Fuzzy congruences on non-associative semigroups

Abstract

We give the notion of fuzzy congruences in a non-associative semigroup briefly known as Abel-Grassmann’s groupoid. First, we study fuzzy full, fuzzy self-conjugate, fuzzy normal subgroupoids and show that fuzzy kernel κ and fuzzy trace τ of a congruence make up a congruence pair (κ, τ) . Then, we investigate fuzzy idempotent-separating congruence and show that μ_(κ,τ) is a unique fuzzy congruence. Next, we study the lattice of fuzzy congruences, fuzzy congruence relations φ_min and φ_max and show a relation between fuzzy kernel and fuzzy trace of ρ_max in a completely inverse AG^**-groupoid.

Keywords

fuzzy kernel of a congruence fuzzy trace of congruence fuzzy congruence pair lattice of fuzzy congruences

1 Introduction

With the start of fuzzy set theory by Zadeh [20] in 1965 many contributions have been made using the concept of fuzzy sets from purely theoretical to scientific and technological disciplines since then. 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 [16] made a valuable contribution in generalizing groupoids and groups using fuzzy set theory. Sanchez [18], Bhattacharya and N.P Munkherjee [1] and Nemitz [14] investigated fuzzy relations and fuzzy equivalence relations in general and in groups. Murali [12] studied fuzzy relations on sets and lattices properties of fuzzy equivalence relations. Kuroki [10] investigated fuzzy congruence on groupoids and give a brief knowledge of fuzzy congruences on groups via their fuzzy normal subgroups. Samhan [17] studied fuzzy congruences on semigroups along with some other characteristics of congruences.

A left almost semigroup which is very common and meaningful to call Abel-Grassmann’s groupoid is a groupoid S satisfying the left invertive law (ab) c = (cb) a for all a, b, c ∈ S . In a number of research articles, (cf. [5–7 , 13]), it has been manifested that the theory of AG-groupoids is parallel to the theory of commutative semigroups and this rarity is the motivation for this study. The essential characteristic of the left invertive law places an AG-groupoid into the class of non-associative algebraic structures. Some other classes of AG-groupoids like, inverse AG^**-groupoids and completely inverse AG^**-groupoids constitute the most important classes of AG-groupoids (see for example [2–4 , 15]).

The theory of fuzzy sets has attracted the attention of many researchers in AG-groupoids. A number of articles covering different aspects of fuzzy AG-groupoids, fuzzy ordered AG-groupoids, fuzzy ideals in AG-groupoids and regularity classes of fuzzy AG-groupoids have been published (cf. [19]).

In this paper, fuzzy relations, fuzzy equivalence relations and fuzzy congruences in a completely inverse AG^**-groupoid are introduced. In Section 2, we give preliminary and sufficient knowledge leading to the concept of fuzzy congruences. Proofs of some basic lemmas and propositions are provided in this section. Section 3, comprises of fuzzy idempotent-separating congruence and AG-group fuzzy congruence on a completely inverse AG^**-groupoid. In Section 4, we introduce the concepts of fuzzy full, fuzzy self-conjugate and fuzzy normal AG^**-groupoids and fuzzy normal congruence on the set of idempotent E_S . We further show that (κ_φ, τ_φ) constitutes a fuzzy congruence pair. Moreover, section 5 deals with the lattice of fuzzy congruences. In this section, we investigate the fuzzy congruences ρ_min and ρ_max and provide a characterization of the fuzzy kernel of ρ_min and ρ_max . We further show that if Θ is a complete homomorphism between the lattice of all fuzzy congruences and the lattice of fuzzy congruences on E_S and if θ is a fuzzy congruence relation induced by Θ, then the interval [ρ_min, ρ_max] for every ρ ∈ C_S, is complete modular sublattice of C_S .

2 Preliminaries and basic results

2.1 Basic results

An AG-groupoid S in which every element has a unique inverse is called inverse AG-groupoid. If a^-1 is the unique inverse of a ∈ S, then a groupoid satisfying the following identities is called a completely inverse AG^**-groupoid, that is, for all a, b, c ∈ S $\begin{matrix} (ab) c = (cb) a, a (bc) = b (ac) \\ a = ({aa}^{- 1}) a, a^{- 1} = (a^{- 1} a) a^{- 1} \\ and {aa}^{- 1} = a^{- 1} a where a^{- 1} \in V (a) . \end{matrix}$

In this part, we give the basic concept of fuzzy relations, fuzzy left and right compatible relations and fuzzy congruences.

Proposition 2.1.[4, Proposition 4.1] Let S be a completely inverse AG^**-groupoid and let a, b ∈ S such that ab ∈ E_S . Then, ab = ba .

Definition 2.2.Let S be an AG-groupoid. A mapping φ : S × S → [0, 1] is called a fuzzy relation on S . Let φ₁ and φ₂ be two fuzzy relations on S . The product φ₁ ∘ φ₂ is defined by

$(φ_{1} \circ φ_{2}) (a, b) = \underset{x \in S}{⋁} {φ_{1} (a, x) \land φ_{2} (x, b)} \forall a, b \in S .$

Definition 2.3.A fuzzy relation φ on an AG-groupoid S is called fuzzy equivalence relation if

(1) φ (a, a) =1 (∀ a ∈ S) (fuzzy reflexive),

(2) φ (a, b) = φ (b, a) (∀ a, b ∈ S) (fuzzy symmetric),

(3) φ ∘ φ ≤ φ (fuzzy transitive).

Definition 2.4.Let S be an AG-groupoid. A fuzzy relation φ on S is said to be a fuzzy left (right) compatible if for all a, b, x ∈ S, $φ (xa, xb) \geq φ (a, b) (φ (ax, bx) \geq φ (a, b)) .$ It is fuzzy compatible if φ (a, b) ∧ φ (c, d) ≤ φ (ab, cd) (∀ a, b, c, d ∈ S) . A fuzzy compatible relation is called fuzzy congruence.

Let φ be a fuzzy equivalence relation on S . For each a ∈ S, the fuzzy subset φ_a of S is defined as $φ_{a} (s) = φ (a, s) (\forall s \in S) .$

Lemma 2.5. Let φ be a fuzzy relation on an AG-groupoid S . Then, $φ_{a} = φ_{b} \Leftrightarrow φ (a, b) = 1 (\forall a, b \in S) .$

Proof. The proof is similar to the proof of Theorem 5.3.2 [11].

Let φ be a fuzzy congruence relation on S. Then, the fuzzy subset φ_a of S is called a fuzzy equivalence class of φ containing a . The set S/φ = {φ_a : a ∈ S} is called fuzzy quotient set by φ . If ⊙ is a binary relation on S/φ, defined by φ_a ⊙ φ_b = φ_ab then, it is easy to see that ⊙ is well-defined on S/φ . If f and g are two fuzzy subsets of S, then for some y, z ∈ S, the product of f and g denoted by f ∘ g is defined by $(f \circ g) (x) = {\begin{matrix} \underset{x = yz}{⋁} f (y) \land f (z) if x = yz, \\ 0 if x \neq yz . \end{matrix}$

Lemma 2.6. Let φ be a fuzzy congruence relation on an AG-groupoid S . Then, $φ^{- 1} (1) = {(a, b) \in S \times S : φ (a, b) = 1}$ is a congruence on S .

Proof. The proof is similar to the proof of Theorem 5.3.4 [11].

3 Idempotent-separating fuzzy congruence

In this section, we study maximum idempotent-separating and minimum AG-group fuzzy congruences in a completely inverse AG^**-groupoid. A fuzzy congruence φ on a completely inverse AG^**-groupoid S is idempotent-separating if each φ-class, denoted by φ_a for all a ∈ S contains at maximum one idempotent.

Theorem 3.1.Let φ be a fuzzy congruence on a completely inverse AG^**-groupoid S . Then, the following statements hold.

(1) φ_a ∈ E_S/φ for some a ∈ S .

(2) φ_a = φ_e for some e ∈ E_S such that Se ⊆ Sa and eS ⊆ aS .

(3) φ_a = φ_e for some e ∈ E_S .

Proof. (1)⇒ (2): Let φ_a ∈ E_S/φ . Then, φ_a = φ_a ⊙ φ_a = φ_{a
²} . Since S is completely inverse, it follows that there exists s ∈ S such that a² = (a²s) a², s = (sa²) s and a²s = sa² . Then, a (sa) ∈ E_S . In fact; $\begin{matrix} (a \cdot sa) (a \cdot sa) & = (as \cdot as) (aa) = a ((ss \cdot aa) a) \\ = a (as \cdot a^{2} s) = (a ({sa}^{2} \cdot sa)) \\ = a ((a^{2} s \cdot s) a) = (a ({sa}^{2} \cdot s)) a \\ = a \cdot sa . \end{matrix}$ Also, (a²s) a² = (a²s) (aa) = (aa) (sa²) = a² (sa²) . Therefore, we obtain $\begin{matrix} φ_{a (sa)} & = φ_{a} ⊙ (φ_{s} ⊙ φ_{a}) = φ_{a^{2}} ⊙ (φ_{s} ⊙ φ_{a^{2}}) \\ = φ_{a^{2} ({sa}^{2})} = φ_{(a^{2} s) a^{2}} = φ_{a^{2}} = φ_{a} . \end{matrix}$ Further, (as · a) S ⊆ aS and S (as · a) ⊆ Sa .

(2)⇒ (3): It is straightforward.

(3)⇒ (1): Since φ_a = φ_e, it is easy to see that φ_a ⊙ φ_a = φ_e ⊙ φ_e = φ_e = φ_a ∈ E_S/φ .□

Theorem 3.2. Let φ be a fuzzy congruence on a completely inverse AG^**-groupoid S . Then, (S/φ, ∗) is a completely inverse AG^**-groupoid with ∗ defined as φ_x ∗ φ_t = φ_xt forany x, t ∈ S .

Proof. If φ_a = φ_b and φ_c = φ_d, then φ (a, b) = φ (c, d) =1 . Thus,

$\begin{matrix} φ (ac, bd) \geq φ \circ φ (ac, bd) & = \underset{x \in S}{⋁} φ (ac, x) \land φ (x, bd) \\ φ (ac, bc) \land φ (bc, bd) \\ φ (a, b) \land φ (c, d) \\ = 1 \end{matrix}$ By Lemma 2.5, we have φ_a ⊙ φ_c = φ_ac = φ_bd = φ_b ⊙ φ_d which shows that ∗ is well-defined. If a ∈ S, then there exists s ∈ S such that a = (as) a and as = sa . Then, φ_a = φ_(as)a = (φ_a ⊙ φ_s) ⊙ φ_a . Also, inverses commute, because φ_a ⊙ φ_s = φ_as = φ_sa = φ_s ⊙ φ_a .

Hence, S/φ is a completely inverse AG^**-groupoid.□

It is useful to be noticed that as a consequence of Theorem 3.2, inverse of an element φ_a ∈ S/φ is expressed as (φ_a) ^-1 = φ_{a
^-1} .

Lemma 3.3. If φ is a congruence on a completely inverse AG^**-groupoid S, then $φ (a^{- 1}, b^{- 1}) = φ (a, b) (\forall a, b \in S) .$

Lemma 3.4. [9, Proposition 5.4] If S is a completely inverse AG^**-groupoid with E_S the set of idempotents of S, then the relation ϱ = {(a, b) ∈ S × S : a^-1 · ea = b^-1 · eb forall e ∈ E_S} is the greatest idempotent-separating congruence on S .

Theorem 3.5. Let φ be a congruence on a completely inverse AG^**-groupoid S . Then, φ is fuzzy idempotent-separating congruence if and only if φ^-1 (1) ⊆ ϱ .

Proof. Suppose that φ is idempotent-separating and (a, b) ∈ φ^-1 (1) . Then, φ (a, b) =1 . So, we obtain

$\begin{matrix} φ (a^{- 1} \cdot ea, b^{- 1} \cdot eb) \geq (φ \circ φ) (a^{- 1} \cdot ea, b^{- 1} \cdot eb) \\ = \underset{x \in S}{⋁} {φ (a^{- 1} \cdot ea, x) \land φ (x, b^{- 1} \cdot eb)} \\ \geq φ (a^{- 1} \cdot ea, b^{- 1} \cdot ea) \land φ (b^{- 1} \cdot ea, b^{- 1} \cdot eb) \\ = φ (a^{- 1}, b^{- 1}) \land φ (a, b) \\ = 1 . \end{matrix}$ Thus, by Lemma 2.5, we have φ_a^-1·ea = φ_b^-1·eb . But φ is idempotent-separating, then a^-1 · ea = b^-1 · eb which implies that (a, b) ∈ ϱ .

Conversely, suppose that φ^-1 (1) ⊆ ϱ . Let φ_e = φ_f for e, f ∈ E_S . Since φ (e, f) =1 . Then, (e, f) ∈ φ^-1 (1) . But ϱ is idempotent-separating, then e = f . Hence, φ is fuzzy idempotent-separating congruence.

□

Notice that if S is a completely inverse AG^**-groupoid and E_S is the set of idempotents of S, then $σ = {(a, b) \in S \times S : ea = eb for some e \in E_{S}}$ is the least AG-group congruence on S (cf. [4]).

It is important to note that an AG-group is a completely inverse AG^**-groupoid with exactly one idempotent. In the following, we investigate this important result in the fuzzy context.

Theorem 3.6.Let S be an inverse AG^**-groupoid. Then, a fuzzy congruence φ is an AG-group congruence if and only if σ ⊆ φ^-1 (1) .

Proof. We show that σ ⊆ φ^-1 (1) . Let (a, b) ∈ σ, then for some e ∈ E_S, ea = eb . On the other hand, S/φ is an AG-group. Then, for φ_e, φ_a ∈ S/φ $φ_{e} ⊙ φ_{a} = φ_{ea} = φ_{eb} = φ_{e} ⊙ φ_{b} .$ Using Lemma 2.5 and since φ is a congruence, we have φ (ea, eb) = φ (a, b) =1 . Hence, (a, b) ∈ φ^-1 (1) .

Conversely, suppose that σ ⊆ φ^-1 (1) . We show that S contains exactly on idempotent. Let e, f ∈ E_S then $\begin{matrix} (ef) e & = (ef) (ee) = (ee) (fe) = f (ee) \\ = (ff) (ee) = (ee) (ff) = (ef) f \end{matrix}$ which implies that (e, f) ∈ σ . Thus, (e, f) ∈ φ^-1 (1) ⇒ φ (e, f) =1 ⇒ φ_e = φ_f . Hence, S/φ is an AG-group.

4 Fuzzy congruence pair

In this section, we discuss fuzzy subgroupoid, fuzzy normal subset and fuzzy congruence pair of a completely inverse AG^**-groupoid. We show that if φ is a fuzzy congruence then (κ_φ, τ_φ) is a congruence pair on S . Further investigation of some important results follows in the rest of this section.

Definition 4.1.A fuzzy subset κ of a completely inverse AG^**-groupoid S is a fuzzy AG^**-subgroupoid of S if $κ (xy) \geq κ (x) \land κ (y) and κ (x^{- 1}) = κ (x) (\forall x, y \in S) .$

Example 1.Let S = {a, b, c, e} shown in the following multiplication table is a completely inverse AG^**-groupoid

.	a	b	c	e
a	a	b	c	e
b	c	b	e	a
c	b	b	c	b
e	e	b	a	e

with (ba) e ≠ b (ae) and x^-1 = x, for all x ∈ S . If κ is a fuzzy subset of S such that κ (a) =0.8, κ (b) = κ (c) =0.5 and κ (e) =1 . Then, κ (xy) ≥ κ (x) ∧ κ (y) and κ (x^-1) = κ (x) for all x, y ∈ S .

Definition 4.2.Let φ be a fuzzy congruence on a completely inverse AG^**-groupoid S . The fuzzy kernel κ_φ is defined as $κ_{φ} (s) = \underset{e \in E_{S}}{⋁} φ (s, e) (\forall s \in S) .$ The fuzzy trace of φ denoted by τ_φ is defined as $τ_{φ} (e, f) = φ (e, f) (\forall e, f \in E_{S}) .$

Definition 4.3.Let S be a completely inverse AG^**-groupoid. A fuzzy subset κ of S is called fuzzy normal AG^**-subgroupoid if it is:

(1) fuzzy full, that is, κ (e) =1 (∀ e ∈ E_S) ,

(2) fuzzy self-conjugate, that is, κ (x · sx^-1) ≥ κ (s) (∀ x, s ∈ S) .

Example 2.This example shows that S = {a, b, c, e} is non-associative completely inverse AG^**-groupoid because (ae) b ≠ a (eb) and x^-1 = x, for all x ∈ S .

.	a	b	c	e
a	c	c	a	b
b	c	e	b	b
c	a	c	c	c
e	a	b	c	e

Given the above multiplication table it is evident that if κ (f) =1, where f ∈ E_S and κ (s) =0.7 for all s ∉ E_S, then κ (x · sx^-1) ≥ κ (s) .

Definition 4.4.We call (κ, τ) a fuzzy congruence pair of S if κ is fuzzy normal subgroupoid of S, τ is a fuzzy congruence and if for all e ∈ E_S, s ∈ S:

(1) κ (s) ≥ κ (se) ∧ τ (e, ss^-1),

(2) τ (ss^-1, s^-1s) ≥ κ (s) .

In the following lemma, we show a relation between the fuzzy kernel and fuzzy trace of a fuzzy congruence of S .

Lemma 4.5. If φ is fuzzy congruence on a completely inverse AG^**-groupoid S. Then, (κ_φ, τ_φ) is a fuzzy congruence pair on S .

Proof. First we show that κ_φ is a fuzzy subgroupoid. For any x, y ∈ S, $κ_{φ} (xy) = \underset{e \in E_{S}}{⋁} φ (xy, e) .$ If e₁, e₂ ∈ E_S, then φ (xy, e₁e₂) ≥ φ (x, e₁) ∧ φ (y, e₂) and consequently $κ_{φ} (xy) \geq κ_{φ} (x) \land κ_{φ} (y) .$ By the definition of fuzzy kernel κ_φ, we have $κ_{φ} (x) = \underset{e \in E_{S}}{⋁} φ (x, e) = \underset{e \in E_{S}}{⋁} φ (x^{- 1}, e) = κ_{φ} (x^{- 1}) .$ Further, we show that κ_φ is self-conjugate. Since x · ex^-1 ∈ E_S and κ_φ (e) =1, we have $\begin{matrix} {κφsx}^{- 1}) = \underset{e \in E_{S}}{⋁} φ (x \cdot {sx}^{- 1}, e) \\ \underset{e \in E_{S}}{⋁} φ (x \cdot {sx}^{- 1}, x \cdot {ex}^{- 1}) \\ \land φ (x \cdot {ex}^{- 1}, e) \\ E_{S} \end{matrix} ⋁ φ (x \cdot {sx}^{- 1}, x \cdot {ex}^{- 1}) \land \underset{e \in E_{S}}{⋁} φ (x \cdot {ex}^{- 1}, e) \dots \dots E_{S} ⋁ φ (x \cdot {sx}^{- 1}, x \cdot {ex}^{- 1}) (as x \cdot {ex}^{- 1} \in E_{S}) \dots \dots E_{S} ⋁ φ (s, e) (since φ is congruence) .$ Hence, κ_φ is a fuzzy normal subgroupoid of S . Further, we show conditions (1) and (2) of Definition 4.4. Thus $\begin{matrix} κ_{φ} (s) & \geq φ (s, e) \geq φ ({ss}^{- 1} \cdot s, fs) \land φ (fs, e) \\ \geq φ ({ss}^{- 1}, f) \land φ (fs, e) . \end{matrix}$ Since ss^-1 ∈ E_S, it follows that φ (ss^-1, f) = τ_φ (ss^-1, f) . Thus, for all e, f ∈ E_S and s ∈ S, we have $\begin{matrix} κ_{φ} (s) & \geq τ_{φ} ({ss}^{- 1}, f) \land \underset{e \in E_{S}}{⋁} φ (fs, e) \\ \geq τ_{φ} ({ss}^{- 1}, f) \land κ_{φ} (fs) . \end{matrix}$ Furthermore, if e ∈ E_S and s ∈ S, and since φ (s^-1, e^-1) = φ (s, e), then $\begin{matrix} τ_{φ} ({ss}^{- 1}, s^{- 1} s) & = φ ({ss}^{- 1}, s^{- 1} s) \\ \geq φ ({ss}^{- 1}, e) \land φ (e, s^{- 1} s) \\ \geq φ (s, e) \land φ (s^{- 1}, e) \land φ (e, s^{- 1}) \\ \land φ (e, s) = φ (s, e) . \end{matrix}$ Hence τ_φ (ss^-1, s^-1s) ≥ κ_φ (s) . This completes the proof. The following result is important because it establishes a relation between a fuzzy kernel and fuzzy trace of a fuzzy congruence of a completely inverse AG^**-groupoid. Proposition 4.6. Let (κ, τ) be a fuzzy congruence pair on a completely inverse AG^**-groupoid S . Then, for all a, b ∈ S, e ∈ E_S, (1) κ (ab) ≥ κ (ab · e) ∧ τ (e, aa^-1) , (2) κ (ab · e) ≥ κ (ab) , (3) τ (a · ea^-1, b · eb^-1) ≥ τ (aa^-1, bb^-1) ∧ κ (a^-1b) . Proof. (1). Using the definition of a congruence pair, we have $κ (ab) \geq κ ((ab) (b \cdot {eb}^{- 1})) \land τ ((b \cdot {eb}^{- 1}), (ab \cdot a^{- 1} b^{- 1})) .$ Now, (ab) (b · eb^-1) = e (ab · bb^-1) = e (bb^-1 · ba) = ab · e and since τ is a fuzzy congruence, we have τ (b · eb^-1, (ab · a^-1b^-1)) = τ (b · eb^-1, b (a^-1a · b^-1)) ≥ τ (e, aa^-1) . Thus, $\begin{matrix} κ (ab) & \geq κ ((ab) (b \cdot {eb}^{- 1})) \land τ (b \cdot {eb}^{- 1}, ab \cdot a^{- 1} b^{- 1}) \\ \geq κ (ab \cdot e) \land τ (e, a^{- 1} a) . \end{matrix}$ (2). Since b · eb^-1 ∈ E_S and κ is fuzzy full, κ (b · eb^-1) =1 . Then, $\begin{matrix} κ (ab \cdot e) & = κ (({bb}^{- 1} \cdot ab) e) = κ ((ab) ({bb}^{- 1} \cdot e)) \\ \geq κ (ab) \land κ ({bb}^{- 1} \cdot e) \\ \geq κ (ab) (since κ ({bb}^{- 1} \cdot e) = 1) . \end{matrix}$ (3). Similarly, since a · ea^-1 ∈ E_S, it follows that $\begin{matrix} a \cdot {ea}^{- 1} = (a \cdot {ea}^{- 1}) (({aa}^{- 1}) (a^{- 1} \cdot ea)) . \end{matrix}$ Also, $\begin{matrix} (a^{- 1} (ea \cdot b^{- 1})) & (a ({ea}^{- 1} \cdot b)) \\ = & ((ea \cdot a^{- 1} a)) (({ea}^{- 1} \cdot b^{- 1} b)) \\ = & (a (ea \cdot a^{- 1})) (({bb}^{- 1} \cdot a^{- 1} e)) \\ = & (a \cdot {ea}^{- 1}) (({bb}^{- 1}) (a^{- 1} \cdot ea)) . \end{matrix}$ So $\begin{matrix} τ (a \cdot {ea}^{- 1}, (a^{- 1} (ea \cdot b^{- 1})) (a ({ea}^{- 1} \cdot b))) \\ = τ ((a \cdot {ea}^{- 1}) (({aa}^{- 1}) (a \cdot {ea}^{- 1})), \\ (a \cdot {ea}^{- 1}) (({bb}^{- 1}) (a \cdot {ea}^{- 1}))) \\ \geq τ ({aa}^{- 1}, {bb}^{- 1}) (τ is a fuzzy congruence) . \end{matrix}$ Thus, since τ is fuzzy congruence $\begin{matrix} τ [(a^{- 1} & (ea \cdot b^{- 1})) (a ({ea}^{- 1} \cdot b)), \\ (a^{- 1} (eb \cdot a^{- 1})) (a ({eb}^{- 1} \cdot a)] \\ = τ [({ea}^{- 1}) ((a^{- 1} b \cdot {ab}^{- 1}) (ea)), \\ ({ea}^{- 1}) (({ab}^{- 1} \cdot a^{- 1} b)) (ea)] \\ \geq τ ((a^{- 1} b) ({ab}^{- 1}), ({ab}^{- 1}) (a^{- 1} b)) \\ = τ ((a^{- 1} b) (a^{- 1} b)^{- 1}, (a^{- 1} b)^{- 1} (a^{- 1} b)) \\ \geq κ (a^{- 1} b) . (by Definition 4.4) \end{matrix}$ Using the above two inequalities, we have $\begin{matrix} τ (a \cdot & {ea}^{- 1}, (a^{- 1} (eb \cdot a^{- 1})) (a ({eb}^{- 1} \cdot a))) \\ \geq τ [a \cdot {ea}^{- 1}, (a^{- 1} (ea \cdot b^{- 1})) (a ({ea}^{- 1} \cdot b)) \\ \land τ ((a^{- 1} (ea \cdot b^{- 1})) (a ({ea}^{- 1} \cdot b)), \\ (a^{- 1} (eb \cdot a^{- 1})) (a ({eb}^{- 1} \cdot a))] \\ \geq τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 1} b) . \end{matrix}$ (1) Moreover, $\begin{matrix} τ [(a^{- 1} & (eb \cdot a^{- 1})) (a ({eb}^{- 1} \cdot a)), \\ (a^{- 1} ({eb}^{- 1} \cdot a)) (a (eb \cdot a^{- 1}))] \\ = τ [(a^{- 1} a) ((eb \cdot a^{- 1}) ({eb}^{- 1} \cdot a)), \\ (a^{- 1} a) (({eb}^{- 1} \cdot a) (eb \cdot a^{- 1}))] \\ \geq τ [((eb \cdot a^{- 1}) ({eb}^{- 1} \cdot a)), \\ (({eb}^{- 1} \cdot a) (eb \cdot a^{- 1}))] \\ (τ is a congruence) \\ \geq τ [((eb \cdot a^{- 1}) (eb \cdot a^{- 1})^{- 1}), \\ ((eb \cdot a^{- 1}) (eb \cdot a^{- 1})^{- 1})] \\ \geq κ (eb \cdot a^{- 1}) (by Definition 4.4) . \end{matrix}$ (2) Combining (1) and (2), and since κ (e) =1, we get $\begin{matrix} τ [(a \cdot & {ea}^{- 1}), (a^{- 1} \cdot {ab}^{- 1}) (e (a \cdot a^{- 1} b))] \\ = & τ [(a \cdot {ea}^{- 1}), (a^{- 1} (ea \cdot b^{- 1})) (a ({ea}^{- 1} \cdot b)] \\ \geq & [τ (a \cdot {ea}^{- 1}, (a^{- 1} (eb \cdot a^{- 1})) (a ({eb}^{- 1} \cdot a))) \\ \land τ (a^{- 1} (eb \cdot a^{- 1})) (a ({eb}^{- 1} \cdot a), \\ (a^{- 1} (ea \cdot b^{- 1})) (a ({ea}^{- 1} \cdot b))] \\ \geq & τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 1} b) \land κ (eb \cdot a^{- 1}) \\ \geq & τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 1} b) \land κ (a^{- 1} b) \land κ (e) \\ = & τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 1} b) . \end{matrix}$ (3) Similarly, $\begin{matrix} τ [ & (a^{- 1} \cdot {ab}^{- 1}) (e (a \cdot a^{- 1} b)), b \cdot {eb}^{- 1}] \\ = τ [(a^{- 1} (ea \cdot b^{- 1}) (a ({ea}^{- 1} \cdot b))), \\ ({bb}^{- 1} \cdot b) (e (b^{- 1} b \cdot b^{- 1})] \\ \geq τ (a^{- 1} a, b^{- 1} b) \land τ (b \cdot {eb}^{- 1}, b \cdot {eb}^{- 1}) \\ = τ ({aa}^{- 1}, {bb}^{- 1}) \end{matrix}$ (4) Combining (3) and (4), we get $\begin{matrix} τ ( & a \cdot {ea}^{- 1}, b \cdot {eb}^{- 1}) \\ \geq τ (a \cdot {ea}^{- 1}, (a^{- 1} \cdot {ab}^{- 1}) (e (a \cdot a^{- 1} b)) \\ \land τ ((a^{- 1} \cdot {ab}^{- 1}) (e (a \cdot a^{- 1} b)), (b \cdot {eb}^{- 1})) \\ \geq τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 1} b) \land τ ({aa}^{- 1}, {bb}^{- 1}) \\ = τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 1} b) . \end{matrix}$ This completes the proof. Proposition 4.7. If (κ, τ) is a fuzzy congruence pair on S, then $μ_{(κ, τ)} (a, b) = τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 1} b) (\forall a, b \in S)$ is a fuzzy congruence on S. Proof. We have to show first that μ_(κ,τ) is reflexive, symmetric and transitive. (1). Since a^-1a ∈ E_S and κ is full, it is clear that for all a ∈ S, κ (a^-1a) =1 . Thus, $μ_{(κ, τ)} (a, a) = τ ({aa}^{- 1}, {aa}^{- 1}) \land κ (a^{- 1} a) = 1 .$ (2). μ_(κ,τ) (a, b) is symmetric, because for a, b ∈ S, $μ_{(κ, τ)} (b, a) = τ ({bb}^{- 1}, {aa}^{- 1}) \land κ (b^{- 1} a) .$ Since κ is a fuzzy AG^**-subgroupoid of S, it follows that κ (b^-1a) = κ ((b^-1a)) ^-1 = κ (ba^-1) . By Proposition 4.6, we get κ (ba^-1) = κ (a^-1b) . Hence, $\begin{matrix} μ_{(κ, τ)} (b, a) & = τ ({bb}^{- 1}, {aa}^{- 1}) \land κ (b^{- 1} a) \\ = τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 1} b) \\ = μ_{(κ, τ)} (a, b) . \end{matrix}$ (3). For transitivity, let a, b, c ∈ S . Then, by Proposition 4.6 $\begin{matrix} μ_{(κ, τ)} (a, c) = τ ({aa}^{- 1}, {cc}^{- 1}) \land κ (a^{- 1} c) \\ \geq τ ({aa}^{- 1}, {bb}^{- 1}) \land τ ({bb}^{- 1}, {cc}^{- 1}) \\ \land κ (a^{- 1} c \cdot {bb}^{- 1}) \land τ ({bb}^{- 1}, {aa}^{- 1}) \\ \geq τ ({aa}^{- 1}, {bb}^{- 1}) \land τ ({bb}^{- 1}, {cc}^{- 1}) \\ \land κ (a^{- 1} b) \land κ ({cb}^{- 1}) \\ \geq τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 1} b) \land τ ({bb}^{- 1}, {cc}^{- 1}) \\ \land κ (b^{- 1} c) (by Proposition 2.1) \\ = μ_{(κ, τ)} (a, b) \land μ_{(κ, τ)} (b, c) . \end{matrix}$ Hence, μ_(κ,τ) (a, b) is an equivalence relation. Further, we show that μ_(κ,τ) (a, b) is a compatible. Since τ is fuzzy congruence, then for any c ∈ S, $τ ((ac) (ac)^{- 1}, (bc) (bc)^{- 1}) \geq τ ({aa}^{- 1}, {bb}^{- 1}) .$ By Proposition 4.6, we have κ (a^-1b · c^-1c) ≥ κ (a^-1b) . Thus, $\begin{matrix} μ_{(κ, τ)} (ac, bc) & = τ (ac \cdot a^{- 1} c^{- 1}, bc \cdot b^{- 1} c^{- 1}) \\ \land κ (a^{- 1} c^{- 1} \cdot bc) \\ \geq τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 1} b) \\ = μ_{(κ, τ)} (a, b) . \end{matrix}$ Similarly, again by Proposition 4.61 $\begin{matrix} μ_{(κ, τ)} (ca, cb) & = τ (ca \cdot c^{- 1} a^{- 1}, cb \cdot c^{- 1} b^{- 1}) \\ \land κ (c^{- 1} a^{- 1} \cdot cb) \\ = τ (a (c^{- 1} c \cdot a^{- 1}), b (c^{- 1} c \cdot b^{- 1})) \\ \land κ (c^{- 1} c \cdot a^{- 1} b) \\ \geq τ (a^{- 1} a, b^{- 1} b) \land κ (a^{- 1} b) \\ = μ_{(κ, τ)} (a, b) . \end{matrix}$ Hence, μ_(κ,τ) (a, b) is a fuzzy congruence.□

Corollary 4.8. If (κ, τ) is a fuzzy congruence pair on S, then

μ_{(κ, τ)} (a, b) = τ (a^{- 1} a, b^{- 1} b) \land κ ({ab}^{- 1}) (\forall a, b \in S)

is a fuzzy congruence on S. Proposition 4.9. If (κ, τ) is a fuzzy congruence on S, then $ν_{(κ, τ)} (a, b) = τ ({aa}^{- 1}, {bb}^{- 1}) \land κ ((a^{- 1})^{2} b^{2}) (\forall a, b \in S)$ is a fuzzy congruence on S. Proof. Both reflexivity and symmetry are clear. We show that ν_(κ,τ) is transitive. Let a, b, c ∈ S . Then, $\begin{matrix} ν_{(κ, τ)} (a, c) & = τ ({aa}^{- 1}, {cc}^{- 1}) \land κ (a^{- 2} c^{2}) \\ \geq τ ({aa}^{- 1}, {bb}^{- 1}) \land τ ({aa}^{- 1}, {cc}^{- 1}) \\ \land κ (a^{- 2} c^{2} \cdot b^{2} b^{- 2}) \\ \geq τ ({aa}^{- 1}, {bb}^{- 1}) \land τ ({aa}^{- 1}, {cc}^{- 1}) \\ \land κ (a^{- 2} b^{2}) \land κ (c^{2} b^{- 2}) \\ \geq τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 2} b^{2}) \\ \land τ ({aa}^{- 1}, {cc}^{- 1}) \land κ (b^{- 2} c^{2}) \\ = ν_{(κ, τ)} (a, b) \land ν_{(κ, τ)} (b, c) . \end{matrix}$ Hence, ν_(κ,τ) is equivalence relation. For compatibility, let c ∈ S, then $\begin{matrix} ν_{(κ, τ)} (ac, bc) & = & τ (ac \cdot a^{- 1} c^{- 1}, bc \cdot b^{- 1} c^{- 1}) \\ \land κ ((a^{- 1} c^{- 1})^{2} \cdot b^{2} c^{2}) \\ = & τ ({aa}^{- 1} \cdot {cc}^{- 1}, {bb}^{- 1} \cdot {cc}^{- 1}) \\ \land κ (a^{- 2} b^{2} \cdot c^{- 2} c^{2}) (τ is a congruence) \\ = & τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 2} b^{2}) \\ \land κ (c^{- 2} c^{2}) \\ = & τ ({aa}^{- 1}, {bb}^{- 1}) \land κ ((a^{- 1})^{2} b^{2}) \\ (since (c)^{- 2} c^{2} \in E_{S}) . \end{matrix}$ Similarly, since (c^-1) ²c² ∈ E_S and τ is a congruence, we have $\begin{matrix} ν_{(κ, τ)} (ca, cb) & = τ (ca \cdot c^{- 1} a^{- 1}, cb \cdot c^{- 1} b^{- 1}) \\ \land κ ((c^{- 1} a^{- 1})^{2} \cdot c^{2} b^{2}) \\ = τ ({cc}^{- 1} \cdot {aa}^{- 1}, {cc}^{- 1} \cdot {bb}^{- 1}) \\ \land κ (c^{- 2} c^{2} \cdot a^{- 2} b^{2}) \\ = τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (c^{- 2} c^{2}) \\ \land κ (a^{- 2} b^{2}) \\ = τ ({aa}^{- 1}, {bb}^{- 1}) \land κ ((a^{- 1})^{2} b^{2}) . \end{matrix}$ Hence, ν_(κ,τ) is a fuzzy congruence.□ Corollary 4.10.If (κ, τ) is a fuzzy congruence pair on S, then for all a, b ∈ S $ν_{(κ, τ)} (a, b) = τ (a^{- 1} a, b^{- 1} b) \land κ (a^{2} (b^{- 1})^{2})$ is a fuzzy congruence on S. Theorem 4.11.Let (κ, τ) be a fuzzy congruence pair on a completely inverse AG^**-groupoid S . Then, μ_(κ,τ) (a, b) is the unique fuzzy congruence relation on S such that κ_φ = κ and τ_φ = φ . Conversely, if μ is a fuzzy congruence on S, then (κ_μ, τ_μ) is a fuzzy congruence pair on S and μ_{(κ_μ,τ_μ)} = μ . Proof. Let (κ, τ) be fuzzy congruence pair. The first part is clear from Proposition 4.7, that μ_(κ,τ) is a fuzzy congruence relation on S. To show that κ_φ = κ and τ_φ = τ, let x ∈ S, then κ_φ (x) ≤ κ (x). In fact, $κ_{φ} (x) = \underset{e \in E_{S}}{⋁} φ (x, e)$ and using Definition 4.4, $\begin{matrix} φ (x, e) & = τ ({xx}^{- 1}, {ee}^{- 1}) \land κ (x^{- 1} e) \\ = τ (x^{- 1} x, e) \land κ (xe) \leq κ (x) . \end{matrix}$ For the converse inclusion, we have $\begin{matrix} κ (x) & = τ (x^{- 1} x, {xx}^{- 1}) \land κ (x) \\ = τ (x^{- 1} x, {xx}^{- 1} \cdot (x^{- 1} x)^{- 1}) \\ \land κ ({xx}^{- 1} \cdot x) (since {xx}^{- 1} \in E_{S}) \\ = φ (x, x^{- 1} x) \leq κ_{φ (x)} . \end{matrix}$ Thus, κ_φ = κ . Also, let e, f ∈ E_S . Then, κ (ef) =1, because ef ∈ E_S . Thus, $\begin{matrix} τ_{φ} (e, f) & = φ (e, f) = τ ({ee}^{- 1}, {ff}^{- 1}) \land κ (e^{- 1} f) \\ = τ (e, f) . \end{matrix}$ Further, we show that μ_(κ,τ) is a unique such congruence on S . Suppose that π is another congruence on S such that κ_π = κ and τ_π = τ . Then, for a, b ∈ S, we have $\begin{matrix} π (a, b) & = π (a, b) \land π (a, b) \\ = π (a, b) \land π (a^{- 1}, b^{- 1}) \\ \leq π ({aa}^{- 1}, {bb}^{- 1}) \\ = τ ({aa}^{- 1}, {bb}^{- 1}) . \end{matrix}$ And since bb^-1 ∈ E_S $\begin{matrix} κ (a^{- 1} b) & = κ_{π} (a^{- 1} b) = \underset{e \in E_{S}}{⋁} π (a^{- 1} b, e) \\ \geq \underset{e \in E_{S}}{⋁} π (a^{- 1} b, {bb}^{- 1}) \land π ({bb}^{- 1}, e) \\ = \underset{e \in E_{S}}{⋁} π (a^{- 1} b, {bb}^{- 1}) \land \underset{e \in E_{S}}{⋁} π ({bb}^{- 1}, e) \\ \geq π (a^{- 1} b, {bb}^{- 1}) = π (a, b) . \end{matrix}$ Thus, $\begin{matrix} φ (a, b) & = τ ({aa}^{- 1}, {bb}^{- 1}) \land κ (a^{- 1} b) \\ \geq π (a, b) \land π (a, b) \\ = π (a, b) . \end{matrix}$ On the other hand, $\begin{matrix} π (a, b) & \geq π (a, ea) \land π (ea, b) \\ \geq π (a, eb) \land π (eb, ea) \land π (b, a^{- 1} a \cdot b) \\ \land π (a^{- 1} a \cdot b, ea) \\ = π (a, eb) \land π (eb, a) \land π (b, a^{- 1} a \cdot b) \\ \land π (a^{- 1} a \cdot b, ea) . \end{matrix}$ (5) Moreover, using the fact that π is a congruence, we have $\begin{matrix} π (a, eb) & \geq π ({aa}^{- 1} \cdot a, {bb}^{- 1} \cdot a) \land π ({bb}^{- 1} \cdot a, eb) \\ \geq π ({aa}^{- 1}, {bb}^{- 1}) \land π ({ab}^{- 1} \cdot b, eb) \\ \geq π (a^{- 1} a, b^{- 1} b) \land π ({ab}^{- 1}, e) . \end{matrix}$ (6) $\begin{matrix} π (eb, a) \geq π (eb, {bb}^{- 1} \cdot a) \land π ({bb}^{- 1} \cdot a, {aa}^{- 1} \cdot a) \\ π (eb, {ab}^{- 1} \cdot b) \land π ({bb}^{- 1} \cdot a, {aa}^{- 1} \cdot a) \\ b^{- 1} b) \land π ({ab}^{- 1}, e) . \end{matrix}$ (7) Similarly, we show $\begin{matrix} π (a^{- 1} a \cdot b, b) & = π (a^{- 1} a \cdot b, b^{- 1} b \cdot b) \\ \geq π (a^{- 1} a, b^{- 1} b) \\ = τ (a^{- 1} a, b^{- 1} b) . \end{matrix}$ (8) Finally, by Proposition 2.1 and Lemma 3.3, we get $\begin{matrix} π ({aa}^{- 1} \cdot b, ea) \geq π ({ba}^{- 1} \cdot a, ea) \\ = π ({ba}^{- 1}, e) = π ({ab}^{- 1}, e) \end{matrix}$ (9) Inserting in (5) from (6), (7), (8) and (9), we have $\begin{matrix} π (a^{- 1} a, b^{- 1} b) \land π ({ab}^{- 1}, e) \land π (a^{- 1} a, b^{- 1} b) \\ π ({ab}^{- 1}, e) \geq π (a^{- 1} a, b^{- 1} b) \land π ({ab}^{- 1}, e) . \end{matrix}$ Since τ (a^-1a, b^-1b) = π (a^-1a, b^-1b) and $\underset{e \in E_{S}}{⋁} π ({ab}^{- 1}, e) = κ_{φ} ({ab}^{- 1}, e),$ we have $π (a, b) \geq τ (a^{- 1} a, b^{- 1} b) \land κ_{φ} ({ab}^{- 1}) = φ (a, b) .$ Thus, φ (a, b) = π (a, b) . Conversely, suppose that μ is a fuzzy congruence, then by Lemma 4.5, (κ_μ, τ_μ) is fuzzy congruence pair. As a consequence of uniqueness of μ_(κ,τ), we have μ (κ_μ, τ_μ) = μ .

5 Lattice of fuzzy congruences

Let C_S be the family of all fuzzy congruences of a completely inverse AG^**-groupoid. Then (C_S, ⊆) is a lattice partially ordered by ordinary inclusion. If ρ, σ ∈ C_S, then ρ ∩ σ ∈ C_S and is the greatest lower bound of ρ and σ, while the intersection of all the family of fuzzy congruences containing ρ ∪ σ is the least upper bound. The join of ρ and σ denoted by ρ ⊔ σ is the transitive closure of ρ ∪ σ . Let {ρ_i} _i∈J be a family of fuzzy congruences, then (C_S, ⊆ , ∩ , ⊔) is a complete lattice.

Proposition 5.1. If for ρ, σ ∈ C_S, ρ ∘ σ ∈ C_S, then ρ ∘ σ = ρ ⊔ σ .

Theorem 5.2.Let S be a completely inverse AG^**-groupoid. The lattice (C_S, ⊆ , ∩ , ∘) of all fuzzy congruences such that ρ ∘ σ = σ ∘ ρ for all ρ, σ ∈ C_S, is modular.

Proof. Let ρ, σ, υ ∈ C_S such that ρ ≤ υ . We need to show that (ρ ∘ σ) ∩ υ ≤ ρ ∘ (σ ∩ υ) . Let a, b ∈ S . Then $\begin{matrix} = ((ρ \circ σ) \cap υ) (a, b) = \underset{z \in S}{⋁} (ρ (a, z) \land σ (z, b)) \land υ (a, b) \\ = \underset{z \in S}{⋁} (ρ (a, z) \land ρ (a, z) \land σ (z, b)) \land υ (a, b) \\ \leq \underset{z \in S}{⋁} (ρ (a, z) \land υ (a, z) \land σ (z, b)) \land υ (a, b) \\ (since ρ \leq σ) \\ \leq \underset{z \in S}{⋁} (ρ (a, z) \land σ (z, b) \land υ (z, b)) \\ = (ρ \circ (σ \cap υ)) (a, b) . \end{matrix}$

Theorem 5.3.Let φ be a fuzzy congruence on E_S . Then, the relation μ defined as $μ (e, f) = \underset{a \in S}{⋀} φ (a^{- 1} e \cdot a, a^{- 1} f \cdot a); (\forall e, f \in E_{S})$ is the greatest fuzzy congruence on E_S contained in φ .

Proof. Clearly μ is reflexive and symmetric. For transitivity, let e, f, g ∈ E_S and a ∈ S . Since $\begin{matrix} φ (a^{- 1} e \cdot a, a^{- 1} f \cdot a) & \geq φ (a^{- 1} e \cdot a, a^{- 1} g \cdot a) \\ \land φ (a^{- 1} g \cdot a, a^{- 1} f \cdot a) . \end{matrix}$ Then $\begin{matrix} μ (e, f) & = & a \in S ⋀ φ (a^{- 1} e \cdot a, a^{- 1} f \cdot a) \\ \geq & \underset{a \in S}{⋀} (φ (a^{- 1} e \cdot a, a^{- 1} g \cdot a) \\ \land & φ (a^{- 1} g \cdot a, a^{- 1} f \cdot a) \\ = & \underset{a \in S}{⋀} (a^{- 1} e \cdot a, a^{- 1} g \cdot a) \\ \land & \underset{a \in S}{⋀} (a^{- 1} g \cdot a, a^{- 1} f \cdot a) \\ = & μ (e, g) \land μ (g, f) \\ = & μ \circ μ (e, f) \end{matrix}$ Thus μ is a fuzzy equivalece relation.

For e, f, g ∈ E_S, and since μ is a congruence, we have $\begin{matrix} μ (ge, gf) & = μ (eg, fg) = μ (g^{- 1} e \cdot g, g^{- 1} f \cdot g) \\ \geq μ (e, f) \end{matrix}$ showing that μ is compatible. Hence, μ is a fuzzy congruence on E_S .

Let e, f ∈ E_S, then μ ≤ φ because $φ (e, f) \geq φ (e, ef) \land φ (ef, f) \geq μ (e, f) .$ To show that μ is the greatest such congruence, let σ be any fuzzy normal congruence relation such that φ ≥ σ . Then $μ (e, f) \geq \underset{e \in E_{S}}{⋀} σ (a^{- 1} \cdot ea, a^{- 1} \cdot fa) \geq σ (e, f) .$ Hence, μ ≥ σ .□

Theorem 5.4.Let φ be a fuzzy congruence relation on S . Then, the fuzzy relation φ_max defined as $φ_{\max} = \underset{e \in E_{S}}{⋀} φ (a^{- 1} e \cdot a, b^{- 1} e \cdot b) (\forall a, b \in S)$ is a fuzzy congruence relation on S .

Proof. We first show that φ_max is an equivalence relation. Reflexivity and symmetry are immediate from the definition of φ_max . For transitivity, let a, b, c ∈ S . Then, $\begin{matrix} \underset{e \in E_{S}}{⋁} [ & φ (a^{- 1} e \cdot a, c^{- 1} e \cdot c) \land φ (c^{- 1} e \cdot c, b^{- 1} e \cdot b)] \\ \geq \underset{e \in E_{S}}{⋁} [\underset{e \in E_{S}}{⋀} φ (a^{- 1} e \cdot a, c^{- 1} e \cdot c) \\ \land φ (c^{- 1} e \cdot c, b^{- 1} e \cdot b)] . \end{matrix}$ $(because \underset{e \in E_{S}}{⋁} φ (a^{- 1} e \cdot a, c^{- 1} e \cdot c) \land φ (c^{- 1} e \cdot c, b^{- 1} e \cdot b) \geq \underset{e \in E_{S}}{⋀} φ (a^{- 1} e \cdot a, c^{- 1} e \cdot c) \land φ (c^{- 1} e \cdot c, b^{- 1} e \cdot b)) .$ Then $\begin{matrix} \underset{e \in E_{S}}{⋁} & [φ (a^{- 1} e \cdot a, c^{- 1} e \cdot c) \land φ (c^{- 1} e \cdot c, b^{- 1} e \cdot b)] \\ \geq \underset{e \in E_{S}}{⋁} [\underset{e \in E_{S}}{⋀} φ (a^{- 1} e \cdot a, c^{- 1} e \cdot c) \\ \land φ (c^{- 1} e \cdot c, b^{- 1} e \cdot b)] \\ = \underset{e \in E_{S}}{⋁} [\underset{e \in E_{S}}{⋀} φ (a^{- 1} e \cdot a, c^{- 1} e \cdot c) \\ \land \underset{e \in E_{S}}{⋀} φ (c^{- 1} e \cdot c, b^{- 1} e \cdot b)] \\ E_{S} \end{matrix} ⋁ φ_{\max} (a, c) \land φ_{\max} (c, b) \dots φ_{\max}) (a, b) .$ Using the fact that φ ∘ φ ≤ φ, thus $\begin{matrix} (φ_{\max} \circ φ_{\max}) (a, b) & \leq & \underset{e \in E_{S}}{⋀} [\underset{e \in E_{S}}{⋁} φ (a^{- 1} e \cdot a, \\ c^{- 1} e \cdot c) \land φ (c^{- 1} e \cdot c, b^{- 1} e \cdot b)] \\ \leq & \underset{e \in E_{S}}{⋀} φ (a^{- 1} e \cdot a, b^{- 1} e \cdot b) \\ \leq & φ_{\max} (a, b) \end{matrix}$ which shows that φ_max is transitive. Hence, φ_max is a fuzzy equivalence relation. For fuzzy compatibility, let e ∈ S and a, b, c ∈ S . Then $\begin{matrix} φ_{\max} (ca, cb) & = \underset{e \in E_{S}}{⋀} φ (((ca)^{- 1} \cdot e) (ca), \\ ((cb)^{- 1} \cdot e) (cb)) \\ = \underset{e \in E_{S}}{⋀} φ (c ((a^{- 1} e \cdot c^{- 1} a), \\ c ((b^{- 1} e \cdot c^{- 1} b)) \\ \geq \underset{e \in E_{S}}{⋀} φ (a^{- 1} e \cdot a, b^{- 1} e \cdot b) \\ (since φ is congruence) \\ = φ_{\max} (a, b) . \end{matrix}$ In a similar manner, we have $\begin{matrix} φ_{\max} (ac, bc) & = \underset{e \in E_{S}}{⋀} φ ((ac)^{- 1} e \cdot (ac), \\ (bc)^{- 1} e \cdot (bc)) \\ = \underset{e \in E_{S}}{⋀} φ (c^{- 1} ((a^{- 1} e \cdot a) c), \\ c^{- 1} ((b^{- 1} e \cdot b) c)) \\ \geq \underset{e \in E_{S}}{⋀} φ (a^{- 1} e \cdot a, b^{- 1} e \cdot b) \\ (since φ is congruence) \\ = φ_{\max} (a, b) . \end{matrix}$ Hence, φ_max is a fuzzy congruence relation on S .□ Lemma 5.5. A fuzzy relation on a completely inverse AG^**-groupoid S is transitive if and only if φ_t = {(x, y) ∈ S × S : φ (x, y) > t} . Theorem 5.6.If φ is a fuzzy congruence on a completely inverse AG^**-groupoid S, then $φ_{\min} (a, b) = {\begin{matrix} \underset{x \in S}{⋁} (e, {aa}^{- 1}) \land ({aa}^{- 1}, {bb}^{- 1}) if \\ S (a, b) = {e \in E_{S} : ea = eb} \neq \emptyset, \\ 0 otherwise \end{matrix}$ is a fuzzy congruence relation on S . Proof. The relation φ_min is reflexive and symmetric. We show that φ_min is transitive. Let a, b, c ∈ S and 0 ≤ t < 1, then there exist e, f ∈ E_S such that ea = eb, fb = fc and φ (e, aa^-1) ∧ φ (aa^-1, bb^-1) > t and φ (f, bb^-1) ∧ φ (bb^-1, cc^-1) > t . Also ea · f = eb · f = ec · f and by Lemma 5.5, we have $\begin{matrix} φ (ef, {aa}^{- 1}) & \land φ ({aa}^{- 1}, {cc}^{- 1}) \geq φ (e, {aa}^{- 1}) \\ \land φ (f, {aa}^{- 1}) \land φ ({aa}^{- 1}, {bb}^{- 1}) \\ \land φ ({bb}^{- 1}, {cc}^{- 1}) > t . \end{matrix}$ Thus (a, c) ∈ (φ_min) _t . Hence (φ_min) _t is transitive and by Lemma 5.5, φ_min is a fuzzy equivalence relation. Also, if a, b, c ∈ S and φ_min (a, b) =0, then φ_min (ac, bc) ≥ φ_min (a, b) . Let φ_min (a, b) >0 . Then for some e ∈ E_S, ea = eb and e (ac) = (ea) c = (eb) c = e (bc) . Since (ac) (ac) ^-1 = a^-1 (ac · c^-1) = (a^-1a · a^-1) (ac · c^-1) = (aa^-1) (ac · a^-1c^-1) . Then $\begin{matrix} φ ((e \cdot ac) (ac)^{- 1}, & (ac) (ac)^{- 1}) \\ = φ (e (ac \cdot a^{- 1} c^{- 1}), \\ ({aa}^{- 1}) (ac \cdot a^{- 1} c^{- 1})) \\ \geq φ (e, {aa}^{- 1}) . \end{matrix}$ Thus $\begin{matrix} φ ((ac) (ac)^{- 1}, (bc) (bc)^{- 1}) \\ \geq φ (ac \cdot a^{- 1} c^{- 1}, e (ac \cdot a^{- 1} c^{- 1})) \\ \land (e (ac \cdot a^{- 1} c^{- 1}), bc \cdot b^{- 1} c^{- 1}) \\ \geq φ (e, {aa}^{- 1}) \land φ (e (ac \cdot a^{- 1} c^{- 1}), bc \cdot b^{- 1} c^{- 1}) \\ = φ (e, {aa}^{- 1}) \land φ (e (bc \cdot b^{- 1} c^{- 1}), \\ ({bb}^{- 1}) (bc \cdot b^{- 1} c^{- 1})) (since ea = eb) \\ \geq φ (e, {aa}^{- 1}) \land φ (e, {bb}^{- 1}) . \end{matrix}$ But since $\begin{matrix} {aa}^{- 1}) \land φ (e, {bb}^{- 1}) \geq φ (e, {aa}^{- 1}) \land φ (e, {aa}^{- 1}) \\ \land ({aa}^{- 1}, {bb}^{- 1}) = φ (e, {aa}^{- 1}) \\ \land φ ({aa}^{- 1}, {bb}^{- 1}) . \end{matrix}$ Then φ ((ac) (ac) ^-1, (bc) (bc) ^-1) ≥ φ (e, aa^-1) ∧ φ (aa^-1, bb^-1) . So $\begin{matrix} φ_{\min} (a, b) \leq \underset{e \in S (a, b)}{⋁} φ ((e \cdot ac) (a^{- 1} c^{- 1}), \\ (ac \cdot a^{- 1} c^{- 1})) \land φ ((ac) (ac)^{- 1}, (bc) (bc)^{- 1}) \\ = φ_{\min} (ac, bc) \\ (since e ({aa}^{- 1} \cdot {cc}^{- 1}) \in S (ac, bc)) . \end{matrix}$ If φ_min (a, b) =0, then φ_min (ca, cb) ≥ φ_min (a, b) . If φ_min (a, b) >0, then for some e ∈ E_S, ea = eb and e (ca) = c (ea) = c (eb) = e (cb) . And $\begin{matrix} φ (c^{- 1} e \cdot c, (ca) (ca)^{- 1}) \\ \land φ ((ca) (ca)^{- 1}, (bc) (bc)^{- 1}) \\ = φ (c^{- 1} e \cdot c, c^{- 1} ({aa}^{- 1} \cdot c) \\ \land φ (c^{- 1} ({aa}^{- 1} \cdot c), c^{- 1} ({bb}^{- 1} \cdot c)) \\ \geq φ (e, {aa}^{- 1}) \land φ ({aa}^{- 1}, {bb}^{- 1}) . \end{matrix}$ Thus φ_min (a, b) ≤ φ_min (ca, cb) and is compatible. Hence φ_min is a fuzzy congruence relation.□ Theorem 5.7.Let S be a completely inverse AG^**-groupoid. The mapping Θ : C_S ⟶ C_{E
_S}, defined by $Θ (φ) = τ_{φ} (\forall φ \in C_{S})$ is a complete homomorphism of C_S onto C_{E
_S} . If θ is a congruence on C_S induced by Θ and φθ denotes the congruence class of θ containing φ . Then for every φ ∈ C_S, φθ = [φ_min, φ_max] is a modular sublattice of C_S . Proof. Let $F$ be a nonempty family of fuzzy congruences on S . For any $φ \in F$ and e, f ∈ E_S, we have $\begin{matrix} Θ (\cap F) (e, f) & = τ_{(\cap F)} (e, f) = \underset{φ \in F}{⋀} φ (e, f) \\ = & \underset{φ \in F}{⋀} τ_{φ} (e, f) = \cap τ_{φ} (e, f) \\ = & \cap (Θ (φ); φ \in F) (e, f) . \end{matrix}$ Thus $Θ (\cap φ; φ \in F) = \cap (Θ (φ); φ \in F) .$ Since the join of any nonempty family $F$ of fuzzy congruences is $\cup_{n = 1}^{\infty} (\cup F)^{n},$ where $n \in ℕ .$ Thus for e, f ∈ E_S, we have $\begin{matrix} Θ & (⊔ F) (e, f) = τ_{⊔ F} (e, f) \\ = \underset{φ \in F}{⋁} [\underset{φ \in F}{⋁} φ_{1} (e, x_{1}) \land φ_{2} (x_{1}, x_{2}) \\ . . . \land φ_{n} (x_{n - 1}, f)] for some x_{i} \in S, n \in ℕ \\ \leq \underset{φ \in F}{⋁} [\underset{φ \in F}{⋁} φ_{1} (e, x_{1} x_{1}^{- 1}) \land φ_{2} (x_{1} x_{1}, x_{2} x_{2}^{- 1}) \\ . . . \land φ_{n} (x_{n - 1} x_{n - 1}^{- 1}, f)] \\ \leq \underset{φ \in F}{⋁} {(\cup τ_{φ})}^{n} (e, f) \\ = ⊔ Θ (φ) (e, f) . \end{matrix}$ We now show the converse inclusion. So $\begin{matrix} ⊔ (Θ (φ)) (e, f) \\ = \underset{φ \in F}{⋁} [\underset{φ \in F}{⋁} τ_{φ_{1}} (e, e_{1}) \land τ_{φ_{2}} (e_{1}, e_{2}) \\ . . . \land τ_{φ_{n}} (e_{n - 1}, f)] where e_{i} \in E_{S}, n \in ℕ \\ \leq \underset{φ \in F}{⋁} [\underset{φ \in F}{⋁} (\cup F (e, x_{1}) \land \cup F (x_{1}, x_{2})) \\ . . . (\land \cup F (x_{n - 1}, f)]; where x_{i} \in S \\ = \underset{φ \in F}{⋁} {(\cup F)}^{n} (e, f) \\ = Θ (⊔ F) (e, f) . \end{matrix}$ Thus $Θ (⊔ F) = ⊔ (Θ (φ); φ \in F) .$ Consequently, Θ is a complete homomorphism. Further, we show that Θ maps C_S onto C_{E
_S} . Let φ be a fuzzy congruence on E_S . Define a fuzzy relation ϑ on S by $ϑ (a, b) = \underset{e \in E_{S}}{⋀} φ (a^{- 1} e \cdot a, b^{- 1} e \cdot b) (\forall a, b \in S) .$ It is clear that ϑ is a fuzzy congruence. If e, f ∈ E_S, then τ_φ (e, f) ≤ φ (e, f) . Since $\begin{matrix} τ_{ϑ} (e, f) & = ϑ (e, f) = \underset{g \in E_{S}}{⋀} φ (e^{- 1} g \cdot e, f^{- 1} g \cdot f) \\ \leq φ (e, f^{- 1} e \cdot f) \land φ (f^{- 1} e \cdot f, f) \\ \leq φ (e, ef) \land φ (ef, f) \leq φ (e, f) . \end{matrix}$ Also for all e ∈ E_S, φ (e^-1g · e, f^-1g · f) ≥ φ (e, f) . So τ_ϑ (e, f) ≥ φ (e, f) . Thus Θ (ϑ) = φ . Further, we show that φΘ is a modular lattice. If φ ∈ C_S . Then by Theorem 5.4, φ_max is a fuzzy congruence. Now for all e, f ∈ E_S, φ_max (e, f) ≤ φ (e, ef) ∧ φ (ef, f) ≤ φ (e, f) . Again φ_max ≥ φ, because for all e ∈ E_S and a, b ∈ S, φ (a^-1e · a, b^-1e · b) ≥ φ (a, b) . Thus τ_{φ
_max} = τ_φ and φ_max ∈ φθ . Now if σ ∈ C_S such that τ_φ = τ_σ . Then φ_max = σ_max ≥ σ . Hence φ_max is the greatest element of φθ . Let φ be any congruence in C_S . Then by Theorem 5.6, φ_min is a fuzzy congruence relation on S . Now for a, b ∈ S there exists e ∈ E_S such that ea = eb . So $\begin{matrix} φ (a, b) & \geq φ (a, ea) \land φ (ea, b) \\ = φ ({aa}^{- 1} \cdot a, ea) \land φ (eb, {bb}^{- 1} \cdot b) \\ (since ea = eb) \\ \geq φ ({aa}^{- 1}, e) \land φ ({aa}^{- 1}, {bb}^{- 1}) . \end{matrix}$ Thus φ ≥ φ_min . Also for all e, f ∈ E_S, τ_{φ
_min} (e, f) ≥ τ_φ (e, f) . So τ_{φ
_min} = τ_φ . But φ_min depends only on the set of idempotents E_S, thus φ_min ∈ φθ . Consequently φ_min is the least element of φθ . Thus φθ has the least and greatest elements and hence is a complete sublattice of (C_S, ≤ , ∩ , ∘) . Finally, we show the complete lattice φθ = [φ_min, φ_max] is modular. Let φ, ϱ ∈ C_S be such that τ_φ = τ_ϱ . Let a, b, c ∈ S . Then, since τ_φ = τ_ϱ $\begin{matrix} ϱ ({aa}^{- 1}, {bc}^{- 1}) & \geq ϱ ({aa}^{- 1}, {cc}^{- 1}) \land ϱ ({cc}^{- 1}, {bc}^{- 1}) \\ = φ ({aa}^{- 1}, {cc}^{- 1}) \land ϱ ({cc}^{- 1}, {bc}^{- 1}) \\ \geq φ (a, c) \land ϱ (c, b) . \end{matrix}$ So, ϱ (a, bc^-1 · a) ≥ ϱ (aa^-1, bc^-1) ≥ φ (a, c) ∧ ϱ (c, b) . Also, φ (cc^-1, bb^-1) = ϱ (cc^-1, bb^-1) ≥ ϱ (cc^-1, bc^-1) ∧ ϱ (bc^-1, bb^-1) ≥ ϱ (c, b) . Thus $\begin{matrix} φ ({bc}^{- 1} \cdot a, b) & \geq φ ({ac}^{- 1}, {bb}^{- 1}) \geq φ ({ac}^{- 1}, {cc}^{- 1}) \\ \land φ ({cc}^{- 1}, {bb}^{- 1}) \geq φ (a, c) \land ϱ (c, b) . \end{matrix}$ Hence, $\begin{matrix} (φ \circ ϱ) (a, b) & = \underset{c \in S}{⋁} φ (a, c) \land ϱ (c, b) \\ \leq \underset{c \in S}{⋁} ϱ (a, {bc}^{- 1} \cdot a) \land φ ({bc}^{- 1} \cdot a, b) \\ \leq (ϱ \circ φ) (a, b) . \end{matrix}$ Similarly, the converse ϱ ∘ φ ≤ φ ∘ ϱ . Thus ϱ ∘ φ = φ ∘ ϱ . Hence φθ is a complete modular lattice.□ Theorem 5.8.Let S be a competely inverse AG^**-groupoid. If φ, ϱ ∈ C_S, then $φ_{\min} \leq ϱ_{\min} τ_{φ} \leq τ_{ϱ} \Leftrightarrow φ_{\max} \leq ϱ_{\max} .$ Also if φ ≤ ϱ, then φ_min ≤ ϱ_min and φ_max ≤ ϱ_max . Proof. The proof follows from the definition of φ_min and φ_max .□ Theorem 5.9.Let S be a completely inverse AG^**-groupoid. If $F$ is a nonempty subset of C_S . Then (i) . $⊔ (φ_{\min}; φ \in F) = {(⊔ (φ_{\min}); φ \in F)}_{\min}$ (ii) . $⊓ (φ_{\max}; φ \in F) = {(⊓ (φ_{\max}); φ \in F)}_{\max} .$ Proof. The proof is simple. Theorem 5.10.Let S be a completely inverse AG^**-groupoid. If φ ∈ C_S . Then (i) . $κ_{φ_{\max}} (a) = \underset{e \in E_{S}}{⋀} φ (ea, ae) (\forall a \in S) .$ (ii) . κ_{φ
_min} = G, where G : S ⟶ [0, 1] , defined by $G(a) = {\begin{matrix} \underset{e \in S (a)}{⋁} φ (e, {aa}^{- 1}) \\ if S (a) = {e \in E_{S} : ea = e} \neq \emptyset, \\ 0 otherwise . \end{matrix}$ Proof. (i) . Let φ ∈ C_S . Then, φ (aa^-1, e) = φ_max (aa^-1, e) ≥ φ_max (a, e) , because τ_φ = τ_{φ
_max} . Also $\begin{matrix} φ_{\max} (a, e) = & \underset{e \in E_{S}}{⋀} φ (a^{- 1} f \cdot a, e^{- 1} f \cdot e) \\ \leq φ ({aa}^{- 1}, e) \\ = φ ({aa}^{- 1} \cdot a, (e \cdot {aa}^{- 1}) a) \\ \leq φ (a, (e \cdot {aa}^{- 1}) a) . \end{matrix}$ Again $\begin{matrix} φ (ea, a) & \geq φ (ea, (e \cdot {aa}^{- 1}) a) \\ \land φ ((e \cdot {aa}^{- 1}) a, a) \\ \geq φ (e, {aa}^{- 1}) \land φ_{\max} (a, e) \\ = φ_{\max} (a, e) . \end{matrix}$ So, if e, f ∈ E_S, we have $\begin{matrix} φ (af, & fa) = φ (af, f ({aa}^{- 1} \cdot fa)) \\ \geq & φ (af, ae \cdot f) \land φ (ae \cdot f, {aa}^{- 1} \cdot fa) \\ \geq & φ (a, ae) \land φ (fe \cdot a, ({aa}^{- 1} \cdot f) a) \\ \geq & φ (a, ae) \land φ (e, {aa}^{- 1}) \\ \geq & φ_{\max} (a, e) . \end{matrix}$ Thus $\underset{f \in E_{S}}{⋀} φ (af, fa) \geq \underset{e \in E_{S}}{⋁} φ_{\max} (a, e) = κ_{φ_{\max}} (a) .$ Further, if f ∈ E_S, then φ (a^-1f · a, a^-1a · f) ≥ φ (af, fa) ≥ $\underset{f \in E_{S}}{⋀}$ φ (af, fa) . Thus κ_{φ
_max} (a) ≥ φ_max (a, aa^-1) = $\underset{f \in E_{S}}{⋀}$ φ (a^-1f · a, (a^-1a · f) (aa^-1)) ≥ $\underset{f \in E_{S}}{⋀}$ φ (af, fa) . This completes the proof of (i). (ii) . The case κ_{φ
_max} (a) =0, is clear. Let κ_{φ
_max} (a) >0 . Then φ_max (a, e) >0, for some e ∈ E_S . Also, for any f ∈ E_S, if fa = fe, then ef · a = fe . We further have φ (fe, aa^-1) ≥ φ (fe, e) ∧ φ (e, aa^-1) ≥ φ (f, e) ∧ φ (e, aa^-1) . So, $\begin{matrix} φ_{\min} (e, a) & = \underset{f \in S (e, a)}{⋁} φ (f, e) \land φ (e, {aa}^{- 1}) \\ \leq \underset{f \in S (e, a)}{⋁} φ (fe, {aa}^{- 1}) \leq G(a) . \end{matrix}$ Thus κ_{φ
_min} ≤ G . For the converse inclusion, G (a) ≤ κ_{φ
_min} (a) , since G (a) =0 . Let G (a) >0 . Then ea = e for some e ∈ E_S . Now φ (e, aa^-1) = φ (e, e) ∧ φ (e, aa^-1) ≤ φ_min (e, a) . And $G(a) \leq \underset{e \in S (a)}{⋁} κ_{φ_{\min}} (a) .$ Thus G ≤ κ_{φ
_min} . Hence κ_{φ
_min} = G . Theorem 5.11.Let S be a completely inverse AG^**-groupoid. If φ ∈ C_S, then $φ_{e} = κ_{φ} \cap {(φ_{\max})}_{e} = κ_{φ} \cap {((τ_{φ})_{\max})}_{e} (\forall e \in E_{S}) .$ Proof. Let φ ∈ C_S . Then φ ≤ φ_max and φ_e (x) ≤ (φ_max) _e (x) for x ∈ S . Since φ_e (x) = φ (e, x) ≤ κ_φ (x) , it follows that φ_e ≤ κ_φ ∩ (φ_max) _e . For some e, f ∈ E_S, $\begin{matrix} φ ({xx}^{- 1}, ef) & \geq φ ({xx}^{- 1}, x^{- 1} \cdot fx) \\ \land φ (x^{- 1} \cdot fx, ef) \\ = φ ({xx}^{- 1} \cdot {xx}^{- 1}, f \cdot {xx}^{- 1}) \\ \land φ ({xx}^{- 1} \cdot f, ef) \\ \geq φ (x, f) \land φ_{\max} (x, e) . \end{matrix}$ And φ (ef, e) ≥ φ (ef, x^-1 · ex) ∧ φ (x^-1 · ex, e) ≥ φ (f, x) ∧ φ_max (x, e) . Now $\begin{matrix} φ (x, e) & \geq φ ({xx}^{- 1} \cdot x, xe \cdot f) \land φ (xe \cdot f, e) \\ \geq φ ({xx}^{- 1} \cdot x, fe \cdot x) \land φ (xf, fe \cdot f) \\ \land φ (fe \cdot f, e) \geq φ ({xx}^{- 1}, fe) \land φ (x, fe) \\ \land φ (f, e) \geq φ (x, f) \land φ_{\max} (x, e) . \end{matrix}$ Thus $\begin{matrix} φ_{e} (x) & = φ (e, x) \\ \geq \underset{e, f \in E_{S}}{⋁} (φ (x, f) \land φ_{\max} (x, e)) \\ = \underset{e, f \in E_{S}}{⋁} φ (x, f) \land φ_{\max} (e, x) \\ = κ_{φ} (x) \land (φ_{\max})_{e} (x) . \end{matrix}$ That is, φ_e ≥ κ_φ ∩ (φ_max) _e . Hence φ_e = κ_φ ∩ (φ_max) _e . By the definition of τ_φ and φ_e, we have φ_e = κ_φ ∩ ((τ_φ) _max) _e . This completes the proof.

6 Conclusion

Fuzzy set theory which is an extension of the classical set theoretic concept has many applications in the modern mathematical, scientific and technological society. The current study is a first attempt in the line of fuzzy relations and fuzzy congruences in the class of non-associative algebraic structure, a completely inverse AG^**-groupoid. Our results provide a new area of research which indeed will set a new trend for mathematicians working in group theory and generalizations.

Footnotes

Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant No. 11671324).

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