(Fuzzy) strongly regular equivalence relations on semihypergroups

Abstract

In this paper, we study the least strongly regular equivalence relation containing a given binary relation on a semihypergroup. Also, we discuss the least fuzzy strongly regular equivalence relation greater than or equal to a given fuzzy relation on a semihypergroup. As an application of the results of this paper, the fundamental relation β^∗ and the least fuzzy strongly regular equivalence relation $β_{f}^{*}$ on a semihypergroup are obtained. Furthermore, we respectively give descriptions of the greatest strongly regular equivalence relation contained in a given equivalence relation containing β^∗ and the greatest fuzzy strongly regular equivalence relation less than or equal to a given fuzzy equivalence relation greater than or equal to $β_{f}^{*}$ on a semihypergroup.

Keywords

Semihypergroup strongly regular equivalence relation fuzzy strongly regular equivalence relation fundamental relation

1 Introduction

The algebraic hyperstructure theory was introduced in 1934 by Marty [27]. Hyperstructures have many applications in several branches of both pure and applied sciences [3 , 24]. In particular, a semihypergroup is the simplest algebraic hyperstructure which is a generalization of the concept of a semigroup. Nowadays many authors have studied different aspects of semihypergroups, for instance, Bonansinga and Corsini [6], Corsini [9], Davvaz [14], Davvaz and Poursalavati [17], De Salvo et al. [19], Fasino and Freni [20], Gutan [21], Hila et al. [22, 23], Leoreanu [26], and many others.

The concept of a fuzzy set was first introduced by L.A. Zadeh [39] in 1965. Later on, A. Rosenfeld [33] introduced the concept of a fuzzy group in 1971. Since then many researchers are engaged in extending the concepts of abstract algebra to the framework of the fuzzy setting (see [28 –30]). Recently, the fuzzy set theory has been well developed in the context of algebraic hyperstructure theory (see [2 , 13, 40]).

Fuzzy relations on a set have been studied since the concept of fuzzy relations was introduced by Nemitz [32]. Fuzzy equivalence relations were studied in [31]. Samhon [34] discussed the fuzzy equivalence relation generated by a given fuzzy relation on a set and the fuzzy congruence relation generated by a given fuzzy relation on a semigroup. Moreover, he obtained that the lattice of fuzzy congruence relations on a semigroup is a complete lattice.

Strongly regular equivalence relations on semihypergroups were introduced and studied in [10, 16]. Similar to the congruence relations on semigroups, strongly regular equivalence relations on semihypergroups play an important role in studying the algebraic structures of semihypergroups. Davvaz [15] introduced and studied fuzzy strongly regular relations on semihypergroups. In [1], Ameri obtained that both the set of all strongly regular equivalence relations and the set of all fuzzy strongly regular equivalence relations on a semihypergroup consist complete lattices.

The fundamental relation on a semihypergroup was introduced by Koskas [25] and studied then by many researchers (see [35 –38]).

This paper is organized as follows. In Section 2, we recall some definitions and results in semihypergroups and fuzzy sets which will be used in this paper. In Section 3, we study the least strongly regular equivalence relation containing a given binary relation on a semihypergroup and the greatest strongly regular equivalence relation contained in a given equivalence relation containing the fundamental relation on a semihypergroup. In Section 4, we respectively give descriptions of the fuzzy strongly regular equivalence relation generated by a given fuzzy relation on a semihypergroup and the greatest fuzzy strongly regular equivalence relation less than or equal to a given fuzzy equivalence relation greater than or equal to the least fuzzy strongly regular equivalence relation on a semihypergroup.

2 Preliminaries and notations

A hypergroupoid (S, ∘) is a nonempty set S together with a hyperoperation, that is a mapping $\circ : S \times S \to P^{*} (S)$ , where $P^{*} (S)$ denotes the family of all nonempty subsets of S. If x ∈ S and A, B are nonempty subsets of S, then we denote A ∘ B = ⋃ _a∈A,b∈Ba ∘ b, x ∘ A = {x} ∘ A and A ∘ x = A ∘ {x}.

A hypergroupoid (S, ∘) is called a semihypergroup if ∘ is associative, that is x ∘ (y ∘ z) = (x ∘ y) ∘ z for every x, y, z ∈ S. If a semihypergroup S contains an element 1 with the property that, for all x ∈ S, x ∘ 1 =1 ∘ x = {x}, then we say that 1 is an absolute identity. Clearly, a semihypergroup has at most an absolute identity. If a semihypergroup S has no absolute identity, then we adjoin an extra element 1 to S and define 1 ∘ 1 = {1} , 1 ∘ s = s ∘ 1 = {s} for all s ∈ S. Thus S ∪ {1} becomes a semihypergroup containing an absolute identity. We now define $S^{1} = {\begin{matrix} S & if S has an absolute identity, \\ S \cup {1} & otherwise . \end{matrix}$

Let (S, ∘) be a semihypergroup and ρ a binary relation on S. If A and B are nonempty subsets of S, then we set $A \bar{ρ} B \Leftrightarrow {\begin{matrix} a ρ b & (\forall a \in A, \exists b \in B) \\ a^{'} ρ b^{'} & (\forall b^{'} \in B, \exists a^{'} \in A) \end{matrix}$ and $A \bar{\bar{ρ}} B \Leftrightarrow a ρ b (\forall a \in A, \forall b \in B) .$ A binary relation ρ on S is called left strongly compatible if $(\forall a, b, x \in S) a ρ b \Rightarrow x \circ a \bar{\bar{ρ}} x \circ b,$ and right strongly compatible if $(\forall a, b, x \in S) a ρ b \Rightarrow a \circ x \bar{\bar{ρ}} b \circ x .$ An equivalence relation ρ is called strongly regular if it is both left and right strongly compatible.

Given a semihypergroup (S, ∘), the fundamental relation [25] β^∗ = β^∞ = ∪ _m≥1β^m is the transitive closure of the relation β = ∪ _n≥1β_n, where β₁ = 1_S = {(x, x) | x ∈ S} is the diagonal relation on S, and for every integer n > 1, β_n is defined as follows: $x β_{n} y \Leftrightarrow (\exists (x_{1}, x_{2}, \dots, x_{n}) \in S^{n}) {x, y} \subseteq \prod_{i = 1}^{n} x_{i} .$

Let X be a nonempty set. A fuzzy subset of X is a map from X to the real closed interval [0, 1]. A fuzzy relation on X is a fuzzy subset of X × X. For fuzzy relations ρ and σ on X and x, y ∈ X, we denote (ρ ∨ σ) (x, y) = max {ρ (x, y) , σ (x, y)} , (ρ ∧ σ) (x, y) = min {ρ (x, y) , σ (x, y)} , ρ^-1 (x, y) = ρ (y, x)and (ρ ∘ σ) (x, y) = ⋁ _z∈X (ρ (x, z) ∧ σ (z, y)). We say that ρ = ρ^-1 if ρ (x, y) = ρ^-1 (x, y), and ρ ≤ σ if ρ (x, y) ≤ σ (x, y) for all x, y ∈ X. The fuzzy relations _vartriangleX and ▿_X on X are defined by _vartriangleX (x, y) =0 if x ≠ y and _vartriangleX (x, y) =1 if x = y and ▿_X (x, y) =1 for all x, y ∈ X. A fuzzy relation ρ on X is called a fuzzy equivalence relation on X if (1) ρ is fuzzy reflexive, that is _vartriangleX ≤ ρ; (2) ρ is fuzzy symmetric, that is ρ = ρ^-1; (3) ρ is fuzzy transitive, that is ρ ∘ ρ ≤ ρ. The transitive closure of a fuzzy relation ρ, denoted by ρ^∞, is defined as ρ^∞ = ⋁ _n≥1ρⁿ where $ρ^{n} = \underset{n}{\underset{︸}{ρ \circ ρ \circ \dots \circ ρ}}$ . It is well known that ρ^∞ is the least fuzzy transitive relation on X greater than or equal to ρ.

Let (S, ∘) be a semihypergroup. A fuzzy relation ρ on S is called fuzzy left strongly compatible if $(\forall x, y, z \in S) ⋀_{a \in z \circ x, b \in z \circ y} ρ (a, b) \geq ρ (x, y)$ and fuzzy right strongly compatible if $(\forall x, y, z \in S) ⋀_{a \in x \circ z, b \in y \circ z} ρ (a, b) \geq ρ (x, y) .$

A fuzzy equivalence relation ρ on S is said to be fuzzy strongly regular if it is both fuzzy left and right strongly compatible.

3 Strongly regular equivalence relations

Let (S, ∘) be a semihypergroup and ρ a binary relation on S. We denote the relation ρ ∪ ρ^-1 ∪ 1_S on S by ρ^rs. It is obvious that ρ^rs is reflexive and symmetric. Let {ρ_i | i ∈ I} be the family of all strongly regular equivalence relations on S containing ρ. Then the relation ⋂_i∈Iρ_i denoted by ρ^♯ is clearly the least strongly regular equivalence relation on S containing ρ. ρ^♯ is called the strongly regular equivalence relation on S generated by ρ.

Definition 3.1. If ρ is a binary relation on a semihypergroup (S, ∘), then we define the relation ρ^sc on S as

$u ρ^{sc} v \Leftrightarrow (\exists (x_{1}, x_{2}, \dots, x_{m}) \in (S^{1})^{m}, (y_{1}, y_{2}, \dots, y_{n}) \in (S^{1})^{n}) (\exists (a, b) \in ρ) u \in \prod_{i = 1}^{m} x_{i} \circ a \circ \prod_{j = 1}^{n} y_{j}, v \in \prod_{i = 1}^{m} x_{i} \circ b \circ \prod_{j = 1}^{n} y_{j} .$

Lemma 3.2. Let ρ be a binary relation on a semihypergroup (S, ∘). Then ρ^sc is the least left and right strongly compatible relation on S containing ρ.

Proof. (1) If (u, v) ∈ ρ, then (u, v) ∈ (1 ∘ u ∘ 1) × (1 ∘ v ∘ 1). Thus (u, v) ∈ ρ^sc and so ρ ⊆ ρ^sc.

(2) Let (u, v) ∈ ρ and w ∈ S. Then $(u, v) \in (\prod_{i = 1}^{m} x_{i} \circ a \circ \prod_{j = 1}^{n} y_{j}) \times (\prod_{i = 1}^{m} x_{i} \circ b \circ \prod_{j = 1}^{n} y_{j})$ for some x₁, x₂, ⋯ , x_m, y₁, y₂, ⋯ , y_n ∈ S¹ and some (a, b) ∈ ρ. Hence $w \circ u \subseteq (w \circ \prod_{i = 1}^{m} x_{i}) \circ a \circ \prod_{j = 1}^{n} y_{j}, w \circ v \subseteq (w \circ \prod_{i = 1}^{m} x_{i}) \circ b \circ \prod_{j = 1}^{n} y_{j}$ , and so $w \circ u \bar{\bar{ρ^{sc}}} w \circ v$ . Therefore, ρ^sc is left strongly compatible. In a similar way, we obtain that ρ^sc is right strongly compatible.

(3) Suppose that σ is a left and right strongly compatible relation on S containing ρ. Then for all x₁, x₂, ⋯ , x_m, y₁, y₂, ⋯ , y_n ∈ S¹ and all (a, b) ∈ ρ, it follows that $\prod_{i = 1}^{m} x_{i} \circ a \circ \prod_{j = 1}^{n} y_{j} \bar{\bar{σ}} \prod_{i = 1}^{m} x_{i} \circ b \circ \prod_{j = 1}^{n} y_{j}$ . Therefore, ρ^sc ⊆ σ.

Lemma 3.3. Let ρ and σ be two relations on a semihypergroup (S, ∘). Then

ρ ⊆ σ ⇒ ρ^sc ⊆ σ^sc;

(ρ^sc) ^-1 = (ρ^-1) ^sc;

(ρ ∪ σ) ^sc = ρ^sc ∪ σ^sc.

Proof. It is straightforward.

Lemma 3.4. Let ρ be a left and right strongly compatible relation on a semihypergroup (S, ∘). Then so is ρ^∞.

Proof. Let (u, v) ∈ ρ^∞. Then (u, v) ∈ ρⁿ for some integer n ≥ 1. Thus there exist a₁, a₂, ⋯ , a_n-1 ∈ S such that uρa₁ρa₂ρ ⋯ ρa_n-1ρv. It follows that $w \circ u \bar{\bar{ρ}} w \circ a_{1} \bar{\bar{ρ}} w \circ a_{2} \bar{\bar{ρ}} \dots \bar{\bar{ρ}} w \circ a_{n - 1} \bar{\bar{ρ}} w \circ v$ for all w ∈ S. Hence $w \circ u \bar{\bar{ρ^{n}}} w \circ v$ and so $w \circ u \bar{\bar{ρ^{\infty}}} w \circ v$ . Similarly, $u \circ w \bar{\bar{ρ^{\infty}}} v \circ w$ . Therefore, ρ^∞ is left and right strongly compatible.

Theorem 3.5. Let ρ be a binary relation on a semihypergroup (S, ∘). Then ρ^♯ = ((ρ^rs) ^sc) ^∞.

Proof. (1) It is easy to show that ((ρ^rs) ^sc) ^∞ is an equivalence relation on S containing ρ. Moreover, by Lemma 3.2 and Lemma 3.4, we have ((ρ^rs) ^sc) ^∞ is left and right strongly compatible. Therefore, ((ρ^rs) ^sc) ^∞ is a strongly regular equivalence relation containing ρ.

(2) Suppose that σ is a strongly regular equivalence relation on S containing ρ. Then ρ^rs = ρ ∪ ρ^-1 ∪ 1_S ⊆ σ ∪ σ^-1 ∪ 1_S = σ. Hence (ρ^rs) ^sc ⊆ σ^sc = σ from Lemma 3.2. Therefore, ((ρ^rs) ^sc) ^∞ ⊆ σ^∞ = σ. By now, we obtain our conclusion.

Lemma 3.6. The fundamental relation $β^{*} = (1_{S}^{sc})^{\infty}$ on a semihypergroup (S, ∘).

Proof. It is sufficient to prove $β = 1_{S}^{sc}$ .

(1) Let (u, v) ∈ β. If (u, v) ∈ β₁, then $(u, v) \in 1_{S} \subseteq 1_{S}^{sc}$ . If (u, v) ∈ β_n for some integer n > 1, then there exist x₁, x₂, ⋯ , x_n ∈ S such that $(u, v) \in (\prod_{i = 1}^{n} x_{i})^{2} = (x_{1} \circ x_{2} \circ \prod_{i = 3}^{n} x_{i})^{2}$ . Since (x₂, x₂) ∈1_S, we have $(u, v) \in 1_{S}^{sc}$ .

(2) If $(u, v) \in 1_{S}^{sc}$ , then $(u, v) \in (\prod_{i = 1}^{m} x_{i} \circ a \circ \prod_{j = 1}^{n} y_{j})^{2}$ for some x₁, x₂, ⋯ , x_m, y₁, y₂, ⋯ , y_n ∈ S¹ and some a ∈ S. Hence there exists $k \in ℕ_{+}$ such that (u, v) ∈ β_k. Thus (u, v) ∈ β.

For a semihypergroup S, the least strongly regular equivalence relation on S clearly contains 1_S. From Theorem 3.5, we know that $(1_{S}^{sc})^{\infty}$ is the least strongly regular equivalence relation on S containing 1_S. Thus $(1_{S}^{sc})^{\infty}$ is the least strongly regular equivalence relation on S. Furthermore, by Lemma 3.6, we obtain the following conclusion given in [16].

Corollary 3.7. The fundamental relation β^∗ on a semihypergroup S is the least strongly regular equivalence relation.

Let S be a semihypergroup. We denote the set of all strongly regular equivalence relations on S by SR (S). Then SR (S) is a partially ordered set under the set inclusion. Moreover, it is well known from [1] that (SR (S) , ⊆ , ∩ , ∨) is a complete lattice where ∨ is defined as ρ ∨ σ = (ρ ∪ σ) ^♯. The fundamental relation β^∗ and the universal relation S × S are respectively the minimum element and the maximum element in SR (S).

Theorem 3.8. Let S be a semihypergroup and ρ, σ ∈ SR (S). Then ρ ∨ σ = (ρ ∘ σ) ^∞.

Proof. By Theorem 3.5, we have ρ ∨ σ = (ρ ∪ σ) ^♯ = (((ρ ∪ σ) ^rs) ^sc) ^∞. Since ρ, σ ∈ SR (S), we have ((ρ ∪ σ) ^rs) ^sc = ((ρ ∪ σ) ∪ (ρ ∪ σ) ^-1 ∪ 1_S) ^sc = (ρ ∪ σ) ^sc = ρ ∪ σ from Lemma 3.2. Thus ρ ∨ σ = (ρ ∪ σ) ^∞.

Now, since ρ ⊆ ρ ∘ σ and σ ⊆ ρ ∘ σ, ρ ∪ σ ⊆ ρ ∘ σ. Hence (ρ ∪ σ) ^∞ ⊆ (ρ ∘ σ) ^∞. On the other hand, since ρ ⊆ ρ ∪ σ and σ ⊆ ρ ∪ σ, we have ρ ∘ σ ⊆ (ρ ∪ σ) ² and so (ρ ∘ σ) ^∞ ⊆ (ρ ∪ σ) ^∞.

Corollary 3.9. Let S be a semihypergroup and ρ, σ ∈ SR (S). If ρ ∘ σ = σ ∘ ρ, then ρ ∨ σ = ρ ∘ σ.

Proof. It is straightforward.

In the later of this section, we study the greatest strongly regular equivalence relation contained in a given equivalence relation containing β^∗ on a semihypergroup. Since β^∗ is the least strongly regular equivalence relation on a semihypergroup, it is natural to require that the given equivalence relation contains β^∗.

Definition 3.10. Let E be an equivalence relation on a semihypergroup (S, ∘). Then we define the relation E^♭ on S as

$(a, b) \in E^{♭} \Leftrightarrow (\forall m, n \in ℕ_{+}) (\forall (x_{1}, x_{2}, \dots, x_{m}) \in (S^{1})^{m}, (y_{1}, y_{2}, \dots, y_{n}) \in (S^{1})^{n}) \prod_{i = 1}^{m} x_{i} \circ a \circ \prod_{j = 1}^{n} y_{j} \bar{\bar{E}} \prod_{i = 1}^{m} x_{i} \circ b \circ \prod_{j = 1}^{n} y_{j}$ .

Theorem 3.11. Let E be an equivalence relation on a semihypergroup (S, ∘) containing the fundamental relation β^∗. Then E^♭ is the greatest strongly regular equivalence relation on S contained in E.

Proof. (1) Since E is an equivalence relation on S, it is easy to show that $\bar{\bar{E}}$ is symmetric and transitive on $P^{*} (S)$ . Hence, E^♭ is symmetric and transitive on S. Since β^∗ is strongly regular, we have $\prod_{i = 1}^{m} x_{i} \circ a \circ \prod_{j = 1}^{n} y_{j} \bar{\bar{β^{*}}} \prod_{i = 1}^{m} x_{i} \circ a \circ \prod_{j = 1}^{n} y_{j}$ for all $m, n \in ℕ_{+}$ and all x₁, x₂, ⋯ , x_m, y₁, y₂, ⋯ , y_n ∈ S¹, a ∈ S. From β^∗ ⊆ E, we obtain $\prod_{i = 1}^{m} x_{i} \circ a \circ \prod_{j = 1}^{n} y_{j} \bar{\bar{E}} \prod_{i = 1}^{m} x_{i} \circ a \circ \prod_{j = 1}^{n} y_{j}$ . Thus (a, a) ∈ E^♭ and so E^♭ is reflexive. Furthermore, if (a, b) ∈ E^♭ then $1 \circ a \circ 1 \bar{\bar{E}} 1 \circ b \circ 1$ . Hence ${a} \bar{\bar{E}} {b}$ and so (a, b) ∈ E. Therefore, E^♭ is an equivalence relation on S contained in E.

(2) Let (a, b) ∈ E^♭ and c ∈ S. Then $\prod_{i = 1}^{m} x_{i} \circ (c \circ a) \circ \prod_{j = 1}^{n} y_{j} \bar{\bar{E}} \prod_{i = 1}^{m} x_{i} \circ (c \circ b) \circ \prod_{j = 1}^{n} y_{j}$ for all $m, n \in ℕ_{+}$ and all x₁, x₂, ⋯ , x_m, y₁, y₂, ⋯ , y_n ∈ S¹. It follows that $\prod_{i = 1}^{m} x_{i} \circ u \circ \prod_{j = 1}^{n} y_{j} \bar{\bar{E}} \prod_{i = 1}^{m} x_{i} \circ v \circ \prod_{j = 1}^{n} y_{j}$ for any u ∈ c ∘ a, v ∈ c ∘ b. Therefore, $c \circ a \bar{\bar{E^{♭}}} c \circ b$ . In the same way, we obtain $a \circ c \bar{\bar{E^{♭}}} b \circ c$ . Hence, E^♭ is left and right strongly compatible.

(3) Let F be a strongly regular equivalence relation on S contained in E and (a, b) ∈ F. Then $\prod_{i = 1}^{m} x_{i} \circ a \circ \prod_{j = 1}^{n} y_{j} \bar{\bar{F}} \prod_{i = 1}^{m} x_{i} \circ b \circ \prod_{j = 1}^{n} y_{j}$ for all $m, n \in ℕ_{+}$ and all x₁, x₂, ⋯ , x_m, y₁, y₂, ⋯ , y_n ∈ S¹. Thus $\prod_{i = 1}^{m} x_{i} \circ a \circ \prod_{j = 1}^{n} y_{j} \bar{\bar{E}} \prod_{i = 1}^{m} x_{i} \circ b \circ \prod_{j = 1}^{n} y_{j}$ and so (a, b) ∈ E^♭. Hence F ⊆ E^♭.

4 Fuzzy strongly regular equivalence relations

Let (S, ∘) be a semihypergroup and α a fuzzy relation on S. We set $α_{f}^{rs} = α \lor α^{- 1} \lor △_{S}$ . It is easy to see that $α_{f}^{rs}$ is fuzzy reflexive and fuzzy symmetric. Let {α_i | i ∈ I} be the set of all strongly regular equivalence relations on S greater than or equal to α. Then the fuzzy relation ⋀_i∈Iα_i denoted by $α_{f}^{♯}$ is clearly the least strongly regular equivalence relation on S greater than or equal to α. We call $α_{f}^{♯}$ the fuzzy strongly regular equivalence relation on S generated by α.

Definition 4.1. Let (S, ∘) be a semihypergroup and α a fuzzy relation on S. Then for all a, b ∈ S, we define the fuzzy relation $α_{f}^{sc}$ on S as $α_{f}^{s c} (a, b) = \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ a \in \prod_{i = 1}^{m} x_{i} ° c ° \prod_{j = 1}^{n} y_{j} \\ b \in \prod_{i = 1}^{m} x_{i} ° d ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\lor} α (c, d) .$

Lemma 4.2. Let α be a fuzzy left and right strongly compatible relation on a semihypergroup (S, ∘). If there exist x₁, x₂, ⋯ , x_m, y₁, y₂, ⋯ , y_n ∈ S¹ such that $a \in \prod_{i = 1}^{m} x_{i} \circ c \circ \prod_{j = 1}^{n} y_{j}, b \in \prod_{i = 1}^{m} x_{i} \circ d \circ \prod_{j = 1}^{n} y_{j}$ , then α (a, b) ≥ α (c, d).

Proof. We need only prove the following two statements:

If $a \in \prod_{i = 1}^{m} x_{i} \circ c, b \in \prod_{i = 1}^{m} x_{i} \circ d$ , then α (a, b) ≥ α (c, d);

If $a \in c \circ \prod_{j = 1}^{n} y_{j}, b \in d \circ \prod_{j = 1}^{n} y_{j}$ , then α (a, b) ≥ α (c, d).

Indeed, if

a \in \prod_{i = 1}^{m} x_{i} \circ c \circ \prod_{j = 1}^{n} y_{j}, b \in \prod_{i = 1}^{m} x_{i} \circ d \circ \prod_{j = 1}^{n} y_{j}

, then there exist

a^{'} \in \prod_{i = 1}^{m} x_{i} \circ c, b^{'} \in \prod_{i = 1}^{m} x_{i} \circ d

such that

a \in a^{'} \circ \prod_{j = 1}^{n} y_{j}, b \in b^{'} \circ \prod_{j = 1}^{n} y_{j}

. Thus by (1) and (2), we have α (a, b) ≥ α (a′, b′) ≥ α (c, d).

Next we prove the statement (1). Since α is fuzzy left strongly compatible, the conclusion holds for m = 1. By induction, suppose that the result is true for m = k (k ≥ 1). When m = k + 1, we have a ∈ x₁ ∘ u, b ∈ x₁ ∘ v for some $u \in \prod_{i = 2}^{m} x_{i} \circ c, v \in \prod_{i = 2}^{m} x_{i} \circ d$ . Hence α (a, b) ≥ α (u, v) ≥ α (c, d) from the fuzzy left compatibility and our hypothesis.

Similarly, we can obtain that the statement (2) is true from the fuzzy right compatibility of α.

Lemma 4.3. Let α, γ be two fuzzy relations on a semihypergroup (S, ∘). Then

$α \leq α_{f}^{sc}$ ;

$α \leq γ \Rightarrow α_{f}^{sc} \leq γ_{f}^{sc}$ ;

$(α_{f}^{sc})^{- 1} = (α^{- 1})_{f}^{sc}$ ;

$(α \lor γ)_{f}^{sc} = α_{f}^{sc} \lor γ_{f}^{sc}$ ;

$(α_{f}^{sc})_{f}^{sc} = α_{f}^{sc}$ ;

$α = α_{f}^{sc}$ if and only if α is fuzzy left and right strongly compatible.

Proof. It is straightforward to prove (1), (2) and (3).

(4) For all a, b ∈ S, we have $\begin{array}{l} {(α \lor γ)}_{f}^{sc} (a, b) \\ = \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ a \in \prod_{i = 1}^{m} x_{i} ° c ° \prod_{j = 1}^{n} y_{j} \\ b \in \prod_{i = 1}^{m} x_{i} ° d ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\lor} (α \lor γ) (c, d) \\ = \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ a \in \prod_{i = 1}^{m} x_{i} ° c ° \prod_{j = 1}^{n} y_{j} \\ b \in \prod_{i = 1}^{m} x_{i} ° d ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\lor} (α (c, d) \lor γ (c, d)) \\ = \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ a \in \prod_{i = 1}^{m} x_{i} ° c ° \prod_{j = 1}^{n} y_{j} \\ b \in \prod_{i = 1}^{m} x_{i} ° d ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\lor} α (c, d) \lor \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ a \in \prod_{i = 1}^{m} x_{i} ° c ° \prod_{j = 1}^{n} y_{j} \\ b \in \prod_{i = 1}^{m} x_{i} ° d ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\lor} γ (c, d) \\ = α_{f}^{s c} (c, d) \lor γ_{f}^{s c} (c, d) . \end{array}$ (5) By (1) and (2), we have $α_{f}^{sc} \leq (α_{f}^{sc})_{f}^{sc}$ . Conversely, for all a, b ∈ S, $\begin{array}{l} (α_{f}^{sc})_{f}^{sc} (a, b) \\ = \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ a \in \prod_{i = 1}^{m} x_{i} ° c ° \prod_{j = 1}^{n} y_{j} \\ b \in \prod_{i = 1}^{m} x_{i} ° d ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\lor} α_{f}^{s c} (c, d) \\ = \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ a \in \prod_{i = 1}^{m} x_{i} ° c ° \prod_{j = 1}^{n} y_{j} \\ b \in \prod_{i = 1}^{m} x_{i} ° d ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\lor} \underset{\begin{matrix} k, l \in ℕ_{+} \\ p_{i}, q_{j} \in S^{1} \\ c \in \prod_{i = 1}^{k} p_{i} ° s ° \prod_{j = 1}^{l} q_{j} \\ d \in \prod_{i = 1}^{k} p_{i} ° t ° \prod_{j = 1}^{l} q_{j} \end{matrix}}{\lor} α (s, t) \\ \leq \underset{\begin{matrix} m, n, k, l \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ a \in \prod_{i = 1}^{m} x_{i} ° \prod_{i = 1}^{k} p_{i} ° s ° \prod_{j = 1}^{l} q_{j} ° \prod_{j = 1}^{n} y_{j} \\ b \in \prod_{i = 1}^{m} x_{i} ° \prod_{i = 1}^{k} p_{i} ° t ° \prod_{j = 1}^{l} q_{j} ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\lor} α (s, t) \\ = α_{f}^{s c} (a, b) . \end{array}$ (6) The sufficiency can be obtained from Lemma 4.2 and (1). Next we prove the necessity. Suppose that $α = α_{f}^{sc}$ . Then for all a, b, x ∈ S, we have $\begin{array}{l} \underset{\begin{matrix} u \in x ° a \\ v \in x ° b \end{matrix}}{\land} α (u, v) = \underset{\begin{matrix} u \in x ° a \\ v \in x ° b \end{matrix}}{\land} α_{f}^{s c} (u, v) \\ = \underset{\begin{matrix} u \in x ° a \\ v \in x ° b \end{matrix}}{\land} \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ u \in \prod_{i = 1}^{m} x_{i} ° c ° \prod_{j = 1}^{n} y_{j} \\ v \in \prod_{i = 1}^{m} x_{i} ° d ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\lor} α (c, d) \\ \geq α (a, b) . \end{array}$ Similarly, $\underset{\begin{matrix} u \in a ° x \\ v \in b ° x \end{matrix}}{\land} α (u, v) \geq α (a, b) .$ Hence α is fuzzy left and right strongly compatible.

Lemma 4.4. Let α be a fuzzy left and right strongly compatible relation on a semihypergroup (S, ∘). Then so is α^∞.

Proof. For all a, b, c ∈ S and every integer n ≥ 1, $\begin{array}{l} α^{n} (a, b) = \underset{x_{1}, x_{2}, \dots, x_{n - 1} \in S}{\lor} (α (a, x_{1}) \land α (x_{1}, x_{2}) \land \dots \\ \land α (x_{n - 1}, b)) \\ \leq \underset{x_{1}, x_{2}, \dots, x_{n - 1} \in S}{\lor} (\underset{\begin{matrix} u \in c ° a \\ y_{1} \in c ° x_{1} \end{matrix}}{\land} α (u, y_{1}) \land \underset{\begin{matrix} y_{1} \in c ° x_{1} \\ y_{2} \in c ° x_{2} \end{matrix}}{\land} \\ α (y_{1}, y_{2}) \land \dots \land \underset{\begin{matrix} y_{n - 1} \in c ° x_{n - 1} \\ v \in c ° b \end{matrix}}{\land} α (y_{n - 1}, b)) \\ \leq \underset{\begin{matrix} u \in c ° a \\ v \in c ° b \end{matrix}}{\land} (\underset{y_{1}, y_{2}, \dots, y_{n - 1} \in S}{\lor} (α (u, y_{1}) \land α (y_{1}, y_{2}) \\ \land \dots \land α (y_{n - 1}, v))) \\ = \underset{\begin{matrix} u \in c ° a \\ v \in c ° b \end{matrix}}{\land} α^{n} (u, v) . \end{array}$ In the same way, we have $α^{n} (a, b) \leq \underset{\begin{matrix} u \in a ° c \\ v \in b ° c \end{matrix}}{\land} α^{n} (u, v) .$ Thus αⁿ is fuzzy left and right strongly compatible for every integer n ≥ 1. Therefore, α^∞ is fuzzy left and right strongly compatible.

Theorem 4.5. Let α be a fuzzy relation on a semihypergroup (S, ∘). Then $α_{f}^{♯} = ((α_{f}^{rs})_{f}^{sc})^{\infty}$ .

Proof. (1) It is clear that ((α^rs) ^sc) ^∞ is a fuzzy equivalence relation on S and $α \leq ((α_{f}^{rs})_{f}^{sc})^{\infty}$ . Furthermore, from Lemma 4.3 (5),(6) and Lemma 4.4, we have $((α_{f}^{rs})_{f}^{sc})^{\infty}$ is fuzzy left and right strongly compatible. Hence $((α_{f}^{rs})_{f}^{sc})^{\infty}$ is fuzzy strongly regular.

(2) Let θ is a fuzzy strongly regular equivalence relation on S and α ≤ θ. Then $α_{f}^{rs} = α \lor α^{- 1} \lor △_{S} \leq θ \lor θ^{- 1} \lor △_{S} = θ$ . Thus $(α_{f}^{rs})_{f}^{sc} \leq θ_{f}^{sc} = θ$ from Lemma 4.3 (2) and (6). It follows that $((α_{f}^{rs})_{f}^{sc})^{\infty} \leq θ^{\infty} = θ$ . Therefore, we obtain our conclusion.

Corollary 4.6. Let S be a semihypergroup. Then $((△_{S})_{f}^{sc})^{\infty}$ is the least fuzzy strongly regular equivalence relation on S.

In order to corresponding to the least strongly regular equivalence relation β^∗, we denote the fuzzy relation $((△_{S})_{f}^{sc})^{\infty}$ by $β_{f}^{*}$ .

Let S be a semihypergroup. We denote the set of all fuzzy strongly regular equivalence relations on S by FSR (S). It is well known from [1] that (FSR (S) , ≤ , ∧ , ⊔) is a complete lattice, where ⊔ is defined as $α ⊔ γ = (α \lor γ)_{f}^{♯}$ .

Theorem 4.7. Let S be a semihypergroup and α, γ ∈ FSR (S). Then α ⊔ γ = (α ∘ γ) ^∞.

Proof. It can be obtained from Lemma 4.3,4.4 and Theorem 4.5. We omit it.

By Theorem 4.7, we can also obtain the following result given in [1].

Corollary 4.8. (Proposition 2,3 [1]) Let S be a semihypergroup and α, γ ∈ FSR (S). If α ∘ γ = γ ∘ α, then α ⊔ γ = α ∘ γ. In the end of this section, we describe the greatest fuzzy strongly regular equivalence relation less than or equal to a given fuzzy equivalence relation on a semihypergroup. By corollary 4.6, we know that $β_{f}^{*}$ is the least fuzzy strongly regular equivalence relation on a semihypergroup. Therefore, if there exists a fuzzy strongly regular equivalence relation less than or equal to a given fuzzy equivalence relation FE, then $FE \geq β_{f}^{*}$ .

Definition 4.9. Let FE be a fuzzy equivalence relation on a semihypergroup (S, ∘). Then for all a, b ∈ S, we define the fuzzy relation FE^♭ on S as $F E^{♭} (a, b) = \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ c \in \prod_{i = 1}^{m} x_{i} ° a ° \prod_{j = 1}^{n} y_{j} \\ d \in \prod_{i = 1}^{m} x_{i} ° b ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} F E (c, d)$

Theorem 4.10. Let FE be a fuzzy equivalence relation on a semihypergroup (S, ∘) and $β_{f}^{*} \leq FE$ . Then FE^♭ is the greatest fuzzy strongly regular equivalence relation on S less than or equal to FE.

Proof. (1) By Lemma 4.2, we can easily see that FE ≤ FE^♭. Clearly, FE^♭ is fuzzy symmetric. Next we prove that FE^♭ is fuzzy reflexive and fuzzy transitive. For all a, b ∈ S, $\begin{array}{l} F E^{♭} (a, b) = \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ c \in \prod_{i = 1}^{m} x_{i} ° a ° \prod_{j = 1}^{n} y_{j} \\ d \in \prod_{i = 1}^{m} x_{i} ° b ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} F E (c, d) \\ \geq \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ c \in \prod_{i = 1}^{m} x_{i} ° a ° \prod_{j = 1}^{n} y_{j} \\ d \in \prod_{i = 1}^{m} x_{i} ° b ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} β^{*} (c, d) \\ \geq β^{*} (a, b) (By Lemma 4 .2) \\ \geq △_{S} (a, b), \end{array}$ and $\begin{array}{l} (F E^{♭} ° F E^{♭}) (a, b) \\ = \underset{x \in S}{\lor} (F E^{♭} (a, x) \land F E^{♭} (x, b)) \\ = \underset{x \in S}{\lor} (\underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ u \in \prod_{i = 1}^{m} x_{i} ° a ° \prod_{j = 1}^{n} y_{j} \\ w \in \prod_{i = 1}^{m} x_{i} ° x ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} F E (u, w) \land \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ w \in \prod_{i = 1}^{m} x_{i} ° x ° \prod_{j = 1}^{n} y_{j} \\ v \in \prod_{i = 1}^{m} x_{i} ° b ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} \\ F E (w, v)) \\ \leq \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ u \in \prod_{i = 1}^{m} x_{i} ° a ° \prod_{j = 1}^{n} y_{j} \\ v \in \prod_{i = 1}^{m} x_{i} ° b ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} (\underset{w \in S}{\lor} (F E (u, w) \land F E (w, v))) \\ = \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ u \in \prod_{i = 1}^{m} x_{i} ° \\ a ° \prod_{j = 1}^{n} y_{j} \\ v \in \prod_{i = 1}^{m} x_{i} ° b ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} F E^{2} (u, v) \\ \leq \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ u \in \prod_{i = 1}^{m} x_{i} ° a ° \prod_{j = 1}^{n} y_{j} \\ v \in \prod_{i = 1}^{m} x_{i} ° b ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} F E (u, v) \\ = F E^{♭} (a, b) . \end{array}$ Therefore, FE^♭ is a fuzzy equivalence relation on S less than or equal to FE.

(2) For all a, b, c ∈ S, we have $\begin{array}{l} \underset{\begin{matrix} u \in c ° a \\ v \in c ° b \end{matrix}}{\land} F E^{♭} (u, v) = \underset{\begin{matrix} u \in c ° a \\ v \in c ° b \end{matrix}}{\land} \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ s \in \prod_{i = 1}^{m} x_{i} ° u ° \prod_{j = 1}^{n} y_{j} \\ t \in \prod_{i = 1}^{m} x_{i} ° v ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} F E (s, t) \\ \geq \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ s \in \prod_{i = 1}^{m} x_{i} ° c ° a ° \prod_{j = 1}^{n} y_{j} \\ t \in \prod_{i = 1}^{m} x_{i} ° c ° b ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} F E (s, t) \\ \geq \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ s \in \prod_{i = 1}^{m} x_{i} ° a ° \prod_{j = 1}^{n} y_{j} \\ t \in \prod_{i = 1}^{m} x_{i} ° b ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} F E (s, t) \\ = F E^{♭} (a, b) . \end{array}$ Hence FE^♭ is fuzzy left strongly compatible. Similarly, we have FE^♭ is fuzzy right strongly compatible.

(3) Let γ be a fuzzy strongly regular equivalence relation on S and γ ≤ FE. Then for all a, b ∈ S, $\begin{array}{l} F E^{♭} (a, b) = \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ c \in \prod_{i = 1}^{m} x_{i} ° a ° \prod_{j = 1}^{n} y_{j} \\ d \in \prod_{i = 1}^{m} x_{i} ° b ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} F E (c, d) \\ \geq \underset{\begin{matrix} m, n \in ℕ_{+} \\ x_{i}, y_{j} \in S^{1} \\ c \in \prod_{i = 1}^{m} x_{i} ° a ° \prod_{j = 1}^{n} y_{j} \\ d \in \prod_{i = 1}^{m} x_{i} ° b ° \prod_{j = 1}^{n} y_{j} \end{matrix}}{\land} γ (c, d) \\ \geq γ (a, b) . (By Lemma 4 .2) \end{array}$

5 Conclusion

Similar to the theory of (fuzzy) congruences on semigroups, (fuzzy) strongly regular equivalence relations on semihypergroups paly an important role in studying the structure of semihypergroups. In this paper, we respectively study the least strongly regular equivalence relation containing a given binary relation and the greatest strongly regular equivalence relation contained in a given equivalence relation containing the fundamental relation β^∗ on a semihypergroup. Furthermore, we discuss the least fuzzy strongly regular equivalence relation greater than or equal to a given fuzzy relation and obtain the least fuzzy strongly regular equivalence relation $β_{f}^{*}$ on a semihypergroup. Finally, we describe the greatest fuzzy strongly regular equivalence relation less than or equal to a given fuzzy equivalence relation greater than $β_{f}^{*}$ on a semihypergroup. In the future, we will continue our research along this direction and hopefully we can respectively give descriptions of the least regular equivalence relation containing a given binary relation on a semihypergroup and the least fuzzy equivalence relation greater than or equal to a given fuzzy relation on a semihypergroup.

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