Hypercongruences in fuzzy AG-hypergroupoids

Abstract

We introduce the notion of fuzzy Abel-Grassmann’s hypergroupoid, hypercongruence, fuzzy hypercongruence, fuzzy strong hypercongruence, compatible relations in an Abel-Grassmann’s hypergroupoid. This paper is aimed to study fuzzy hyperideals, smallest fuzzy hyperideals, fuzzy equivalence relations, fuzzy compatible fuzzy strong compatible, fuzzy hypercongruences, fuzzy strong hypercongruences, fuzzy regular, fuzzy strong regular relations and fuzzy hypercongruences in Abel-Grassmann’s hypergroupoids. Characterizations of hypercongruences, their corresponding quotient structure, homomorphisms and an important theorem on embedding Abel-Grassmann’s hypergroupoids by means of fuzzy sets. We show that each hypergroupoid is embedded into a poe-hypergroupoid of all fuzzy subsets of an Abel-Grassmann’s hypergroupoid.

Keywords

Abel-Grassmann’s hypergroupoid fuzzy equivalence relation fuzzy hypercongruence homomorphism fuzzy strong hypercongruence cartesian product regular Abel-Grassmann’s hypergroupoid

1 Introduction

An Abel-Grassmann’s groupoid (briefly as AG-groupoid) is a groupoid S satisfying the left invertive law (ab) c = (cb) a for all a, b, c ∈ S . The essential characteristic of the left invertive law places an AG-groupoid into the class of non-associative algebraic structures such as quasigroups and loops [21 , 32]. It is generally considered a structure midway between a groupoid and a commutative semigroup. Every commutative semigroup is an AG-groupoid, but the converse is not true.

After the introduction of the concept of fuzzy sets by Zadeh [44], in 1965, it has found multiple applications in the field of mathematics and related areas. Several researches conducted the researches on the generalizations of the notions of fuzzy sets with huge applications in computer, logics and many branches of pure and applied mathematics. The theoretical aspect of fuzzy set theory served as a tool in broadening the concepts of classical algebraic structures into a new version of algebraic structures such as fuzzy sets, fuzzy relations, fuzzy equivalence relations and fuzzy congruence relations, fuzzy groups and fuzzy semigroups. Rosenfeld [33], defined the concept of fuzzy group. Many research articles have been published in the field of fuzzy algebra. Nemitz and Sanchez [31, 37], considered fuzzy relations and fuzzy equivalence relations. Kuroki [26], investigated fuzzy congruence on groupoids and gave a brief knowledge of fuzzy congruences on groups via their fuzzy normal subgroups. A concise work on fuzzy semigroups can be found in [28]. Abel-Grassmann’s groupoid for its unique structure has attracted a great attention in fuzzy set theory over time and many researchers have shown their contributions in this area. The reader is referred to articles covering different basic properties of fuzzy AG-groupoids, fuzzy ordered AG-groupoids, fuzzy ideals in AG-groupoids and regularity classes of fuzzy AG-groupoids [24 , 42].

Hyperstructure theory was introduced in 1934 by the French mathematician Marty [27], as a generalization of an operation by the hyperoperation, from the single-valued operation to the multi-valued one. Since then, hyperstructure has got a tremendous attention among the mathematics society. A great contribution has been made in developing the theory of hyperstructures for the last few decades. In his book, Corsini [7] presents the basic ideas in terms of hypergroup theory. In their review Corsini & Leoreanu [10], provide sufficient knowledge of hyperstructure theory and particularly a thorough detail on the applications of hyperstructure theory in the fields of geometry, graphs, fuzzy sets, cryptography, automata, lattices, binary operations, codes and artificial intelligence is categorized. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Some important notions of hyperstructures and semihypergroups can be found in [10 , 40].

Fuzzy hyperstructure is an important research direction in recent years. Fuzzy hyperoperation which uses fuzzy sets connected through a pair of elements is widely studied with many applications. A considerable number of articles have been published aiming a variety of areas in mathematical sciences. Fuzzy equivalence relations, fuzzy congruences of hypergroups, fuzzy congruences of semihypergroups, fuzzy lattices and fuzzy hypergroups through fuzzy relations are a few of them [1 , 39].

In this contribution, we present some new important characterizations of fuzzy hyperideals in fuzzy AG-hypergropoid. We introduce the concept of hypercongruence in AG-hypergroupoid, fuzzy AG-hypergroupid, fuzzy hypercongruence, fuzzy strong hypercongruence, compatible relations in an AG-hypergroupoid. Some necessary results in section 2 which encompasses fuzzy left (right) hyperideals, fuzzy bi-hyperideal and the smallest fuzzy left hyperideal are obtained. In section 3, fuzzy equivalence relations, fuzzy compatible fuzzy strong compatible, fuzzy hypercongruence and fuzzy strong hypercongruences are considered. Some further work on composition of fuzzy relations is provided and we give a few interesting results. Section 4 deals with hypercongruences, quotient AG-hypergroupoid, homomorphisms and an important theorem on embedding an AG-hypergroupoid into a poe-AG-groupoid of the family of all fuzzy subsets of AG-hypergroupoid, is proved. Finally, in section 5, we discuss fuzzy strong regular relations and propose some new results.

2 Fuzzy AG-hypergroupoids

In this section, we recall basic definitions for a hypergroupoid, give examples supporting the notion of fuzzy AG-hypergroupoids and characterize fuzzy right (left, bi)-hyperideals of fuzzy AG-hypergroupoids. We prove for a fuzzy subset f, the smallest fuzzy left hyperideal of the form f ∪ (S ∘ f) of an AG-hypergroupoid that contains f.

Definition 2.1. { [7] A nonempty set X together with a hyperoperation “∘” given as $\begin{matrix} \circ : X & \times X \to P^{*} (X) \\ (a, b) \to a \circ b \end{matrix}$ where $P^{*} (X)$ denotes the set of all nonempty subsets of X, is called a hypergroupoid. If $A, B \in P^{*} (X),$ then, A ∘ B conveniently denoted as AB is denoted by AB = ⋃ {a ∘ b ∣ a ∈ A, b ∈ B} . The singleton {a} is generally expressed as a . Then aA and Aa show {a} A and A {a} .

If (X, ∘) is a hypergroupoid, a mapping f of X into the closed interval [0, 1] , that is f : X → [0, 1] is called a fuzzy subset of X . Let X_f denotes the set of all fuzzy subsets of X . A fuzzy hyperoperation on X is a mapping ∘ : X × X → X_f defined as (a, b) → a ∘ b . The pair (X, ∘) is called a fuzzy hypergroupoid.

Definition 2.2. A fuzzy hypergroupoid (S, ∘) is called a fuzzy AG-hypergroupoid if for all a, b, c ∈ S, (a ∘ b) ∘ c = (c ∘ b) ∘ a .

Let f be any fuzzy subset of S, then $(a \circ f) (x) = {\begin{matrix} ⋁_{y \in S} ((a \circ y) (x) \land f (y)) if f \neq 0 \\ 0 otherwise . \end{matrix}$ and $(f \circ a) (x) = {\begin{matrix} ⋁_{y \in S} (f (y) \land (y \circ a) (x)) if f \neq 0 \\ 0 otherwise . \end{matrix}$

Definition 2.3. [38] Let (S, ∘) be a fuzzy hypergroupoid and f, g are two fuzzy subsets of S . Then f ∘ g denotes the fuzzy subset of S defined by $(f \circ g) (x) = ⋁_{y, z \in S} f (y) \land g (z) \land (y \circ z) (x) for all x \in S .$

Example 1. The following Cayley’s table

∘	a	b	c	d	e
a	a	a	a	a	a
b	a	{a, b}	a	{a, b}	a
c	a	a	a	a	{a, e}
d	a	{a, b}	a	{a, b}	{a, e}
e	a	a	a	a	a

for S = {a, b, c, d, e} , satisfies left invertive law for an AG-hypergroupoid. Moreover, since (dd) e ≠ d (de) , then (S, ∘) is not associative.

Example 2. If for all a, b ∈ S, a ∘ b = ω_{a,b}, where ω_{a,b} is the characteristic function of {a, b}, then (S, ∘) is a fuzzy AG-hypergroupoid. For if a, b, c ∈ S, then $\begin{matrix} ((a \circ b) \circ c) (x) & = & (ω_{{a, b}} \circ c) (x) \\ = & ⋁_{y \in S} ((ω_{{a, b}}) (y) \land (y \circ c) (x)) \\ = & {\begin{matrix} (y \circ c) (x) if y \in {a, b} \\ 0 if y \notin {a, b} . \end{matrix} \\ = & {\begin{matrix} ω_{{y, c}} (x) if y \in {a, b} \\ 0 if y \notin {a, b} . \end{matrix} \\ = & {\begin{matrix} 1 if x \in {a, b, c} \\ 0 if x \notin {a, b, c} . \end{matrix} \end{matrix}$ $\begin{matrix} ((c \circ b) \circ a) (x) & = (ω_{{c, b}} \circ a) (x) \\ = ⋁_{y \in S} ((ω_{{c, b}}) (y) \land (y \circ a) (x)) \\ = {\begin{matrix} (y \circ a) (x), if y \in {a, b} \\ 0, if y \notin {a, b} . \end{matrix} \\ = {\begin{matrix} ω_{{y, a}} (x), if y \in {a, b} \\ 0, if y \notin {a, b} . \end{matrix} \\ = {\begin{matrix} 1 if x \in {a, b, c} \\ 0 if x \notin {a, b, c} \end{matrix} \end{matrix}$ which shows that (a ∘ b) ∘ c = (c ∘ b) ∘ a .

Also, if ω_a represents the characteristic function of a ∈ S, then it is easy to show that S together with fuzzy hyperoperation defined as a ∘ b = ω_ab for all a, b ∈ S, is a fuzzy AG-hypergroupoid.

Example 3. Let f be a nonzero fuzzy AG-subgroupoid of an AG-groupoid S and a, b ∈ S . Then, S together with the hyperoperation ∘ defined by $(a \circ b) (x) = {\begin{matrix} f (a) \land f (b) if x = ab \\ 0 otherwise . \end{matrix}$ is a fuzzy AG-hypergroupoid. Indeed, if a, b, c ∈ S, then $\begin{matrix} ((a \circ b & ) & \circ c) (x) = ⋁_{y \in S} (a \circ b) (y) \land (y \circ c) (x) \\ = & f (a) \land f (b) \land (ab \circ c) (x) \\ = & {\begin{matrix} f (a) \land f (b) \land f (ab) \land f (c) if x = (ab) c \\ 0 otherwise . \end{matrix} \\ = & {\begin{matrix} f (a) \land f (b) \land f (c) if x = (ab) c \\ 0 otherwise . \end{matrix} \end{matrix}$ On the other hand $\begin{matrix} ((c \circ b & ) & \circ a) (x) = ⋁_{y \in S} (c \circ b) (y) \land (y \circ a) (x) \\ = f (c) \land f (b) \land (cb \circ a) (x) \\ = {\begin{matrix} f (c) \land f (b) \land f (cb) \land f (a), if x = (cb) a \\ 0 otherwise . \end{matrix} \\ = {\begin{matrix} f (c) \land f (b) \land f (a), if x = (cb) a \\ 0 otherwise . \end{matrix} \end{matrix}$

Example 4. Let S be a non-empty set. Define the fuzzy hyperoperation “∘” as follows: $(a \circ b) (x) = {\begin{matrix} \frac{1}{4} if t \in {a, b} \\ 0 otherwise . \end{matrix}$

Then (S, ∘) is a fuzzy AG-hypergroupoid.

Example 5. Let S = Z. Define in S the following fuzzy hyperoperation: ∘ : S × S → S_f (x, y) ⟼ x ∘ y

defined as: $\begin{matrix} (x \circ y) (t) \\ = {\begin{matrix} 0.5 if t, x, y have the same parity \\ 0.3 if t has the same parity with x or y \\ and x, y have opposite parity \\ 1 if x = y \end{matrix} \end{matrix}$ for all x, y ∈ S. The case when x = 0, x will have even parity.

Then (S, ∘) is a fuzzy AG-hypergroupoid.

Theorem 2.4. Let (S, ∘) be a fuzzy AG-hypergroupoid. Then ω_a ∘ ω_b = a ∘ b, for all a, b ∈ S .

Proof. Let a, b ∈ S . Then, for all x ∈ S $\begin{matrix} (ω_{a} \circ ω_{b}) (x) \\ = & ⋁_{y \in S} ω_{a} (y) \land (y \circ ω_{b}) (x) = (a \circ ω_{b}) (x) \\ = & ⋁_{z \in S} (a \circ z) (x) \land ω_{b} (z) = (a \circ b) (x) . \end{matrix}$ Thus, ω_a ∘ ω_b = a ∘ b .□

Definition 2.5. A fuzzy subset f of a fuzzy AG-hypergroupoid (S, ∘) is called fuzzy AG-subhypergroupoid if f ∘ f ⊆ f .

Definition 2.6. A fuzzy subset f of a fuzzy AG-hypergroupoid (S, ∘) is called a left (right) fuzzy hyperideal if a ∘ f ⊆ f (f ∘ a ⊆ f), for all a ∈ S .

Definition 2.7. A fuzzy subset f of a fuzzy AG-hypergroupoid (S, ∘) is called a fuzzy bi-hyperideal if (f ∘ S) ∘ f ⊆ f .

Theorem 2.8. If f and g are two fuzzy AG-subhyper-groupoids on (S, ∘), then f ∩ g is also a fuzzy AG-subhypergroupoid of (S, ∘) .

Theorem 2.9. Let f and g be two fuzzy hyperideals of a fuzzy AG-hypergroupoid (S, ∘) . Then

(1)f ∩ g and f ∪ g are the fuzzy left hyperideals,

(2) f ∩ g is a fuzzy bi-hyperideal,

(3) The intersection of a family of fuzzy left (right) hyperideals of S is a fuzzy left (right) hyperideal of S .

Proof. (1). Let x ∈ S . Then $\begin{matrix} (x \circ (f \cap g)) (a) & = & ⋁_{y \in S} (x \circ y) (a) \land (f \cap g) (y) \\ = & ⋁_{y \in S} ((x \circ y) (a) \land f (y) \land g (y)) \\ \leq & ⋁_{y \in S} (x \circ y) (a) \land f (y) \\ = & (x \circ f) (a) \leq f (a) . \end{matrix}$ Similarly, (x ∘ (f ∩ g)) (a) ≤ g (a) . Then, for all x ∈ S, $(x \circ (f \cap g)) (a) \leq f (a) \land g (a) = (f \cap g) (a) .$ Also, $\begin{matrix} (x \circ (f \cup g)) (a) \\ = ⋁_{y \in S} ((x \circ y) (a) \land (f \cup g) (y)) \\ = ⋁_{y \in S} ((x \circ y) (a) \land f (y) \lor ((x \circ y) (a) \land g (y))) \\ \leq ⋁_{y \in S} ((x \circ z) (a) \land f (z)) \lor \underset{b \in S}{⋁} ((x \circ b) (a) \land g (b)) \\ = (x \circ f) (a) \lor (x \circ g) (a) \leq f (a) \lor g (a) \\ = (f \cup g) (a), for all x \in S . \end{matrix}$

(2) Since f ∩ g ⊆ f and f ∩ g ⊆ g, then ((f ∩ g) ∘ S) ∘ f ∩ g ⊆ (f ∘ S) ∘ f ⊆ f and ((f ∩ g) ∘ S) ∘ f ∩ g ⊆ (g ∘ S) ∘ g ⊆ g . Thus, ((f ∩ g) ∘ S) ∘ f ∩ g ⊆ f ∩ g .

(3). Let {f_i} _i∈I be a family of fuzzy left hyperideals and x ∈ S and let $M = \cap_{i \in I} f_{i}$ . Then $\begin{matrix} (x \circ M) (a) \\ = & ⋁_{y \in S} (x \circ y) (a) \land M) (y) \\ = & ⋁_{y \in S} ((x \circ y) (a) {\land f_{1} (y) \land f_{2} (y) \land . . .}) \\ \leq & ⋁_{y \in S} (x \circ y) (a) \land f_{1} (y) \\ = & (x \circ f_{1}) (a) \leq f_{1} (a) . \end{matrix}$ In the same manner, $(x \circ M) (a) \leq f_{2} (a), (x \circ M) (a) \leq f_{3} (a)$ and so on.□

Theorem 2.10. Let f be a fuzzy subset of (S, ∘) . Then f is a fuzzy bi-hyperideal if and only if (f∘ S) ∘ f ⊆ f .

Proof. If f be a fuzzy bi-hyperideal, then (f ∘ x) ∘ f ⊆ f for all x ∈ S . Therefore, for all p ∈ S, ((f ∘ S) ∘ f) (p) = ⋁ _x,y,z∈S ((x ∘ y) ∘ z) (p) ∧ f (x) ∧ f (z) = ⋁ _y∈S ((f ∘ y) ∘ f) (p) ≤ ⋁ _y∈Sf (p) = f (p) .

Conversely, if (f ∘ S) ∘ f ⊆ f, then for all x ∈ S, (f ∘ x) ∘ f ⊆ f .□

Theorem 2.11. Let (S, ∘) be a fuzzy AG-hypergroupoid. Then the following statements hold true:

(1) ω_S is a fuzzy left hyperideal.

(2) ω_S ∘ a = S ∘ a, for all a ∈ S .

(3) S ∘ a is a fuzzy left hyperideal.

(4) If f is a fuzzy subset of S, then S ∘ f is a fuzzy left hyperideal.

Theorem 2.12. Let (S, ∘) be a fuzzy AG-hypergroupoid. If f is a fuzzy hyperideal of (S, ∘) , then the following statements hold true:

(1) f ∘ a is a fuzzy left hyperideal.

(2) f ∘ S is a fuzzy left hyperideal.

Let f be a fuzzy subset of S . The intersection of all left fuzzy hyperideals of S containing f is a fuzzy left hyperideal of S containing f and it is contained in every other fuzzy left hyperideal. Such a fuzzy left hyperideal is referred as a fuzzy left hyperideal generated by f . We have the following theorem.

Theorem 2.13. Let f be a fuzzy subset of S, then f ∪ (S ∘ f) is the smallest fuzzy left hyperideal of S containing f .

Proof. Let x ∈ S . Then $\begin{matrix} (x \circ (f \cup (S \circ f))) (a) \\ = ⋁_{y \in S} ((x \circ y) (a) \land (f \cup (S \circ f) (y))) \\ = ⋁_{y \in S} ((x \circ y) (a) \land (f (y) \lor (S \circ f) (y))) \\ = ⋁_{y \in S} ((x \circ y) (a) \land f (y)) \lor ((x \circ y) (t) \land (S \circ f) (y)) \\ \leq ⋁_{y \in S} ((x \circ y) (a) \land f (y)) \\ \lor ⋁_{y \in S} ((x \circ y) (a) \land (S \circ f) (y)) \\ = (x \circ f) (a) \lor ⋁_{y, z \in S} ((x \circ y) (a) \land (z \circ f) (y)) \\ = (x \circ f) (a) \\ \lor ⋁_{y, z, u \in S} ((x \circ y) (a) \land (z \circ u) (y) \land f (u)) \\ = (x \circ f) (a) \lor ⋁_{z, u \in S} ((x \circ y) \circ u) (a) \land f (u)) \\ \leq (x \circ f) (a) \lor ⋁_{u \in S} ((S \circ u) \land f (u)) \\ = (x \circ f) (a) \lor (S \circ f) (a) = (S \circ f) (a) \\ \leq f (a) \lor (S \circ f) (a) = (f \cup (S \circ f)) (a) . \end{matrix}$

This shows that f ∪ (S ∘ f) is a fuzzy left hyperideal containing f . If g is a fuzzy left hyperideal containing f, then f ∪ S ∘ f ⊆ g and it is the smallest fuzzy left hyperideal of S containing f. In fact, f ⊆ g, therefore, S ∘ f ⊆ S ∘ g ⊆ g .□

3 Fuzzy Hypercongruence

In this section, fuzzy hypercongruences, fuzzy strong hypercongruences and their different properties in an AG-hypergroupoid are investigated. If S is an AG-hypergroupoid, then the family of all fuzzy strong hypercongruences denoted by $F_{SC} (S)$ on S is a partially ordered set that satisfies the set inclusion property. If $f_{r}, g_{r} \in F_{SC} (S),$ then $f_{r} \cap g_{r} \in F_{SC} (S)$ and it is the greatest lower bound. The intersection of the family of all fuzzy hypercongruences containing f_r ∪ g_r is the least upper bound.

Definition 3.1. [25] Let f_r be a fuzzy binary relation from X into Y and g_r be a fuzzy relation from Y into Z . The composition f_r ∘ _cg_r is defined as (f_r ∘ _cg_r) (x, z) = ⋁ _y∈Y {f_r (x, y) ∧ g_r (y, z) }.

Definition 3.2. [4] Let S be hypergroupoid. A fuzzy binary relation f_r on S is called:

(1) fuzzy compatible if $\begin{matrix} (⋀_{x \in a \circ c} ⋁_{z \in b \circ d} f_{r} (x, z)) \land (⋀_{z \in b \circ d} ⋁_{x \in a \circ c} f_{r} (x, z)) \\ \geq f_{r} (a, b) \forall a, b, c, d \in S, \end{matrix}$

(2) fuzzy strong compatible if $\begin{matrix} ⋀_{x \in a \circ c, z \in b \circ d} f_{r} (x, z) \geq f_{r} (a, b) \\ \land g_{r} (c, d) \forall a, b, c, d \in S . \end{matrix}$ Definition 3.3. [25] A fuzzy binary relation f_r on S is called an equivalence relation iff

(1) f_r (x, x) =1 for all x ∈ S, (fuzzy reflexive),

(2) f_r (x, y) = f_r (y, x) for all x, y ∈ S, (fuzzy symmetric),

(3) f_r (x, z) ≥ ⋀ _y∈S (f_r (x, y) ∧ f_r (y, x)) for all x, z ∈ S (fuzzy transitive).

We give the following definition.

Definition 3.4. A fuzzy equivalence and fuzzy compatible relation f_r on an AG-hypergroupoid S is called a fuzzy hypercongruence. If f_r is fuzzy strong compatible, then f_r is fuzzy strong hypercongruence.

Equivalently, an equivalence relation ρ on (S, ∘) is said to be a fuzzy strong hypercongruence if (a, b) ∈ ρ (or aρb) , (c, d) ∈ ρ (or cρd) implies that if x ∈ S be such that (a ∘ c) (x) >0, then there exists y ∈ S such that (b ∘ d) (y) >0 and (a ∘ c) (x) = (b ∘ d) (y) and (x, y) ∈ ρ (or xρy) . Conversely, if t ∈ S be such that (b ∘ d) (t) >0, then there exists s ∈ S such that (a ∘ c) (s) >0 and (a ∘ c) (s) = (b ∘ d) (t) and (s, t) ∈ ρ (or sρt) .

Theorem 3.5. Let f_r be a fuzzy equivalence relation on AG-hypergroupoid S, then f_r ∘ _cf_r = f_r .

Proof. Since f_r is fuzzy equivalence, f_r (x, y) ≥ ⋁ _z∈S (f_r (x, z) ∧ f_r (z, y)) = f_r ∘ _c f_r (x, y) ≥ (f_r (x, x) ∧ f_r (x, y)) = f_r (x, y) . Thus, f_r ∘ _cf_r = f_r .□

Theorem 3.6. Let f_r be a fuzzy binary relation on AG-hypergroupoid S . Then the following statements hold true.

f_r is a fuzzy hypercongruence if and only if it is both a fuzzy left and a fuzzy right compatible equivalence relation.

f_r is a fuzzy strong hypercongruence if and only if it is both fuzzy left and fuzzy right strong compatible equivalence relation.

Proof. (1) Suppose f_r is fuzzy hypercongruence. Then f_r is a similarity relation. Also, for a, b, t ∈ S, we have (⋀ _x∈t∘a ⋁ _z∈t∘bf_r (x, z)) ∧ (⋀ _z∈t∘b ⋁ _x∈t∘af_r (x, z)) ≥ f_r (a, b) . Similarly, (⋀ _x∈a∘t ⋁ _z∈b∘tf_r (x, z)) ∧ (⋀ _z∈b∘t ⋁ _x∈a∘tf_r (x, z)) ≥ f_r (a, b) . Thus f_r is both fuzzy left and right compatible similarity relation.

Conversely, let a, b, c, d ∈ S . Then (⋀ _x∈b∘c ⋁ _z∈b∘df_r (x, z)) ∧ (⋀ _z∈b∘d ⋁ _x∈b∘cf_r (x, z)) ≥ f_r (c, d) . Also, for every x ∈ a ∘ c, z ∈ b ∘ d, f_r (x, z) ≥ ⋁ _s∈S (f_r (x, s) ∧ f_r (s, z)) ≥ f_r (x, t) ∧ f_r (t, z) ∀t ∈ b ∘ c . Therefore, ⋁_z∈b∘df_r (x, z) ≥ ⋁ _t∈b∘df_r (x, t) ∧ ⋁ _t∈b∘cf_r (t, z) . Thus $\begin{matrix} \land_{x \in a \circ c} ⋁_{z \in b \circ d} f_{r} (x, z) & \geq & (\land_{x \in a \circ c} ⋁_{t \in b \circ c} f_{r} (x, z)) \\ \land & (\land_{z \in b \circ d} ⋁_{t \in b \circ d} f_{r} (t, z)) \\ \geq & f_{r} (a, b) \land f_{r} (d, c) \\ = & f_{r} (a, b) \land f_{r} (c, d) . \end{matrix}$ Hence, f_r is fuzzy hypercongruence.

(2) Suppose f_r is fuzzy strong hypercongruence and a, b, c, t ∈ S . Since f_r is reflexive, f_r (t, t) =1 . Also, by compatibility f_r (a, b) = f_r (a, b) ∧ f_r (t, t) ≤ f_r (a ∘ t, b ∘ t) . So, f_r is fuzzy right compatible. Similarly, it is easy to show that f_r is fuzzy left compatible. Thus, f_r is both fuzzy left and right compatible.

Conversely, if a, b, c, d ∈ S, then by right compatibility of f_r, f_r (a, b) ≤ f_r (a ∘ c, b ∘ c) . Thus, f_r (a, b) ∧ f_r (c, d) ≤ f_r (a ∘ c, b ∘ c) ∧ f_r (b ∘ c, b ∘ d) . For any x ∈ a ∘ c and z ∈ b ∘ d and by transitivity of f_r, $\begin{matrix} f_{r} (x, z) & \geq & ⋁_{t \in S} (f_{r} (x, t) \land f_{r} (t, z)) \\ \geq & f_{r} (x, t) \land f_{r} (t, z) (\forall t \in b \circ c) \\ \geq & f_{r} (a \circ c, b \circ c) \land f_{r} (b \circ c, b \circ d) \\ = & f_{r} (a \circ c, b \circ d) . \end{matrix}$ So, f_r (a, b) ∧ f_r (c, d) ≤ f_r (a ∘ c, b ∘ d) . Hence, f_r is a fuzzy hypercongruence.□

If f_r, g_r, h_r are fuzzy hypercongruence relations on (S, ∘) and composition is symmetric, then, the definition of composition follows the left invertive law of Abel-Grassmann’s groupoid. In fact; for all a, b ∈ S $\begin{matrix} ((f_{r} \circ_{c} g_{r}) \circ_{c} h_{r}) (a, b) \\ = & ⋁_{c \in S} ((f_{r} \circ_{c} g_{r}) (a, c) \land h_{r} (c, b)) \\ = & ⋁_{c \in S} ⋁_{s \in S} (f_{r} (a, s) \land g_{r} (s, c) \land h_{r} (c, b)) \\ = & ⋁_{s \in S} ⋁_{c \in S} (h_{r} (b, c) \land g_{r} (c, s)) \land f_{r} (s, a)) \\ = & ((h_{r} \circ_{c} g_{r}) \circ_{c} f_{r}) (b, a) \\ = & ((h_{r} \circ_{c} g_{r}) \circ_{c} f_{r}) (a, b) \\ (since (h_{r} \circ_{c} g_{r}) \circ_{c} f_{r} is symmetric) . \end{matrix}$

Proposition 3.7. [25, Proposition 1.7] If f_r, g_r are any fuzzy relations on X and f_r ∘ _cg_r = g_r ∘ _cf_r, then $\begin{matrix} (\underset{b \in X}{⋁} \underset{x \in X}{⋁} f_{r} (a, x) \land g_{r} (x, b) \\ ⋀ \underset{y \in X}{⋁} f_{r} (b, y) \land g_{r} (y, c)) \\ = & (\underset{b \in X}{⋁} \underset{x \in X}{⋁} f_{r} (a, x) \land f_{r} (x, b) \\ ⋀ \underset{y \in X}{⋁} g_{r} (b, y) \land g_{r} (y, c)) . \end{matrix}$

Proposition 3.8. If f_r and g_r are strong fuzzy hypercongruences on AG-hypergroupoid S such that f_r ∘ _cg_r = g_r ∘ _cf_r, then f_r ∘ _cg_r is also a fuzzy congruence on S .

Proof. We show first f_r ∘ _cg_r is a similarity relation. Reflexivity is obvious. For symmetry, let x, y ∈ S . Then $\begin{matrix} (f_{r} \circ_{c} g_{r}) (x, y) = ⋁_{s \in S} (f_{r} (x, s) \land g_{r} (s, y)) \\ = & ⋁_{s \in S} (g_{r} (y, s) \land f_{r} (s, x)) \\ (since f_{r}, g_{r} arefuzzyhypercongruences), \\ = & (g_{r} \circ_{c} f_{r}) (y, x) = (f_{r} \circ_{c} g_{r}) (y, x) . \end{matrix}$ Since (f_r ∘ _cg_r) ∘ _c (f_r ∘ _cg_r) ⊂ f_r ∘ _cg_r, then f_r ∘ _cg_r is transitive. Thus, f_r ∘ _cg_r is a similarity relation. For fuzzy left compatibility, let a, b, c, t ∈ S. Then $\begin{matrix} (f_{r} \circ_{c} g_{r}) (a, b) = ⋁_{x \in S} (f_{r} (a, x) \land g_{r} (x, b)) \\ \leq & ⋁_{x \in S} (f_{r} (a \circ_{c} t, x \circ_{c} t) \land g_{r} (x \circ_{c} t, b \circ_{c} t)) \\ \leq & ⋁_{s \in S} (f_{r} (a \circ_{c} t, s) \land g_{r} (s, b \circ_{c} t)) \\ = & (f_{r} \circ_{c} g_{r}) (a \circ_{c} t, b \circ_{c} t) . \end{matrix}$ Similarly, let a, b, c, t ∈ S. Then $\begin{matrix} (f_{r} \circ_{c} g_{r}) (a, b) = ⋁_{x \in S} (f_{r} (a, x) \land g_{r} (x, b)) \\ \leq & ⋁_{x \in S} (f_{r} (t \circ_{c} a, t \circ_{c} x) \land g_{r} (t \circ_{c} x, t \circ_{c} b)) \\ \leq & ⋁_{s \in S} (f_{r} (t \circ_{c} a, s) \land g_{r} (s, t \circ_{c} b)) \\ = & (f_{r} \circ_{c} g_{r}) (t \circ_{c} a, t \circ_{c} b) . \end{matrix}$ Hence, f_r ∘ _cg_r is fuzzy left strong and fuzzy right strong compatible relation. Thus, by Theorem 3.6, f_r ∘ _cg_r is fuzzy hypercongruence.□

Proposition 3.9. If f_r ∘ _cg_r of two fuzzy strong hypercongruences f_r, g_r of AG-hypergroupoid S is also a fuzzy congruence, then f_r ∘ _cg_r = f_r ∨ g_r .

Proof. Let a, b ∈ S . Then $\begin{matrix} (f_{r} \circ_{c} g_{r}) (a, b) & = & ⋁_{z \in S} (f_{r} (a, z) \land g_{r} (z, b)) \\ \geq & (f_{r} (a, b) \land g_{r} (b, b)) = f_{r} (a, b) . \end{matrix}$ Similarly, (f_r ∘ _cg_r) (a, b) ≥ g_r (a, b) . Suppose, M be a fuzzy strong congruence on S such that M ≥ f_r and M ≥ g_r . Then, for any a, b ∈ S, $\begin{matrix} (f_{r} \circ_{c} g_{r}) (a, b) & = & ⋁_{z \in S} (f_{r} (a, z) \land g_{r} (z, b)) \\ \leq & ⋁_{z \in S} (M (a, z) \land M (z, b)) \\ \leq & M (a, b) . \end{matrix}$ Hence, f_r ∘ _cg_r is the least upper bound of f_r and g_r .□

4 Homomorphisms and Hypercongruence

This section deals with hypercongruences, quotient AG-hypergroupoid, homomorphisms and an important theorem on embedding an AG-hypergroupoid into a poe-AG-groupoid of the family of all fuzzy subsets of AG-hypergroupoid.

Let (S, ∘) and (T, ∘) be two AG-hypergroupoids. A mapping θ from S into T is called a strong homomorphism if θ (a ∘ b) = θ (a) ∘ θ (b) , for all a, b ∈ S and it is called inclusion homomorphism if θ (a ∘ b) ⊆ θ (a) ∘ θ (b) , for all a, b ∈ S .

Definition 4.1. Let (S, ∘) and (T, ∘) be AG-hypergroupoids. Then (S, ∘) is embedded in (T, ∘) if there is a mapping θ : S → T which is strong homomorphism and one-one.

Definition 4.2. [43, Definition 18] Let (S, ∘) be AG-hypergroupoid. An element x ∈ (S, ∘) is called regular if there exists a ∈ (S, ∘) such that x ∈ (x ∘ a) ∘ x . If all the elements of (S, ∘) are regular, then it is called regular.

It is of much interest that the image of a strong homomorphism, that is, Imθ, in terms of regularity results in regularity. In the following, we prove this very important result.

Theorem 4.3. If θ : S → T is a strong homomorphism from a regular AG-hypergroupoid (S, ∘) into AG-hypergroupoid (T, ∘), then, Imθ is regular.

Proof. Let x ∈ S. Then, there exists a ∈ S such that x ∈ (x ∘ a) ∘ x . Let t ∈ Imθ. Then there exists x ∈ S such that t = θ (x). We have $\begin{matrix} t = θ (x) \in θ ((x \circ a) \circ x) & = & (θ (x) \circ θ (a)) \circ θ (x) \\ = & (t \circ θ (a)) \circ t . \end{matrix}$ Hence, Imθ is a regular AG-hypergroupoid.□

Let (S, ø) and (T, ⊙) be distinct AG-hypergroupoids. The cartesian product of (S, ø) and (T, ⊙) is an AG-hypergroupoid with the following hyperoperation: $(a_{1}, b_{1}) \otimes (a_{2}, b_{2}) = (a_{1} ø a_{2}) \times (b_{1} ⊙ b_{2}) .$ In fact, $\begin{matrix} ((a_{1}, b_{1}) & \otimes & (a_{2}, b_{2})) \otimes (a_{3}, b_{3}) \\ = & ((a_{1} ø a_{2}) \times (b_{1} ⊙ b_{2})) \otimes (a_{3}, b_{3}) \\ = & (((a_{1} ø a_{2}) ø a_{3}) \times ((b_{1} ⊙ b_{2}) ⊙ b_{3})) \\ = & (((a_{3} ø a_{2}) ø a_{1}) \times ((b_{3} ⊙ b_{2}) ⊙ b_{1})) \\ = & ((a_{3} ø a_{2}) \times (b_{3} ⊙ b_{2})) \otimes (a_{1}, b_{1}) \\ = & ((a_{3}, b_{3}) \otimes (a_{2}, b_{2})) \otimes (a_{1}, b_{1}) \end{matrix}$ shows that ((S, ø) × (T, ⊙) , ⊗) is an AG-hypergroupoid.

Theorem 4.4. Let (S, ø) and (T, ⊙) be two regular AG-hypergroupoids. Then, ((S, ø) × (T, ⊙) , ⊗) is a regular AG-hypergroupoid with the hyperoperation defined as (a₁, b₁) ⊗ (a₂, b₂) = (a₁ ø a₂) × (b₁ ⊙ b₂).

Proof. Let (a, b) ∈ (S, ø) × (T, ⊙) . Then a ∈ (a ø x) ø a and b ∈ (b ⊙ y) ⊙ b for x ∈ (S, ø) and y ∈ (T, ⊙) . We have $\begin{matrix} (a, b) \in ((a ø x) ø a) & \times & ((b ⊙ y) ⊙ b) \\ = & ((a ø x) \times (b ⊙ y)) \otimes (a, b) \\ = & ((a, b) \otimes (x, y)) \otimes (a, b) . \end{matrix}$ Hence, ((S, ø) × (T, ⊙) , ⊗) is regular.□

Definition 4.5. [38] Let ρ be an equivalence relation on a fuzzy AG-hypergroupoid S . Then (f, g) ∈ ρ for two fuzzy subsets f and g of S if for f (a) >0 there exists b ∈ S such that g (b) >0 and (a, b) ∈ ρ and f (x) >0 implies there exists y ∈ S such that g (y) >0 and (x, y) ∈ ρ .

Definition 4.6. [38] An equivalence relation ρ on S is said to be fuzzy hypercongruence if for (a, b) ∈ ρ, there is some c ∈ S such that (c ∘ a, c ∘ b) ∈ ρ and (a ∘ c, b ∘ c) ∈ ρ .

Also, it is worth noticing that if (a, b) ∈ ρ and (c, d) ∈ ρ, then (a ∘ c, b ∘ d) ∈ ρ for all a, b, c, d ∈ S .

In the following, we define a hyperoperation ∗ on a fuzzy AG-hypergroupoid S as a ∗ b = {x ∈ S ∣ (a ∘ b) (x) >0} .

Let x ∈ (a ∗ b) ∗ c. Then (a ∗ b) ∗ c ⊆ (c ∗ b) ∗ a . Certainly, for some y ∈ a ∗ b, we have (a ∘ b) (y) >0 and (y ∘ c) (x) >0, that is, ⋁_z∈S ((a ∘ b) (z) ∧ (z ∘ c) (x)) >0, that is, ((a ∘ b) ∘ c) (x) >0. This implies that ((c ∘ b) ∘ a) (x) >0, that is, x ∈ (c ∗ b) ∗ a . Using similar argument, we can show the converse statement. Hence, (a ∗ b) ∗ c = (c ∗ b) ∗ a . We just showed that (S, ∗) is AG-hypergroupoid.

Further, let ρ be an equivalence relation on (S, ∗) and (a₁, a₂) ∈ ρ and (b₁, b₂) ∈ ρ . If x ∈ a₁ ∗ b₁, then (a₁ ∘ b₁) (x) >0 . Also (a₁ ∘ b₁, a₂ ∘ b₂) ∈ ρ because ρ is fuzzy hypercongruence on (S, ∘) . Thus, there exists y ∈ S such that (a₂, b₂) (y) >0 and (x, y) ∈ ρ . Thus, x ∈ a₁ ∗ b₁ → y ∈ S such that y ∈ a₂ ∗ b₂ and (x, y) ∈ ρ . On the other, if x ∈ a₂ ∗ b₂, then there exists y ∈ a₁ ∗ b₁ such that (x, y) ∈ ρ . Thus, (a₁ ∗ b₁, a₂ ∗ b₂) ∈ ρ and so ρ is a fuzzy hypercongruence on (S, ∗) .

The fuzzy quotient AG-hypergroupoid (S/ρ, ⊙) for any fuzzy hypercongruence ρ is written as S/ρ = {aρ ∣ a ∈ S} . Let ⊙ be a hyperoperation on S/ρ such that aρ ⊙ bρ = {cρ ∣ c ∈ a ∗ b} . In the following, we show that (S/ρ, ⊙) is an AG-hypergroupoid.

To show that ⊙ is well-defined, let a₁ρ = a₂ρ and b₁ρ = b₂ρ . If c₁ρ ∈ a₁ρ ⊙ b₁ρ, then c₁ ∈ a₁ ∗ b₁ which implies that there exists c₂ ∈ a₂ ∗ b₂ such that (c₁, c₂) ∈ ρ. This implies that c₂ρ ∈ a₂ρ ⊙ b₂ρ and c₁ρ = c₂ρ . But then c₁ρ ∈ a₂ρ ⊙ b₂ρ and a₁ρ ⊙ b₁ρ ⊆ a₂ρ ⊙ b₂ρ . In the same way, a₂ρ ⊙ b₂ρ ⊆ a₁ρ ⊙ b₁ρ . Thus, a₁ρ ⊙ b₁ρ = a₂ρ ⊙ b₂ρ .

Further, let xρ ∈ (aρ ⊙ bρ) ⊙ cρ . Then, x ∈ (a ∗ b) ∗ c which further implies that (c ∗ b) ∗ a . Consequently, xρ ∈ (cρ ⊙ bρ) ⊙ aρ . Thus, (aρ ⊙ bρ) ⊙ cρ ⊆ (cρ ⊙ bρ) ⊙ aρ . For the reverse inclusion a similar argument gives, (cρ ⊙ bρ) ⊙ aρ ⊆ (aρ ⊙ bρ) ⊙ cρ . Hence, (aρ ⊙ bρ) ⊙ cρ = (cρ ⊙ bρ) ⊙ aρ .

Theorem 4.7. If ρ is a fuzzy strong hypercongruence on (S, ∘) , then (S/ρ, ∗) is a fuzzy AG-hypergroupoid.

Proof. Let ∗ be a fuzzy hyperoperation defined on S/ρ, for all a₁ρ, b₁ρ, c₁ρ ∈ S/ρ, by $\begin{matrix} (a_{1} ρ * b_{1} ρ) (c_{1} ρ) = ⋁_{a_{2} \in a_{1} ρ, b_{2} \in b_{1} ρ, c_{2} \in c_{1} ρ} (a_{2} \circ b_{2}) (c_{2}) . \end{matrix}$ It is clear that ∗ is well-defined. Let a₁, b₁, c₁ ∈ S. Then $\begin{matrix} ((a_{1} ρ * b_{1} ρ) * c_{1} ρ) (z ρ) \\ = ⋁_{s_{1} ρ \in S / ρ} ((a_{1} ρ * b_{1} ρ) (s_{1} ρ) \land (s_{1} ρ * c_{1} ρ) (z ρ)) \\ = ⋁_{s_{1} ρ \in S / ρ} (⋁_{a_{2} \in a_{1} ρ, b_{2} \in b_{1} ρ, s_{2} \in s_{1} ρ} (a_{2} \circ b_{2}) (s_{2}) \\ \land ⋁_{s_{3} \in s_{1} ρ, c_{2} \in c_{1} ρ, d_{2} \in d_{1} ρ} (s_{3} \circ c_{2}) (s_{3})) \\ = ⋁_{a_{2} \in a_{1} ρ, b_{2} \in b_{1} ρ, s_{2} \in s_{1} ρ, s_{3} \in s_{1} ρ, c_{2} \in c_{1} ρ, d_{2} \in d_{1} ρ} (a_{2} \\ \circ b_{2}) (s_{2}) \land (s_{2} \circ c_{2}) (s_{3}) \\ = ⋁_{a_{2} \in a_{1} ρ, b_{2} \in b_{1} ρ, c_{2} \in c_{1} ρ, s_{3} \in s_{1} ρ} ((a_{2} \circ b_{2}) \circ c_{2}) (s_{3}) \\ = ⋁_{a_{2} \in a_{1} ρ, b_{2} \in b_{1} ρ, c_{2} \in c_{1} ρ, s_{3} \in s_{1} ρ} ((c_{2} \circ b_{2}) \circ a_{2}) (s_{3}) \\ (since (S, \circ) is AG - hypergroupoid), \\ = ((c_{1} ρ * b_{1} ρ) * a_{1} ρ) (z ρ), for all, z ρ \in S / ρ . \end{matrix}$ Thus, (S/ρ, ∗) is a fuzzy AG-hypergroupoid.□

Theorem 4.8. Let (S, ∘) and (T, ∗) be two fuzzy AG-hypergroupoids. If θ : (S, ∘) → (T, ∗) be a fuzzy homomorphism, then Kerθ = {(a, b) ∈ S × S ∣ θ (a) = θ (b)} is a fuzzy hypercongruence.

Proof. It is clear that Kerθ is an equivalence relation. To show that Kerθ is a fuzzy hypercongruence, let (a₁, b₁) , (a₂, b₂) ∈ Kerθ . Therefore, θ (a₁ ∘ a₂) = θ (a₁) ∗ θ (a₂) = θ (b₁) ∗ θ (b₂) = θ (b₁ ∘ b₂) . Further, assume that (a₁ ∘ b₁) >0 . Then θ (a₁ ∘ b₁) (θ (x)) = ⋁ _{x₁∈θ^-1(θ(x))} (a₁ ∘ b₁) (x₁) >0 . Therefore, (θ (a₁ ∘ b₁)) (θ (x)) >0. This implies that θ (a₂ ∘ b₂) (θ (x)) >0, that is, ⋁_{x₁∈θ^-1(θ(x))} (a₂ ∘ b₂) (x₁) >0 . It follows that there exists x₁ ∈ S such that (a₂ ∘ b₂) (x₁) >0 and θ (x) = θ (x₁) . So, (a₁ ∘ b₁, a₂ ∘ b₂) ∈ Kerθ . Thus, Kerθ is a fuzzy hypercongruence on S .□

Theorem 4.9. If ρ is a fuzzy hypercongruence on S, then a homomorphism θ : (S, ∘) → (S/ρ, ∗) exists.

Proof. Let a ∈ S. Define θ : (S, ∘) → (S, ∗) by θ (a) = aρ, for all a ∈ S . Therefore, $\begin{matrix} θ (a \circ b) (c ρ) & = & ⋁_{x \in θ^{- 1} (x ρ)} (a \circ b) (x) \\ = & ⋁_{x ρ = c ρ} (a \circ b) (x) \\ = & ⋁_{x \in c ρ} (a \circ b) (x) \end{matrix}$ and ⋁_x∈cρ (a ∘ b) (x) ≤ ⋁ _{a₂∈aρ,b₂∈bρ,x∈cρ} (a₂ ∘ b₂) (x) = (aρ ∗ bρ) (cρ) . Thus, θ (a ∘ b) (cρ) = (aρ ∗ bρ) (cρ) .□

Theorem 4.10. Let (S, ∗) and (T, ∘) be two AG-hypergroupoids. If θ : S → T be a homomorphism, then Kerθ = {(a, b) ∣ θ (a) = θ (b)} is a hypercongruence on (S, ∗) .

Proof. It is clear that Kerθ is an equivalence relation. Now, let (a₁, b₁) , (a₂, b₂) ∈ Kerθ . Also, let x ∈ a₁ ∗ a₂. Then θ (x) ∈ θ (a₁ ∗ a₂) = θ (a₁) ∘ θ (a₂) = θ (b₁) ∘ θ (b₂) = θ (b₁ ∗ b₂) which implies that there exists y ∈ b₁ ∗ b₂ such that θ (x) = θ (y) . Consequently, (x, y) ∈ Kerθ . Thus, Kerθ is a hypercongruence on (S, ∗) .□

Theorem 4.11. If ρ is a hypercongruence on AG-hypergroupoid (S, ∗), then S/ρ = {aρ : a ∈ S} is AG-hypergroupoid.

Proof. Let a, b ∈ S . Define ⊙ on S/ρ by a ⊙ b = {cρ ∣ c ∈ aρ ∗ bρ} for all aρ, bρ ∈ S/ρ . Indeed, a ⊙ b is nonempty. We first show that ⊙ is well-defined. Assume that a₁ρ = a₂ρ and b₁ρ = b₂ρ . Also, let cρ ∈ a₁ρ ⊙ b₁ρ . Then, c ∈ a₁ ∗ b₁. It implies that there exists c₂ ∈ a₂ ∗ b₂ such that (c₁, c₂) ∈ ρ which implies that c₁ρ ∈ a₂ρ ⊙ b₂ρ . So a₁ρ ⊙ b₁ρ ⊆ a₂ρ ⊙ b₂ρ . Similarly, together with the reverse inclusion, we have a₁ρ ⊙ b₁ρ = a₂ρ ⊙ b₂ρ .

We show that (S/ρ, ⊙) is AG-hypergroupoid. Let xρ ∈ (aρ ⊙ bρ) ⊙ cρ which implies that x ∈ (a ∗ b) ∗ c . Also, x ∈ (c ∗ b) ∗ a . Consequently, xρ ∈ (cρ ⊙ bρ) ⊙ aρ . Hence, (aρ ⊙ bρ) ⊙ cρ ⊆ (cρ ⊙ bρ) ⊙ aρ . Similarly, combined with the reverse inclusion, we get (aρ ⊙ bρ) ⊙ cρ = (cρ ⊙ bρ) ⊙ aρ . This completes the proof.□

Theorem 4.12. If ρ is a fuzzy strong hypercongruence on a fuzzy AG-hypergroupoid (S, ∗), then σ : S → S/ρ defined by σ (a) = aρ for all a ∈ S, is surjective homomorphism such that Kerσ = ρ .

Proof. Evidently, σ is surjective. Suppose a, b ∈ S . Then σ (a ∗ b) = {σ (s) : s ∈ a ∗ b} = {sρ : s ∈ a ∗ b} = aρ ⊙ aρ = σ (a) ⊙ σ (b) . Thus, σ is a strong homomorphism. Also, we have $\begin{matrix} Ker σ & = & {(a, b) : σ (a) = σ (b)} = {(a, b) : a ρ = b ρ} \\ = & {(a, b) : (a, b) \in ρ} = ρ . \end{matrix}$ □

The following theorem is an embedded theorem for a poe-AG-groupoid S into the set of all fuzzy subsets $F (S)$ of S . The set $F (S)$ is nonempty, since f : S → [0, 1] and for some x ∈ S, $f (x) : = \frac{1}{2} \in F (S) .$ A poe-AG-groupoid is an ordered AG-groupoid (S, · , ≤) with the greatest element.

Theorem 4.13. Let $(F (S), \circ, \subseteq)$ be the set of all fuzzy subsets of an AG-hypergroupoid (S, ∘) with hyperoperation ∘ and order relation ⊆ defined as: $\circ : F (S) \times F (S) \to F (S) | (f, g) \to f \circ g,$ where f ∘ g is defined as (f ∘ g) (z) = ⋁ _z∈x∘y {f (x) ∧ g (y)} , f ⊆ g ⇔ f (x) ≤ g (x) forall x ∈ S. Then it is a poe-AG-groupoid having a zero element and S is embedded in $F (S) .$

Proof. The operation ∘ is well-defined. We show that $(F (S), \circ, \subseteq)$ is ordered AG-hypergroupoid. Let z ∈ S and $f, g, h \in F (S)$ . Then there exist x, y ∈ S such that z ∈ x ∘ y and $\begin{matrix} ((f \circ g) \circ h) (z) = ⋁_{z \in x \circ y} {(f \circ g) (x) \land h (y)} \\ ((h \circ g) \circ f) (z) = ⋁_{z \in x \circ y} {(h \circ g) (x) \land f (y)} . \end{matrix}$ Suppose Δ = ⋁ _z∈x∘y {(f ∘ g) (x) ∧ h (y)} and ∇ = ⋁ _z∈x∘y {(h ∘ g) (x) ∧ f (y)} . Let c, d ∈ S, z ∈ d ∘ c and Δ ≥ f (c) . Then, f (c) ≥ f (c) ∧ (h ∘ g) (d) and we have Δ ≥ f (c) ∧ (h ∘ g) (d) . If Δ ≤ f (c) and there do not exist d₁, d₂ ∈ S such that d ∈ d₁ ∘ d₂, then (h ∘ g) (d) =0 and f (c) ≥0 . Therefore, f (c) ∧ (h ∘ g) (d) =0 and since Δ ≥ 0, then it follows Δ ≥ f (c) ∧ (h ∘ g) (d) . Now if there exist d₁, d₂ ∈ S such that d ∈ d₁ ∘ d₂, then, Δ ≥ ⋁ _{d∈d₁∘d₂} {h (d₁) ∧ g (d₂)} = (h ∘ g) (d) . Since (h ∘ g) (d) ≥ f (c) ∧ (h ∘ g) (d) , then this shows that Δ ≥ f (c) ∧ (h ∘ g) (d) . Also, if Δ ≥ h (d₁), then since h (d₁) ≥ h (d₁) ∧ g (d₂) , we have Δ ≥ h (d₁) ∧ g (d₂) . Again, if Δ ≤ h (d₁), then since z ∈ d ∘ c, d ∈ d₁ ∘ d₂, we have z ∈ (d₁ ∘ d₂) ∘ c = (c ∘ d₂) ∘ d₁ . Therefore, Δ ≥ ⋁ _r∈c∘d₂ {(f ∘ g) (r) ∧ h (d₁)} . Moreover, (f ∘ g) (r) ≥ f (c) ∧ g (d₂) . We know that Δ < f (c) and Δ < g (d₂) , then Δ < f (c) ∧ g (d₂) . Thus, (f ∘ g) (r) > Δ . Therefore, h (d₁) ≤ Δ . On contrary, if Δ < h (d₁) , then (f ∘ g) (r) ∧ h (d₁) > Δ, that is, ⋁_r∈c∘d₂ {(f ∘ g) (r) ∧ h (d₁)} > Δ, which is impossible. If there do not exist x, y ∈ S such that z ∈ x ∘ y, then (f ∘ (g ∘ h)) (z) = ((h ∘ g) ∘ f) (z) =0 .

Also, let $f, g, h \in F (S)$ and f ⊆ g . We show that f ∘ h ⊆ g ∘ h and h ∘ f ⊆ h ∘ g . If there do not exist x, y ∈ S such that z ∈ x ∘ y, then f ∘ h = 0 = g ∘ h . Thus, for all z ∈ S, (f ∘ h) (z) ≤ (g ∘ h) (z) .

If there exist x, y ∈ S such that z ∈ x ∘ y, then by definition we have $\begin{matrix} (f \circ h) (z) = ⋁_{z \in x \circ y} {f (x) \land h (y)} \\ (g \circ h) (z) = ⋁_{z \in x \circ y} {g (x) \land h (y)} . \end{matrix}$ Let Δ₁ = ⋁ _z∈x∘y {f (x) ∧ h (y)} and ∇₁ = ⋁ _z∈x∘y {g (x) ∧ h (y)} . Then we show that for all x, y ∈ S, z ∈ x ∘ y, f (x) ∧ h (y) ≤ Δ₁ . Since, Δ₁ ≥ g (x) ∧ h (y) , then Δ₁ ≥ g (x) or Δ₁ ≥ h (y) .

(1): If Δ ≥ h (y) , then as h (y) ≥ f (x) ∧ h (y) , we have Δ₁ ≥ f (x) ∧ h (y) .

(2): If Δ₁ ≥ g (x) , then f (x) ≤ g (x) since f ⊆ g . Also, because f (x) ≥ f (x) ∧ h (y) , it follows Δ₁ ≥ f (x) ∧ h (y) . Thus, f ∘ h ⊆ g ∘ h . This shows right compatibility. Similarly, the left compatibility can be shown.

The mapping 1 : S → [0, 1] |x → 1 (x) =1 is the greatest element of $F (S) .$

We now show that S is embedded in $F (S) .$ Consider A ⊆ S. The function $f : S \to F (S) | x \to f_{x},$ where $f_{A} (x) : S \to [0, 1] | y \to f_{A} (y) = {\begin{matrix} 1 if y \in A \\ 0 otherwise \end{matrix}$ is well-defined, one-one and a strong homomorphism. If a, b ∈ S, then f_a∘b (x) = (f_a ∘ f_b) (x) for x ∈ S .

If there do not exist c, d ∈ S such that x ∈ c ∘ d, then (f_a ∘ f_b) (x) =0 . x ∉ c ∘ d, (f_a∘b) (x) =0 . Also, if there exist c, d ∈ S such that x ∈ c ∘ d, then (f_a ∘ f_b) (x) = ⋁ _x∈c∘d {f_a (c) ∧ f_b (d)} = Δ₂ .

Now let x ∈ a ∘ b. Then f_a∘b (x) =1 . But f_a (a) ∧ f_b (b) =1 ≥ Δ₂ . Since Δ₂ ∈ [0, 1] , Δ₂ ≤ 1, then we have Δ₂ = 1 . Therefore, f_a∘b (x) = (f_a ∘ f_b) (x) =1 . On the other hand, if x ∉ a ∘ b, then f_a∘b (x) =0 . Also, suppose c, d ∈ S, x ∈ c ∘ d . If f_a (c) ≠0 ≠ f_b (d) , then f_a (c) =1 = f_b (d) . That is c = a and b = d and x ∈ a ∘ b, which is impossible.

Thus, we have f_a (c) =0 or f_b (d) =0 . For the first one, f_b (d) ≥0 and f_a (c) ∧ f_b (d) =0 . If f_b (d) =0, then f_a (c) ≥0 and again f_a (c) ∧ f_b (d) =0 . Then ⋁_x∈c∘d {f_a (c) ∧ f_b (d)} =0 and f_a∘b (x) = (f_a ∘ f_b) (x) =0 . □

5 Fuzzy Strong Regular Relations

Let f_r be a fuzzy equivalence relation on AG-hypergroupoid S . Let A₁, A₂ ⊆ S. We write $A_{1} {\dot{f}}_{r} A_{2}$ to denote that for every a₁ ∈ A₁, there exists a₂ ∈ A₂ such that a₁f_ra₂ and for every a₂ ∈ A₂, there exists a₁ ∈ A₁ such that a₁f_ra₂ . We write $A_{1} {\ddot{f}}_{r} A_{2}$ if for every a₁ ∈ A₁ and for every a_a ∈ A₂, then a₁f_ra₂ .

Definition 5.1. [13] A fuzzy equivalence relation f_r is said to be fuzzy strong regular to the right if $f_{r} (a, b) \leq ⋀_{p \in a \circ x, q \in b \circ x} f_{r} (p, q)$ for all a, b, x ∈ S . The fuzzy strong regular to left can be defined as f_r (a, b) ≤ ⋀ _{p∈x∘a, q∈x∘b} (f_r (p, q)) for all a, b, x ∈ S .

We say f_r is strong regular if f_r (a, b) ∧ f_r (x, y) ≤ ⋀ _{p∈a∘x, q∈b∘y}f_r (p, q) for all a, b, x, y ∈ S .

Example 6. Let S = Z. Define the hyperoperation “ ∘ ″ as follows: x ∘ y = y-x + 3Z, for all x, y ∈ S.

Then (S, ∘) is an AG-hypergroupoid.

Now, we define in S the relation ρ as follows: ∀x ∈ S, ρ (x, x) =1. ∀x, y ∈ S, if x, y have the same parity, then ρ (x, y) =0, 5. ∀x, y ∈ S, if x, y have opposite parity, then ρ (x, y) =0, 5. Then, it can be easily verified that ρ is an equivalence relation on S.

Also, it can be seen that it is a fuzzy strong regular relation to left and to right, that is, a fuzzy strong relation on S.

Lemma 5.2. Let f_r be a fuzzy strong regular relation on an AG-hypergroupoid S . Then $f_{r}^{- 1} (1) = {(a, b) \in S \times S ∣ f_{r} (a, b) = 1}$ is a strong regular relation on S .

Proof. Clearly f_r is reflexive and symmetric. For transitivity, let $(a, b), (b, c) \in f_{r}^{- 1} .$ Since f_r is fuzzy strong regular, then $\begin{matrix} f_{r} (a, c) & \geq & ⋁_{t \in S} {f_{r} (a, t) \land f_{r} (t, c)} \\ \geq & f_{r} (a, b) \land f_{r} (b, c) = 1 \land 1 = 1 \end{matrix}$ which shows that $(a, c) \in f_{r}^{- 1} .$ Thus, $f_{r}^{- 1} (1)$ is transitive.

Further, we show that $f_{r}^{- 1}$ is fuzzy strong regular. Let $(a, b) \in f_{r}^{- 1} (1) .$ Then ⋀_{p∈a∘x, q∈b∘x}f_r (p, q) ≥ f_r (a, b) = 1 . Thus, f_r (p, q) =1 for all p ∈ a ∘ x and q ∈ b ∘ x . Consequently, $(a \circ x, b \circ x) \in {\ddot{f}}_{r}^{- 1} (1) .$ Similarly, $(x \circ a, x \circ b) \in {\ddot{f}}_{r}^{- 1} (1) .$ Hence, $f_{r}^{- 1} (1)$ is a fuzzy strong regular relation.□

Proposition 5.3. Let {f_i} _i∈I be fuzzy strong regular relations on S . Then, $\cap_{i = 1}^{n} f_{i}$ is fuzzy strong regular.

Proof. Suppose that f_i are strong regular fuzzy relations and $M = \cap_{i = 1}^{n} f_{i}$ . For a, b ∈ S $M (x, x) = f_{1} (x, x) \land f_{2} (x, x) . . . \land f_{n} (x, x) = 1 .$ Since f_i, for i ∈ I, is symmetric, then $M$ is symmetric. Also, for transitivity, $\begin{matrix} M (x, y) & = & f_{1} (x, y) \land f_{2} (x, y) . . . \land f_{n} (x, y) \\ \geq & [⋁_{t_{1} \in S} {f_{1} (x, t_{1}) \land f_{1} (t_{1}, y)} \\ . . . \land & ⋁_{t_{n} \in S} {f_{n} (x, t_{n}) \land f_{n} (t_{n}, y)}] \\ = & ⋁_{t_{1}, t_{2} . . . t_{n} \in S} [({f_{1} (x, t_{1}) \land f_{1} (t_{1}, y)} \\ . . . \land & (f_{n} (x, t_{n}) \land f_{n} (t_{n}, y)))] \\ \geq & ⋁_{t_{k} \in S} [({f_{1} (x, t_{k}) \land f_{1} (t_{k}, y)} \\ . . . \land & {f_{n} (x, t_{k}) \land f_{n} (t_{k}, y)})] \\ = & ⋁_{t_{k} \in S} [(f_{1} (x, t_{k}) . . . \land f_{n} (x, t_{k})) \\ \land (f_{1} (t_{k}, y) . . . \land f_{n} (t_{k}, y))] \\ = & ⋁_{t_{k} \in S} (M (x, t_{k}) \land M (t_{k}, y)) \end{matrix}$ which shows that $M$ is fuzzy transitive relation. Hence, $M$ is fuzzy equivalence.

Further, let (a, b) , (x, y) ∈ S × S . Then $\begin{matrix} M (a, b) & \land & M (x, y) \\ = & (f_{1} (a, b) \land f_{2} (a, b) . . . \land f_{n} (a, b)) \\ \land & (f_{1} (x, y) \land f_{2} (x, y) . . . \land f_{n} (x, y)) \\ = & (f_{1} (a, b) \land f_{1} (x, y)) \land \\ . . . \land & (f_{n} (a, b) \land f_{n} (x, y)) \\ \leq & ⋀_{p_{1} \in a \circ x, q_{1} \in b \circ y} f_{1} (p_{1}, q_{1}) \\ \land & ⋀_{p_{2} \in a \circ x, q_{2} \in b \circ y} f_{2} (p_{2}, q_{2}) \\ . . . \land & ⋀_{p_{n} \in a \circ x, q_{n} \in b \circ y} f_{n} (p_{n}, q_{n}) \\ = & ⋀_{p_{i} \in a \circ x, q_{i} \in b \circ y} (f_{1} (p_{1}, q_{1}) \\ \land & f_{2} (p_{2}, q_{2}) . . . \land f_{n} (p_{n}, q_{n})) \\ \leq & ⋀_{p_{k} \in a \circ x, q_{k} \in b \circ y s} f_{1} (p_{k}, q_{k}) \\ \land & f_{2} (p_{k}, q_{k}) . . . \land f_{n} (p_{k}, q_{k}) \\ = & ⋀_{p_{k} \in a \circ x, q_{k} \in b \circ y s} \land M (p_{k}, q_{k}) . \end{matrix}$

Thus, $M (p_{k}, q_{k})$ is a fuzzy strong regular relation on S .□

Definition 5.4. Let f_r be a fuzzy equivalence relation on S . Then the t-level set of f_r for each t ∈ [0, 1] , is defined as f_rt = { (a, b) ∈ S × S ∣ f_r (a, b) ≥ t } .

Proposition 5.5. Let f_r be equivalence relation on S . Then

(1) f_r is a strong regular relation to the right

(2) ω_{f
_r} the characteristic function of f_r is a fuzzy strong regular relation to the right.

Proof. (1) → (2) : For a, b, x ∈ S, let (a, b) ∈ f_rt . Then, by the strong regularity of f_r, $(x \circ a, y \circ a) \in \ddot{f_{r}} .$ Then for p ∈ x ∘ a, q ∈ y ∘ a, we have (p, q) ∈ f_r . That is, ω_{f
_r} (p, q) =1 . Thus, $ω_{f_{r}} (a, b) = 1 = {ω_{f_{r}}}_{p \in a \circ x, q \in b \circ x} (p, q) .$ On the other hand, if (a, b) ∉ f_r, then ω_{f
_r} (a, b) =0 ≤ ⋀ _{p∈a∘x, q∈b∘x}ω_{f
_r} (p, q) . Hence, ω_{f
_r} is a fuzzy strong regular relation to the right.

(2) → (1) : This is simple.□

Proposition 5.6. Let f_r be equivalence relation on S . Then

(1): f_r is a strong regular relation to the left

(2): ω_{f
_r} the characteristic function of f_r is a fuzzy strong regular relation to the left.

Theorem 5.7. Let f_r be a fuzzy strong regular relation on S . Then, f_rt is a strong regular relation on S .

Proof. It follows from the definition that f_rt is an equivalence relation. Since f_r is fuzzy strong regular, then $⋀_{p \in a \circ x, q \in b \circ y} f_{r} (p, q) \geq f_{r} (a, b) \land f_{r} (x, y) \geq (t, t) = t .$

This shows that for every p ∈ a ∘ x, q ∈ b ∘ x, (p, q) ∈ f_rt . Therefore $(a \circ x, b \circ y) \in \ddot{f_{rt}} .$ Hence, f_rt is strong regular relation.□

6 Conclusions and future research

In this paper, we have applied fuzzy sets in hyperstructure theory, in particular, in Abel-Grassmann’s Hypergroupoids. We have introduced the notion of fuzzy AG-hypergroupoid, hypercongruence, fuzzy hypercongruence, fuzzy strong hypercongruence and compatible relations in an AG-hypergroupoid. Some characterizations in terms of fuzzy hyperideals, smallest fuzzy hyperideals, fuzzy equivalence relations, fuzzy compatible fuzzy strong compatible, fuzzy hypercongruences, fuzzy strong hypercongruences, fuzzy regular and fuzzy strong regular relations and fuzzy hypercongruences are provided. Also, some properties of hypercongruences and the corresponding quotient AG-hypergroupoid, homomorphisms are obtained. A theorem on embedding an AG-hypergroupoid into a poe-AG-groupoid of the family of all fuzzy subsets of AG-hypergroupoid, is proved. In our future work, we will give characterization of fuzzy ordered regular relations for a fuzzy Abel-Grassmann’s hypergroupoid.

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