplicity is denoted by ab. The hyperoperation is extended to subsets of H in a natural way, that is, for non-empty subsets A, B of H, A · B = ⋃ _a∈A,b∈Ba · b. The notation a · A is used for {a} · A and A · a for A · {a}. Generally, the singleton {a} is identified with its member a. The structure (H, ·) is called a semi-hypergroup if a · (b · c) = (a · b) · c for all (a, b, c) ∈ H³. Let (H, ·) be a semi-hypergroup and A be a non-empty subset of H. We say that A is a complete part of H if for every non-zero natural number n and for all a₁, …, a_n of H, the following implication holds: $\begin{matrix} A \cap 1 \prod_{i = 1}^{n} a_{i} \neq \emptyset \Rightarrow \prod_{i = 1}^{n} a_{i} \subseteq A . \end{matrix}$

Let A be a non-empty part of H. The intersection of the parts of H which are complete and contain A is called the complete closure of A in H, it will be denoted by C (A). A semi-hypergroup is a hypergroup if a · H = H · a = H for all a ∈ H. A non-empty subset K of a hypergroup (H, ·) is called a subhypergroup if it is a hypergroup. An element e of H is called an identity element if, for all x ∈ H, x ∈ x · e ∩ e · x and a′ ∈ H is called an inverse of a in H if e ∈ a · a′ ∩ a′ · a . Suppose that (H, ·) and (H′, ∘) are two semi-hypergroups. A function f : H → H′ is called a homomorphism if f (a · b) ⊆ f (a) ∘ f (b) for all a and b in H. We say that f is a good homomorphism if for all a and b in H, f (a · b) = f (a) ∘ f (b).

If (H, ·) is a hypergroup and ρhs1suhs1H × H is an equivalence, we set $A \overset{=}{ρ} B \Leftrightarrow a 1 ρ 1 b, 5 \forall a \in A, \forall b \in B,$ for all pairs (A, B) of non-empty subsets of H².

The relation ρ is called strongly regular on the left (on the right) if $x 1 ρ 1 y \Rightarrow a \cdot x \overset{=}{ρ} a \cdot y$ ( $x 1 ρ 1 y \Rightarrow x \cdot a \overset{=}{ρ} y \cdot a$ , respectively), for all (x, y, a) ∈ H³. Moreover, ρ is called strongly regular if it is strongly regular on the right and on the left.

If (H, ·) is a semi-hypergroup (hypergroup) and (x, y) ∈ H² then we denote ρ (x) by $\bar{x}$ and instead of $\bar{x} \circ \bar{y}$ we write $\bar{x} \bar{y}$ .

For all n > 1, we define the relation β_n on a semi-hypergroup H, as follows: $\begin{matrix} a 1 β_{n} 1 b \Leftrightarrow \exists (x_{1}, \dots, x_{n}) \in H^{n} 1 : 1 {a, b} \subseteq \prod_{i = 1}^{n} x_{i}, \end{matrix}$ and $β = ⋃_{i = 1}^{n} β_{n}$ , where β₁ = {(x, x) ∣ x ∈ H} is the diagonal relation on H. This relation was introduced by Koskas and studied mainly by Corsini [9]. Suppose that β^* is the transitive closure of β. The relation β^* is a strongly regular relation and it is the smallest equivalence relation on a hypergroup H, such that the quotient H/β^* is a group. The heart ω_H of a hypergroup H is the set of all elements x of H, for which the equivalence class β^* (x) is the identity of the group H/β^*, i.e., if φ : H ⟶ H/β^* is the canonical map, then ω_H = {x ∈ H | φ (x) =1_H/β^*}.

A semi-hypergroup (H, ∘) is called complete if for all natural numbers n, m ≥ 2 and all tuples (x₁, x₂, …, x_n) ∈ Hⁿ and (y₁, y₂, …, y_m) ∈ H^m, we have the following implication: $\prod_{i = 1}^{n} x_{i} \cap \prod_{j = 1}^{m} y_{j} \neq \emptyset \Rightarrow \prod_{i = 1}^{n} x_{i} = \prod_{j = 1}^{m} y_{j},$ where $\prod_{i = 1}^{n} x_{i} = x_{1} \circ x_{2} \circ \cdot \cdot \cdot \circ x_{n}$ and $\prod_{j = 1}^{m} y_{j} = y_{1} \circ y_{2} \circ \cdot \cdot \cdot \circ y_{m}$ . In practice, the next characterization is more useful.

Theorem 2.1. ([9]) A (semi-)hypergroup (H, ∘) is complete if it can be written as the union H = ⋃ _s∈SA_s of its subsets, where S and A_s satisfy the conditions:

(S, ·) is a (semi)group;

for all (s, t) ∈ S², where s ≠ t, we have A_s⋂ A_t = ∅;

if (a, b) ∈ A_s × A_t, then a ∘ b = A_s·t.

A hypergroup H is called regular if it has at least one identity and each element has at least one inverse.

Definition 2.2. ([10]) A regular hypergroup H is called reversible if the following condition holds: x ∈ y · z implies that y ∈ x · z′ and z ∈ y′ · x, where z′ and y′ are the inverse elements of z and y, respectively.

An introductory review for the theory of hypergroup appears in [9] and the book [10] contains a wealth of applications. General structure that satisfies the ring-like axioms is the hyperring. In general sense (R, + , ·) is a hyperring if + and · are two hyperoperations such that (R, +) is a hypergroup and (R, ·) is a semi-hypergroup and also · is distributive with respect to +. There are different kinds of hyperrings. If only the addition + is a hyperoperation and the multiplication · is an usual operation, then we say that R is an additive hyperring. A special case of this type is the hyperring introduced by Krasner [18]. Also, Krasner introduced a new important class of hyperfields that we call them Krasner hyperfields.

Definition 2.3. ([18]) A Krasner hyperring is an algebraic structure (R, + , ·) which satisfies the following axioms:

(R, +) is a canonical hypergroup, i.e.,

for every (x, y, z) ∈ R³, x + (y + z) = (x + y) + z,

for every (x, y) ∈ R², x + y = y + x,

there exists 0 ∈ R such that 0 + x = {x} for every x ∈ R,

for every x ∈ R there exists a unique element x′ ∈ R such that 0 ∈ x + x′; (we shall write -x for x and we call it the opposite of x.)

z ∈ x + y implies y ∈ z - x and x ∈ z - y;

(R, ·) is a semigroup having zero as a bilaterally absorbing element, i.e., x · 0 =0 · x = 0.

The multiplication is distributive with respect to the hyperoperation +.

Definition 2.4. ([18]) A Krasner hyperring (R, + , ·) is called commutative if (R, ·) is a commutative semigroup with unit element; usually is called monoid. A Krasner hyperring is called a Krasner hyperfield if (R - {0} , ·) is a commutative group.

3 Soft int-hpypergroups

In this section, we introduce and analyze the definition of soft int-hypergroups (F, P) as a generalization of Ça $\overset{ˇ}{g}$ man et. al work in [6]. We present some results about this new concept. This new concept is as a bridge among soft sets theory and hypergroup theory shows the effect of soft sets on hypergroups in the sense of intersection of sets.

Definition 3.1. ([6]) A pair (F, A) is called a soft set over U, where F is a mapping given by $F : A ⟶ P (U)$ . If K ⊆ A, then the image of set K under F is defined by F (K) = ∪ _x∈KF (x) . Moreover, the pair (F, A) is called a non-null soft set if F (A) ≠ ∅ .

Definition 3.2. ([6]) Let (F₁, A), (F₂, A) be two soft sets over U . Then, union F₁⋓F₂ and intersection F₁⋒F₂ of F₁ and F₂ are defined by (F₁⋓F₂) (x) = F₁ (x) ∪ F₂ (x) and (F₁⋒F₂) (x) = F₁ (x) ∩ F₂ (x) , for every x ∈ A, respectively.

Definition 3.3. ([6]) Let H be a hypergroup, (F, H) be a non-null soft set over U. Then, (F, H) is called a

left soft int-hypergroup over U if F (x/y) ⊇ F (x) ∩ F (y) , for all (x, y) ∈ H², where x/y = {a ∈ H|x ∈ ay} ,

right soft int-hypergroup over U if F (x ∖ y) ⊇ F (x) ∩ F (y) , for all (x, y) ∈ H², where x ∖ y = {a ∈ H|y ∈ xa} ,

soft int-hypergroup over U if it is a left soft int-hypergroup and a right soft int-hypergroup over U,

special soft int-hypergroup over U if it is a soft int-hypergroup over U and F (x ∖ x) = F (x/x) ⊇ F (y), for every (x, y) ∈ H² .

Note that the set of all left(resp. right) soft int-hypergroups over U will be denoted by S_l (H, U)(resp. S_r (H, U)) . Moreover, the set of all soft int-hypergroup and the set of all special soft int-hypergroup over U will be denoted by S (H, U) and SS (H, U) , respectively.

Remark 3.4. If H is a hypergroup, then SS (H, U) ⊆ S (H, U) = S_l (H, U) ∩ S_r (H, U) .

Proposition 3.5. The pair (S_l (H, U) , ⋒) (resp. (S_r (H, U) , ⋒)) is a semigroup.

Proof. Let (F₁, H), (F₂, H) be two soft sets over U. Then,

(F₁⋒F₂) (x/y) = F₁ (x/y) ∩ F₂ (x/y) ⊇ (F₁ (x) ∩ F₁ (y)) ∩ (F₂ (x) ∩ F₂ (y)) = (F₁⋒F₂) (x) ∩ (F₁⋒F₂) (y) .□

Proposition 3.6. Let H be a hypergroup and (F, H) be a non-null soft set over U . Then,

if x ∈ x/y, or y ∈ x/y, for every (x, y) ∈ H², then (F, H) is a left soft int-hypergroup over U ;

if x ∈ x ∖ y, or y ∈ x ∖ y, for every (x, y) ∈ H², then (F, H) is a right soft int-hypergroup over U .

Proof. The proof is straightforward.□

Proposition 3.7. If H is a commutative hypergroup then S (H, U) = S_l (H, U) = S_r (H, U) .

Proof. Let (a, b) ∈ H² and (F, H) ∈ S_l (H, U). From commutativity of H we have a/b = b ∖ a so F (b ∖ a) = F (a/b) ⊇ F (a) ∩ F (b) . Consequently S_l (H, U) ⊆ S_r (H, U) . Similarly we have S_r (H, U) ⊆ S_l (H, U) , and so S (H, U) = S_l (H, U) = S_r (H, U) .□

Example 3.8. Let H = {e, a, b} , Consider the hypergroup (H, ·) defined as follows:

$\begin{matrix} \cdot & e & a & b \\ e & e & {e, a} & {e, b} \\ a & a & a & H \\ b & b & {a, b} & H \end{matrix}$ Suppose that U = {0, 1, 2, 3} and F (e) = {1, 2} , F (a) = {0, 1, 2, 3} and F (b) = {1} be non-null soft set over U . Then (F, H) is a left soft int-hypergroup over U . which is not a right soft int-hypergroup. Indeed, $F (a ∖ e) ⊋ F (a) \cap F (e) .$

Example 3.9. Suppose that H = {e, a, b, c}. Consider the non-commutative hypergroup (H, ·), where · is defined on H as follows:

$\begin{matrix} \cdot & e & a & b & c \\ e & e & a & b & c \\ a & a & a & H & c \\ b & b & {e, a, b} & b & {b, c} \\ c & c & {a, c} & c & H \end{matrix}$ Now let U = {0, 1, 2, 3} and F (e) = {0} , F (a) = {1, 2} , and F (c) = {3} be non-null soft set over U . Then (F, H) is a soft int-hypergroup over U .

Proposition 3.10. If G is a group and (F, G) ∈ S_l (G, U) then

F (e) ⊇ F (x), for every x ∈ G,

F (x) = F (x^-1), for every x ∈ G,

S_l (G, U) = S (G, U)

Proof. Let G be a group, (F, G) ∈ S_l (G, U) and (x, y) ∈ G². Then, we have x/y = xy^-1 and x ∖ y = x^-1y and so F (e) = F (xx^-1) = F (x/x) ⊇ F (x) ∩ F (x) = F (x) . Moreover, F (x) = F (ex) = F (e/x^-1) ⊇ F (e) ∩ F (x^-1) = F (x^-1) . Hence F (x) = F (x^-1). Now let (F, U) ∈ S_l (G, U), and (x, y) ∈ G². We have F (y^-1 ∖ x^-1) = F (yx^-1) = F (x/y) ⊇ F (x) ∩ F (y) = F (y^-1) ∩ F (x^-1) . Hence F (a ∖ b) ⊇ F (a) ∩ F (b), for every (a, b)∈G², so (F, U) ∈ S_r (G, U). Consequently, S_l (G, U)= S (G, U) .□

Proposition 3.11. If G is a group and (F, U) ∈ S_r (G, U), then

F (e) ⊇ F (x), for every x ∈ G,

F (x) = F (x^-1), for every x ∈ G,

S_r (G, U) = S (G, U)

Corollary 3.12. If G is group then $S_{l} (G, U) = S_{r} (G, U) = S (G, U) = SS (G, U) .$

Theorem 3.13. Let H be a complete hypergroup. Then, S (H, U) = SS (H, U) .

Proof. Let H be a complete hypergroup. According to Theorem 2.1 (H, ∘) can be written as the union H = ⋃ _s∈GA_s of its subsets, where G and A_s satisfy the conditions:

(G, ·) is a group;

for all (s, t) ∈ G², where s ≠ t, we have A_s⋂ A_t = ∅;

if (a, b) ∈ A_s × A_t, then a ∘ b = A_s·t.

Now let (F, H) be a soft int-hypergroup over U and (a, b) ∈ H² . It is easy to see that a/a = a ∖ a = A_e, where e is the identity element of G . Therefore

F (a ∖ a) = F (a / a) = F (A_{e}) = F (b / b) \supseteq F (b) .

□

Let H be a hypergroup and (F, H) ∈ S (H, U). Then, W_F,H is defined as the set {x ∈ H ∣ F (xw_H) = F (w_H)}.

Proposition 3.14. If H is a hypergroup and (F, H) ∈ S (H, U), then

F (w_H) ⊇ F (x) , for every x ∈ H,

if F (xyw_H) ⊇ F (xw_H) ∩ F (yw_H), for every (x, y) ∈ H² then W_F,H is a subsemihypergroup of H .

Proof. (1). Let x ∈ H . Then, we have F (w_H)⊇F (x/x) ⊇ F (x) . (2) . Suppose that $(x, y) \in W_{F, H}^{2}$ and t ∈ xy . Then F (tw_H) = F (xyw_H) ⊇ F (xw_H) ∩ F (yw_H) = F (w_H) . Hence t ∈ W_F,H .□

4 Soft int-polygroups

In [7] the authors introduced soft int-rings and their algebraic applications. In this section first we introduce and analyze the definition of soft int- polygroup (F, P) and we present some results about this concept. Then we investigate soft int-additive hyperrings, soft int-hyperideal as a generalization of soft int-rings. A characterization of soft int-hyperideal for the class of Krasner hyperfields is also investigated.

A polygroup [14] is a system ℘ =< P, · , e, - 1 >, where e ∈ P, ^-1 is a unitary operation on P, · maps P × P into the non-empty subsets of P, and the following axioms hold for all x, y, z in P:

(x · y) · z = x · (y · z),

e · x = x · e = x,

x ∈ y · z implies y ∈ x · z^-1 and z ∈ y^-1 · x .

The following elementary facts about polygroups easily derive from the axioms: e ∈ x · x^-1 ∩ x^-1 · x, e^-1 = e, (x^-1) ^-1 = x, and (x · y) ^-1 = y^-1 · x^-1, where A^-1 = {a^-1| a ∈ A}. A non-empty subset K of a polygroup 〈P, · , e, - 1〉 is a subpolygroup of P if (1) (x, y) ∈ K² implies x · y ∈ K; (2) x ∈ K implies x^-1 ∈ K. A subpolygroup N of a polygroup 〈P, · , e, - 1〉 is normal in P if N · x = x · N (or x^-1 · N · x ⊆ N), for all x ∈ P .

Definition 4.1. Let 〈P, · , e, - 1〉 be a polygroup and U be a non-empty set we mean by $S_{l}^{*} (P, U)$ and $S_{r}^{*} (P, U)$ are the sets {(F, P) ∈ S_l (P, U) |F (e) = F (w_P)} and {(F, P) ∈ S_r (P, U) |F (e) = F (w_P)}, respectively.

Remark 4.2. If G is a group then $S_{r}^{*} (G, U) = S_{r} (G, U) = S_{l} (G, U) = S_{l}^{*} (G, U) .$

Proposition 4.3. If P is a polygroup and $(F, P) \in S_{l}^{*} (P, U)$ (resp. $(F, P) \in S_{r}^{*} (P, U))$ then

F (x) = F (x^-1) , for every x ∈ P,

$(F, P) \in S_{r}^{*} (P, U)$ (resp. $(F, P) \in S_{l}^{*} (P, U))$ .

Proof. (1). Suppose that P is a polygroup, $(F, P) \in S_{l}^{*} (P, U)$ and x ∈ P then F (x^-1) = F (e/x)⊇F (e) ∩ F (x) = F (x) . Hence F (x^-1) ⊇ F (x) , for every x ∈ P so F (x) = F (x^-1) .

(2). Let (x, y) ∈ P² then x/y = x^-1 ∖ y^-1 so F (x^-1∖y^-1) = F (x/y) ⊇ F (x) ∩ F (y) = F (x^-1) ∩ F (y^-1) . Thus $(F, P) \in S_{r}^{*} (P, U) .$ □ Corollary 4.4. If P is a polygroup, then $S_{r}^{*} (P, U) = S_{l}^{*} (P, U) .$

Proposition 4.5. If P is a polygroup and (F, P) ∈ S_l (P, U) (resp. (F, P) ∈ S_r (P, U)) then

F (xx^-1) ⊇ F (x), for every x ∈ P,

F (x^-1w_P) ⊇ F (x), for every x ∈ P,

F (x^-1w_P) = F (xw_P) , for every x ∈ P .

Proof. (1). Let P be a polygroup, (F, P) ∈ S_l (P, U) and (x, y) ∈ P². Then, we have x/y = xy^-1 so F (xx^-1) = F (x/x) ⊇ F (x) . (2). $\begin{matrix} F (x^{- 1} w_{P}) = & F (w_{P} / x) \\ = & F (\cup_{a \in w_{P}} a / x) \\ = & \cup_{a \in w_{P}} F (a / x) \\ \supseteq & \cup_{a \in w_{P}} F (a) \cap F (x) \\ = & F (w_{P}) \cap F (x) \\ = & F (x) \end{matrix}$ (3). Let x ∈ P and a ∈ xw_P . Then, F (a^-1w_P) ⊇ F (a) so $\cup_{a \in {xw}_{P}} F (a^{- 1} w_{P}) \supseteq \cup_{a \in {xw}_{P}} F (a) .$ Consequently F (x^-1w_P) ⊇ F (xw_P) . Similarly we have F (xw_P) ⊇ F (x^-1w_P) and so F (x^-1w_P) = F (xw_P) .□

Corollary 4.6. If P is a polygroup and (F, P) ∈ S_l (P, U) then W_F,P is a subpolygroup of P .

Theorem 4.7. Let P be a polygroup, (F, P) ∈ S_l (P, U) (resp. (F, P) ∈ S_r (P, U)) and F (xy^-1w_P) = F (w_P) for some (x, y) ∈ P². Then, F (xw_P) = F (yw_P) .

Proof. Suppose that (x, y) ∈ P². In one hand we have F (xw_P) = F (xy^-1yw_P) ⊇ F (xy^-1w_P) ∩ F (yw_P) = F (yw_P) . On the other hand $\begin{matrix} F ({yw}_{P}) = & F (y^{- 1} w_{P}) \\ = & F (x^{- 1} {xy}^{- 1} w_{P}) \\ \supseteq & F (x^{- 1} w_{P}) \cap F ({xy}^{- 1} w_{P}) \\ = & F ({xw}_{P}) . \end{matrix}$ Thus F (xw_P) = F (yw_P) .□

Definition 4.8. Let P be a polygroup and (F, P) ∈ S (P, U) . Then, NS (P, U) is called the set of all normal soft int-polygroups in P, where NS (P, U) = {(F, P) ∈ S (P, U) |F (y^-1xy) = F (x) , ∀ (x, y) ∈ P²}.

Definition 4.9. Suppose that P is a polygroup and (F, P) ∈ S (P, U) then CS (P, U) is called the set of all abelian soft int-polygroups in P, where CS (P, U) = {(F, P) ∈ S (P, U) |F (xy) = F (yx) , ∀ (x, y) ∈ P²}.

Example 4.10. Suppose that P = {e, a, b}. Consider the polygroup (P, ·), where · is defined on P as follows:

$\begin{matrix} \cdot & e & a & b \\ e & e & a & b \\ a & a & {e, a} & b \\ b & b & b & {e, a} \end{matrix}$ Now let U = {0, 1, 2, 3} and F (e) = {0, 1} , F (a) = {1, 2} , and F (b) =∅ be non-null soft set over P . Then F (a^-1 · a · a) = F ({e, a}) = {0, 1, 2} ≠ F (a) . Hence CS (P, U) ≠ NS (P, U).

Definition 4.11. Let P be a polygroup. We define (1) N^*S (P, U) = {(F, P) ∈ S (P, U) |F (y^-1xyw_P) = F (xw_P) , ∀ (x, y) ∈ P²}, (2) C^*S (P, U) = {(F, P) ∈ S (P, U) |F (xyw_P) = F (yxw_P) , ∀ (x, y) ∈ P²}.

Proposition 4.12. Let P be a polygroup and (x, y) ∈ P² . Then, C^*S (P, U) = N^*S (P, U).

Proof. Suppose that F ∈ C^*S (P, U) . In one hand we have $\begin{matrix} F (y^{- 1} {xyw}_{P}) = & F (y^{- 1} w_{P} {xyw}_{P}) \\ = & ⋃_{t \in xy} F (y^{- 1} w_{P} {tw}_{P}) \\ = & ⋃_{t \in xy} F (y^{- 1} {tw}_{P}) \\ = & ⋃_{t \in xy} F ({ty}^{- 1} w_{P}) \\ = & F ({xyy}^{- 1} w_{P}) \\ = & F ({xw}_{P}) . \end{matrix}$ On the other hand let (F, P) ∈ N^*S (P, U) . Then $\begin{matrix} F ({xyw}_{P}) = & F ({xyxx}^{- 1} w_{P}) \\ = & ⋃_{t \in yx} F ({xtx}^{- 1} w_{P}) \\ = & ⋃_{t \in yx} F ({tw}_{P}) \\ = & F ({yxw}_{P}) . \end{matrix}$ Thus C^*S (P, U) = N^*S (P, U).□

Remark 4.13. If P is a group then CS (P, U) = C^*S (P, U) = N^*S (P, U) = NS (P, U) .

Proposition 4.14. Let P be a polygroup and (F, P) ∈ S (P, U). Then, the following assertions are equivalent:

F (xyw_P) = F (yxw_P), for all x, y ∈ P,

F (xyx^-1w_P) = F (yw_P), for all x, y ∈ P,

F (xyx^-1w_P) ⊇ F (yw_P), for all x, y ∈ P,

F (xyx^-1w_P) ⊆ F (yw_P) , for all x, y ∈ P .

Proof. (1 ⇒2). Let (x, y) ∈ P². Then $\begin{matrix} F ({xyx}^{- 1} w_{P}) = & F ({xw}_{P} {yx}^{- 1} w_{P}) \\ = & ⋃_{t \in {yx}^{- 1}} F ({xw}_{P} {tw}_{P}) \\ = & ⋃_{t \in {yx}^{- 1}} F ({xtw}_{P}) \\ = & ⋃_{t \in {yx}^{- 1}} F ({txw}_{P}) \\ = & F ({yx}^{- 1} {xw}_{P}) \\ = & F ({yw}_{P}) . \end{matrix}$ (2 ⇒3) Immediate. (3 ⇒4) $\begin{matrix} F ({xyx}^{- 1} w_{P}) = & ⋃_{t \in {xyx}^{- 1}} F ({tw}_{P}) \\ \subseteq & ⋃_{t \in {xyx}^{- 1}} F (x^{- 1} {txw}_{P}) \\ = & ⋃ F (x^{- 1} {xyx}^{- 1} {xw}_{P}) \\ = & F ({yw}_{P}) . \end{matrix}$ (4 ⇒1) We have F (xyw_P) = F (xyxx^-1w_P) ⊆ F (yxw_P) . On the other hand $F ({yxw}_{P}) = F ({yxyy}^{- 1} w_{P}) \subseteq F ({xyw}_{P}) .$ Thus F (xyw_P) = F (yxw_P) .□

Theorem 4.15. Let P be a polygroup. (F, P) ∈ C^*S (P, U) if and only if F (x^-1y^-1xyw_P) ⊇ F (xw_P) , for every (x, y) ∈ P² .

Proof. Assume that (F, P) ∈ C^*S (P, U) and (x, y) ∈ P². Then $\begin{matrix} F (x^{- 1} y^{- 1} {xyw}_{P}) = & F ({xw}_{P} \ y^{- 1} {xyw}_{P}) \\ \supseteq & F ({xw}_{P}) \cap F (y^{- 1} {xyw}_{P}) \\ = & F ({xw}_{P}) \cap F ({xw}_{P}) \\ = & F ({xw}_{P}) . \end{matrix}$ Conversely $\begin{matrix} F (x^{- 1} {yxw}_{P}) = & F ({yy}^{- 1} x^{- 1} {yxw}_{P}) \\ = & F (y^{- 1} w_{P} \ y^{- 1} x^{- 1} {yxw}_{P}) \\ \supseteq & F (y^{- 1} w_{P}) \cap F (y^{- 1} x^{- 1} {yxw}_{P}) \\ = & F (y^{- 1} w_{P}) \cap F ({yw}_{P}) \\ = & F ({yw}_{P}) . \end{matrix}$ Thus (F, P) ∈ C^*S (P, U) .□

Theorem 4.16. Let P be a polygroup, (F, P) ∈ C^*S (P, U) and F (x^-1y^-1xyw_P) = F (w_P) , for every (x, y) ∈ P². Then, (F, P) ∈ C^*S (P, U) .

Proof. Let (x, y) ∈ P² . Then we have $\begin{matrix} F ({xyw}_{P}) = & F ((yx) (yx)^{- 1} (xy) w_{P}) \\ \supseteq & F ({yxw}_{P}) \cap F (x^{- 1} y^{- 1} {xyw}_{P}) \\ = & F ({yxw}_{P}) . \end{matrix}$ Thus F (xyw_P) ⊇ F (yxw_P). Similarly we have F (yxw_P) ⊇ F (xyw_P) so (F, P) ∈ C^*S (P, U) .□

5 Left and right soft int-additive hyperrings

In this section, we introduce and analyze the definition of left and right soft int-additive hyperrings (F, R) and we present some results about this concept.

Definition 5.1. Let (R, + , ·) be an additive hyperring and (F, R) be a non-null soft set over U such that F (x · y) ⊇ F (x) ∩ F (y) , for all (x, y) ∈ R² . Then

(F, R) is called left soft int-additive hyperring over U, if (F, R) is a left soft int-hypergroup over U for the hyperoperation + .

(F, R) is called right soft int-additive hyperring over U, if (F, R) is a right soft int-hypergroup over U for the hyperoperation + .

(F, R) is called soft int-additive hyperring over U, if (F, R) is a soft int-hypergroup over U for the hyperoperation + .

(F, R) is called special soft int-additive hyperring over U, if (F, R) is a special soft int-hypergroup over U for the hyperoperation + .

The set of all left(resp. right) soft int-additive hyperrings over U will be denoted by SA_l (R, U)(resp. SA_r (R, U)) . Moreover, the set of all soft int-additive hyperring and the set of all special soft int-additive hyperring over U will be denoted by SA (R, U) and SSA (R, U) , respectively.

Remark 5.2. If (R, + , ·) is an additive hyperring, then SSA (R, U) ⊆ SA (R, U) = SA_l (R, U) ∩ SA_r (R, U) .

Remark 5.3. If R is a krasner hyperring, then SA_l (R, U) = SA_r (R, U) = SA (R, U) .

Definition 5.4. Let (R, + , ·) be an additive hyperring. Then, a soft-int additive hyperring (F, R) is called a soft left int- hyperideal over U if F (x · y) ⊇ F (y) for all (x, y) ∈ R² and it is called a soft right int- hyperideal over U, if F (x · y) ⊇ F (x) for all (x, y) ∈ R² . If (F, R) is a soft left and right int-hyperideal over U, then it is called to be a soft int-hyperideal over U .

Proposition 5.5. Let (R, + , ·) be a Krasner hyperring and (F, R) ∈ SA (R, U). Then, (F, R) is a soft int-hyperideal over U if and only if

F (x - y) ⊇ F (x) ∩ F (y) ,

F (x · y) ⊇ F (x) ∪ F (y) , for all (x, y) ∈ R² .

Proof. By considering that in a Krasner hyperring R, we have x - y = x/y = y ∖ x for all (x, y) ∈ R² the proof is strightforward.□

Theorem 5.6. Let (R, + , ·) be a Krasner hyperfield and (F, R) be a soft set over U. Then, (F, R) is a soft int-hyperideal over U if and only if F (x · y^-1) ⊇ F (x) ∩ F (y) and F (1_R) = F (x) ⊆ F (0_R) for all xy ≠ 0_R, where 0_R is the bilaterally absorbing element of R and 1_R is the identity element of (R - {0_R} , ·).

Proof. (⇒) Suppose that (R, + , ·) be a Krasner hyperfield and (F, R) is a soft int- hyperideal over U. If x ≠ 0_R then F (1_R) = F (x · x^-1) ⊇ F (x) . Also F (x) = F (x · 1_R) ⊇ F (1_R) . Consequently F (1_R) = F (x) . In addition F (0_R) = F (0_R · x) ⊇ F (x) , for all x ∈ R . Moreover, F (x · y^-1) ⊇ F (x) ∩ F (y^-1) = F (x) ∩ F (y) . Because F (y) = F (y^-1) , for all (x, y) ∈ R × R - {0_R} . (⇐) . Suppose that (x, y) ∈ R² . Case 1. If x · y ≠ 0_R then F (x · y) ⊇ F (x) ∩ F (y^-1) = F (x) ∩ F (y) . Notice that F (y) = F (y^-1) = F (1_R) . Now suppose that there exists 0_R ≠ t ∈ x - y then F (x - y) ⊇ F (t) = F (1_R) = F (x) ⊇ F (x) ∩ F (y) . If 0_R = x - y then F (x - y) = F (0_R) ⊇ F (x) ∩ F (y) . Moreover F (x · y) = F (1_R) = F (x) = F (y) , hence F (x · y) ⊇ F (x) ∪ F (y) . Similarly for the case 2; x · y = 0_R the conditions (1) and (2) of Proposition 4.5 are valid. Thus (F, R) is a soft int-hyperideal over U .□

6 Conclusion

In this paper using soft sets we have defined the notions of left and right soft int-hyperstructures, left and right soft int-hypergroups, soft int-polygroups, soft int-additive hyperrings and soft int-hyperideal. We made theoretical studies on these concepts. A characterization of soft int-hyperideal for the class of Krasner hyperfields is investigated. These new concepts are as a bridge among soft sets theory, set theory and hyperstructure theory show the effect of soft sets on hyperstructures in the sense of intersection of sets. To extend our work further research can be done in other hyperstructures such as H_v-groups and H_v-rings.

References

Acar

, Koyuncu

, Tanay

, Soft sets and soft rings, Comput Math Appl 59 (2010), 3458–3463.

Aktas

, Çağman

, Soft sets and soft groups, Inform Sci 177 (2007), 2726–2735.

Ali

M.I.

, Feng

, Liu

, Min

W.K.

, Shabir

, On some new operations in soft set theory, Comput Math Appl 57 (2009), 1547–1553.

Ameri

, Hoskova-Mayerova

, Fuzzy Continuous Polygroups, In: Aplimat - 15th Conference on Applied Mathematics 2016 Proceedings. Bratislava: Vydavatelstvo STU –Slovak University of Technoology in Bratislava, (2016), 13–19.

Çağman

and Enginoglu

, Soft theory and uni-int decision making, European J Oper Res 207 (2010), 848–855.

Çağman

, Çitak

and Aktas

, Soft int-group and its applications to group theory, Neural Comput Appl. doi:10.1007/s00521-011-0752-x.

Çitak

and Çağman

, Soft int-rings and its algebraic applications, Journal of Intelligent and Fuzzy Systems 28 (2015), 1225–1233.

Comer

S.D.

, Polygroups derived from cogroups, J Algebra 89 (1984), 397–405.

Corsini

, Prolegomena of Hypergroup Theory, 2 edition, Aviani Editore, 1993.

10.

Corsini

, Leoreanu

, Applications of Hyperstructures Theory, Advanced in Mathematics. Kluwer Academic Publishers, 2003.

11.

Cristea

, Jančić-Rašović

, Composition of hyperrings, An tiin Univ Al I Cuza Iaşi Mat (N.S.) 21 (2013), 81–94.

12.

Cristea

, Jančić-Rašović

, Operations on fuzzy relations: A tool to construct new hyperrings, J Mult Valued Logic Soft Comput 21 (2013), 183–200.

13.

Davvaz

, Leoreanu-Fotea

, Hyperring Theory and Applications, International Academic Press, Palm Harbor, Fla, USA, 2007.

14.

Davvaz

, Polygroup Theory and Related Systems, World Scientific Publishing Co. Pte. Ltd., London, 2013.

15.

De Salvo

, Hyperrings and hyperfields, Annales Scientifiques de l’Universite de Clermont-Ferrand II 22 (1984), 89–107.

16.

Feng

, Jun

Y.B.

, Zhao

, Soft semirings, Comput Math Appl 56 (2008), 2621–2628.

17.

Freni

, A new characterization of the derived hypergroup via strongly regular equivalences, Comm Algebra 30(8) (2002), 3977–3989.

18.

Krasner

, A class of hyperrings and hyperfields, Int J Math Math Sci 6 (1983), 307–311.

19.

Jantani

, Jafarpour

, Leoreanu

, On Strongly Associative (semi)hypergroups, J Indones Math Soc 23 (2017), 43–53.

20.

Jun

Y.B.

, Park

C.H.

, Applications soft set in ideal theory of BCK / BCI-algebras, Inform Sci 178 (2008), 2466–2475.

21.

Marty

, Sur uni Generalization de la Notion de Group, 8^th Congress Math. Scandenaves, Stockholm, Sweden (1934), 45–49.

22.

, Zhan

, Ali

M.I.

, Mehmood

, A survey of decision making methods based on two classes of hybrid soft set models, Artificial Intelligence Review 49 (2018), 511–529.

23.

, Zhan

, Leoreanu-Fotea

, Rough soft hyperrings and corresponding decision making, Journal of Intelligent & Fuzzy Systems 33 (2017), 14791489.

24.

, Zhan

, Davvaz

, Notes on Approximations in Hyperrings, J Mult-Valued Logic Soft Comput 29 (2017), 389–394.

25.

Mittas

, Hyperanneaux et certaines de leurs propriétés, G. R. Acad Sc t 269 (1969), A623–A626.

26.

Mousavi

S.Sh.

, Leoreanu

and Jafarpour

R

-parts in (semi)hypergroups, Ann Mat Pura Appl 190 (2011), 667–680. doi:10.1007/s10231-010-0168-8).

27.

Phanthawimol

, Punkla

, Kwakpatoon

, Kemprasit

, On homomorphisms of Krasner hyperrings, An tiin Univ Al I Cuza Iaşi Mat (N.S.) 57 (2011), 239–246.

28.

Rota

, Strongly distributive multiplicative hyperrings, J Geom Appl 39 (1990), 130–138.

29.

Sezgin

, Atagun

A.O.

, On operations of soft sets, Comput Math Appl 61(5) (2011), 1457–1467.

30.

Spartalis

, A class of hyperrings, Riv Mat Pura Appl 4 (1989), 55–64.

31.

Vougiouklis

, The fundamental relation hyperrings. The general hyperfield, Proc Forth Int Congress on Algebraic hyperstructures and Applications, Word scientific, 1991.

32.

Vougiouklis

, Hyperstructures and their representation, Hadronic press, Inc, palm Harber, USA, 115, 1994.

33.

Vougiouklis

, Davvaz

, Commutative rings obtained from hyperrings (H_v-rings) with α^*-relations, Comm Algebra 35 (2007), 3307–3320.

34.

Zhan

, Alcantud

J.C.R.

, A novel type of soft rough covering and its application to multicriteria group decision making, Artificial Intelligence Review, 2018, https:doi.org/10.1007/s10462-018-9617-3.

35.

Zhan

, Alcantud

J.C.R.

, A survey of parameter reduction of soft sets and corresponding algorithms, Artificial Intelligence Review, 2018. doi:10.1007/s10462-017-9592-0.