Fuzzy regular relation on hyper hoop-algebras

1 Introduction

Hoop-algebras, or Hoops, are naturally ordered commutative residuated integral monoids, introduced by B. Bosbach in [5] and [6] under the name of complementary semigroups. It was proved that a hoop is a meet-semilattice. Hoops were investigated by Büchi and Owens in an unpublished manuscript [8], and they have been studied by Blok and Ferreirim [3], and Aglianò, and et al. [1] among others. The study of hoops is motivated by their occurrence both in universal algebra and algebraic logic. In recent years, hoop theory was enriched with deep structure theorems. Many of these results have a strong impact with fuzzy logic. Particularly, from the structure theorem of finite basic hoops one obtains an elegant short proof of the completeness theorem for propositional basic logic introduced by Hájek. The algebraic structures corresponding to Hájek’s propositional (fuzzy) basic logic, BL-algebras, are particular cases of hoops. The main example of BL-algebra is the interval [0,1] endowed with the structure induced by a t-norm. MV-algebras, Product algebras and Gödel algebras are the most known classes of BL-algebras. Hypersructure theory was introduced in (1934) [13], when Marty at the 8th congress of scandinavian mathematiciens, gave the definition of hypergroup and illustrated some applications and showed its utility in the study of groups, algebraic functions and rational fraction. Till now, the hyper structures have been studied from the theoretical point of view for their applications to many subject of pure and applied mathematics. Some fields of applications of the mentioned structures are lattices, graphs, coding, ordered sets, median algebra, automata, and cryptography [10]. Then many researchers have worked on this area. Y.B. Jun et al. [12] applied the hyper structures to BCK-algebras and introduced hyper BCK-algebras. Sh. Ghorbani et al. [11] introduced the concept of hyper MV-algebras as a generalization of MV-algebras. M. Bakhshi et al. studied the fuzzy set theory on hyper BCK-algebras and defined the notion of a fuzzy regular congruence relation on a hyper BCK-algebra. The authors [4] applied hyper structure theory on hoop-algebras and introduced the notion of hyper hoop-algebras.

In this paper, we apply the fuzzy set theory on hyper hoop-algebras and introduce the concepts of fuzzy subalgebra and fuzzy filter on hyper hoops and then we mention some related theorems. Then we define the notion of fuzzy regular relation on hyper hoops and construct a quotient hyper hoop with this relation and we get some isomorphism theorems on these quotient hyper hoops.

2 Preliminaries

A hypergroupoid is a pair (H, ⊙), where ⊙ : H × H ⟶ P (H) - {∅} is a binary hyperoperation on H. If a ⊙ (b ⊙ c) = (a ⊙ b) ⊙ c holds, for all a, b, c ∈ H, then (H, ⊙) is called a semihypergroup, and it is said to be commutative if ⊙ is commutative. An element 1 ∈ H is called a unit if a ∈ (1 ⊙ a) ∩ (a ⊙ 1), for all a ∈ H and is called a scaler unit if {a} =1 ⊙ a = a ⊙ 1, for all a ∈ A. Note that if A, B ⊆ H, then A ⊙ B = ⋃ _a∈A,b∈B (a ⊙ b).

Definition 2.1. [4] A hyper hoop-algebra or briefly, a hyper hoop is a non-empty set A endowed with two binary hyperoperations ⊙, → and a constant 1 such that, for all x, y, z ∈ A satisfying the following conditions,

(HHA1) (A, ⊙ , 1) is a commutative semihypergroup with 1 as the unit,

(HHA2) 1 ∈ x → x,

(HHA3) (x → y) ⊙ x = (y → x) ⊙ y,

(HHA4) x → (y → z) = (x ⊙ y) → z,

(HHA5) 1 ∈ x → 1,

(HHA6) if 1  ∈  x  →  y and 1  ∈  y →  x, then x  =  y,

(HHA7) if 1  ∈  x  →  y and 1  ∈  y →  z, then 1∈x→z.

In the sequel we will refer to the hyper hoop (A, ⊙ , → , 1) by its universe A. On hyper hoop A, we define x ≤ y if and only if 1 ∈ x → y. It is easy to see that ≤ is a partial order relation on A. Moreover, for all B, C ⊆ A we define B ⪡ C iff there exist b ∈ B and c ∈ C such that b ≤ c and define B ≤ C iff for any b ∈ B there exists c ∈ C such that b ≤ c. A hyper hoop A is bounded if there is an element 0 ∈ A such that 0 ≤ x, for all x ∈ A. In any hyper hoop A, the following hold, for all x, y, z, t ∈ A and B, C, D ⊆ A,

(HHA8) x ⊙ y ⪡ z ⇔ x ≤ y → z,

(HHA9) B ⊙ C ⪡ D ⇔ B ⪡ C → D,

(HHA10) x ∈ 1 → x

(HHA11) z → y ≤ (y → x) → (z → x) ,

(HHA12) z → y ⪡ (x → z) → (x → y) ,

(HHA13) 1 ⊙ 1 = {1},

(HHA14) y ≤ x → y .

Let F be a non-empty subset of hyper hoop A. If F is a hyper hoop with respect to the hyperoperations ⊙ and → on A, we say that F is a subalgebra of A and F is called a filter of A, if for all x, y ∈ A, x ∈ F and x ≤ y imply that y ∈ F and for all a, b ∈ F, a ⊙ b ⊆ F (See [4]).

Theorem 2.2. [4] Let F be a non-empty subset of A. Then

F is a subalgebra of A iff x ⊙ y ⊆ F and x → y ⊆ F, for all x, y ∈ F,

Note. Let R be an equivalence relation on hyper hoop A and B, C ⊆ A. Denoted the relations R , R ¯ and R ¯ ¯ on A as follows:

B R C if there exist b ∈ B and c ∈ C such that b R c,

Now, equivalence relation R on hyper hoop A is called a regular relation on A if and only if for all x, y, z ∈ A, if x R y, then x ⊙ z R ¯ y ⊙ z, if x R y, then x → z R ¯ y → z and z → x R ¯ z → y and if x → y R  {1} and y → x R  {1}, then x R y. Moreover, R is called a strong regular relation on A if and only if, for all x, y, z ∈ A, if x R y, then x ⊙ z R ¯ ¯ y ⊙ z, if x R y, then x → z R ¯ ¯ y → z and z → x R ¯ ¯ z → y.

Proposition 2.3. [4] Let R be a regular relation on hyper hoop A. If x → yR {1} and y → zR {1}, then x → zR {1}, for any x, y, z ∈ A.

Theorem 2.4. [4] Let R be a regular relation on hyper hoop A and A R be the set of all equivalence classes respect to R, that is A R = { [ x ] | x ∈ A }. Then ( A R , ⊗ , ↪ , [ 1 ] ) is a hyper hoop, which is called the quotient hyper hoop of A respect to R, where for all [ x ] , [ y ] ∈ A R, [ x ] ⊗ [ y ] = { [ t ] | t ∈ x ⊙ y } and [ x ] ↪ [ y ] = { [ z ] | z ∈ x → y }

Theorem 2.5. [4] (First isomorphism theorem) Let A₁ and A₂ be two hyper hoops and R be a regular relation on A₁. If f : A₁ → A₂ is a homomorphism of hyper hoops and ker (f) = [1], then A 1 R ≅ Im ( f ).

Theorem 2.6. [4] Let R be a strong regular relation on A. Then ( A R , ⊗ , ↪ , [ 1 ] ) is a hoop which is called the quotient hoop of A respect to R.

Let U (A) denote the set of all finite combinations of elements of the hyper hoop A with ⊙ and →. Then, for all a, b ∈ A, we have aβb if and only if {a, b} ⊆ u, where u ∈ U (A), and aβ^*b if and only if there exist z₁, . . . , z_m+1 ∈ A with z₁ = a, z_m+1 = b such that {z _i , z_i+1} ⊆ u _i  ⊆ U (A), for i = 1, . . . , m (In fact β^* is the transitive closure of the relation β).

Theorem 2.7. [4] Let A be a hyper hoop. Then β^* is a strong regular relation on A and A β * is a hoop.

Let A be a non-empty set. A fuzzy subset of A is a mapping μ : A → [0, 1]. For t ∈ [0, 1], the set μ _t  = {a ∈ A|μ (a) ≥ t} is called a level subset of μ. Moreover, μ satisfies the sup-property, if for each non-empty subset B of A there exists b ∈ B such that μ (b) = supμ_x∈B (x) (See [14]).

Note. From now on, we let A be a hyper hoop, unless otherwise is stated. We show inf(A) and sup(A) with ⋀A and ⋁A, respectively, for any non-empty set A. Moreover, for all a, b ∈ A, by a ∧ b, we mean min {a, b} and by a ∨ b we mean max {a, b}.

3 Fuzzy filter on hyper hoops

In this section, we introduce the notions of fuzzy subalgebra and fuzzy filter on hyper hoops and investigate some related properties.

Definition 3.1. A fuzzy subset μ of A is called a fuzzy subalgebra of A, if it satisfies the following conditions, for all x, y ∈ A,

⋀μ (x ⊙ y) ≥ μ (x) ∧ μ (y),

Then (A, ⊙ , → , 1) is a hyper hoop. We define fuzzy set μ : A → [0, 1], such that μ (1) = μ (b) =0.8, μ (a) =0.6. Then μ is a fuzzy subalgebra of A.

Proposition 3.3. Let μ be a fuzzy subalgebra of A. Then μ (1) ≥ μ (x), for all x ∈ A.

Proof. By (HHA2), 1 ∈ x → x and so μ (1) ≥ ⋀ μ (x → x) ≥ μ (x) ∧ μ (x) = μ (x), for all x ∈ A.

Theorem 3.4. μ is a fuzzy subalgebra of A if and only if for any α ∈ [0, 1], ∅ ≠ μ _α is a hyper hoop-subalgebra of A.

Proof. The proof is straightforward.

Definition 3.5. A fuzzy set μ of A is called a fuzzy filter of A, if for all x, y ∈ A, μ (1) ≥ μ (x) and μ (y) ≥ μ (x) ∧ (⋁ μ (x → y)).

Then (A, ⊙ , → , 1) is a hyper hoop. We define fuzzy set μ : A → [0, 1], such that μ (1) =0.6, μ (a) =0. Then μ is a fuzzy filter of A.

Then (A, ⊙ , → , 1) is a hyper hoop. We define fuzzy set μ : A → [0, 1], such that μ (1) = μ (b) =0.7, μ (a) =0.5. Then μ is a fuzzy filter of A.

Note.

In Example 3.2, μ is a fuzzy subalgebra of A,but it is not a fuzzy filter of A. Because μ (a) notgeqμ (b) ∧ (∨ μ (b → a)). Hence, every fuzzy subalgebra may not be a fuzzy filter, in general.

Proposition 3.7. Let μ be a fuzzy filter of A. Then for all x, y ∈ A, if x ≤ y then μ (x) ≤ μ (y).

Proof. Let x ≤ y, for x, y ∈ A. Since A is a hyper hoop, we have 1 ∈ x → y and since μ (1) ≥ μ (x), for all x ∈ A, we get ⋁μ (x → y) = μ (1). Then we have, μ ( y ) ≥ μ ( x ) ∧ ( ⋁ μ ( x → y ) ) = μ ( x ) ∧ μ ( 1 ) = μ ( x ) .

Theorem 3.8. Let μ be a fuzzy filter of A. Then for all t ∈ [0, 1], ∅ ≠ μ _t is a filter of A.

Proof. The proof is straightforward.

Theorem 3.9. Let μ be a fuzzy subset of A with the sup-property. If for all t ∈ [0, 1], ∅ ≠ μ _t is a filter of A, then μ is a fuzzy filter of A.

Proof. For all x ∈ A we have x ∈ μ _μ(x) and so μ _μ(x)≠ ∅. Then by assumption, μ _μ(x) is a filter and so 1 ∈ μ _μ(x) and then μ (1) ≥ μ (x). Now, let μ (x) = α₁, ⋁μ (x → y) = α₂, and α = min {α₁, α₂}. Then x ∈ μ _α and since μ satisfies in the sup-property condition, there exists t ∈ x → y such that ⋁μ (x → y) = μ (t) = α₂ ≥ α and so t ∈ μ _α . Then (x→ y) ∩ μ _α  ≠ ∅ and so μ _α  ⪡ x → y. Now, since μ _α is a filter, by Theorem 2.2, we get y ∈ μ _α and so μ (y) ≥ α = μ (x) ∧ (⋁ μ (x → y)).

Corollary 3.10. Let μ be a fuzzy subset of A with the sup-property. Then μ is a fuzzy filter of A if and only if for any α ∈ [0, 1], ∅ ≠ μ _α is a filter of A.

Proposition 3.11. Let μ be a fuzzy filter of A. Then for all x, y ∈ A, ⋀μ (x ⊙ y) ≥ μ (x) ∧ μ (y).

Proof. For x, y ∈ A, let μ (x) = r, μ (y) = s and min {r, s} = t. Then x ∈ μ _t and y ∈ μ _t and so by Theorem 3.8, x ⊙ y ⊆ μ _t . Hence, ⋀μ (x ⊙ y) ≥ t = μ (x) ∧ μ (y).

Proposition 3.12. Let μ be a fuzzy subset of A with the sup-property and satisfies the following conditions;

if x ≤ y then μ (x) ≤ μ (y),

Then μ is a fuzzy filter of A.

Proof. Let t ∈ [0, 1] and μ _t ≠ ∅. For x, y ∈ μ _t , we get μ (x) ≥ t and μ (y) ≥ t. So by assumption ⋀μ (x ⊙ y) ≥ μ (x) ∧ μ (y) ≥ t and then for all u ∈ x ⊙ y we have μ (u) ≥ t and so x ⊙ y ⊆ μ _t .

Now, Let x, y ∈ A, x ≤ y, and x ∈ μ _t . Then by assumption t ≤ μ (x) ≤ μ (y). Then y ∈ μ _t . Hence, μ _t is a filter of A for all t ∈ [0, 1] and then by Theorem 3.9, μ is a fuzzy filter of A.

Theorem 3.13. Let μ be a fuzzy subset of A with the sup property. Then μ is a fuzzy filter of A if and only if, for all x, y ∈ A, the following conditions hold,

if x ≤ y, then μ (x) ≤ μ (y),

Proof. The proof is clear by Propositions 3.7, 3.11 and 3.12.

Theorem 3.14. Let μ be a fuzzy filter on A and π : A → A β * be the canonical epimorphism. Then there exists a fuzzy filter ν on A β *, such that ν ∘ π ≥ μ.

Proof. For any x ∈ A, we consider the class [ x ] β * ∈ A β * by β^* (x). Now, we define ν : A β * → [ 0 , 1 ] by ν (β^* (x)) = ⋁ _t∈β^*(x) μ (t). First, we show that ν is well-defined. Let x, x′ ∈ A and β^* (x) = β^* (x′). Then xβ^*x′ and so ν ( β * ( x ) ) = ⋁ t ∈ β * ( x ) μ ( t ) = ⋁ t ∈ β * ( x ′ ) μ ( t ) = ν ( β * ( x ′ ) ) .

Now, we show that ν is a fuzzy filter of A β *. Let x ∈ A. Then ν ( β * ( 1 ) ) = ⋁ t ∈ β * ( 1 ) μ ( t ) = μ ( 1 ) ≥ ⋁ t ∈ β * ( x ) μ ( t ) = ν ( β * ( x ) ) .

Moreover, let x, y ∈ A. Since μ is a fuzzy filter, for any t ∈ β^* (y) and t′ ∈ β^* (x), μ ( t ) ≥ μ ( t ′ ) ∧ ( ⋁ μ ( t ′ → t ) ) .

Then we get, ν ( β * ( y ) ) = ⋁ t ∈ β * ( y ) μ ( t ) ≥ ⋁ t ∈ β * ( y ) ( μ ( t ′ ) ∧ ⋁ μ ( t ′ → t ) ) ≥ ⋁ t ′ ∈ β * ( x ) μ ( t ′ ) ∧ ⋁ t ′ ∈ β * ( x ) ⋁ t ∈ β * ( y ) μ ( t ′ → t ) ≥ ⋁ t ′ ∈ β * ( x ) μ ( t ′ ) ∧ ⋁ u ∈ β * ( x ) → β * ( y ) μ ( u ) ≥ ν ( β * ( x ) ) ∧ ν ( β * ( x ) → β * ( y ) ) .

Then, by Theorem 2.7, since A β * is a hoop, we get ν ( β * ( y ) ≥ ν ( β * ( x ) ) ∧ ( ⋁ ν ( β * ( x ) → β * ( y ) ) ) .

Hence, ν is a fuzzy filter of A β *. Now, for all x ∈ A, we have, ( ν ∘ π ) ( x ) = ν ( β * ( x ) ) = ⋁ t ∈ β * ( x ) μ ( t ) ≥ μ ( x ) .

4 Fuzzy regular relation on hyper hoop-algebras

In this section we introdece the notion of fuzzy regular relation on hyper hoops and then we construct a quotient hyper hoop.

Definition 4.1. Let ρ be a fuzzy equivalence relation on A. Then ρ is said to be a fuzzy congruence relation on A if for all u ∈ a ⊙ x there exists v ∈ a ⊙ y such that ρ (u, v) ≥ ρ (x, y) and for all u ∈ a → x (u ∈ x → a) there exists v ∈ a → y (v ∈ y → a) and for all v ∈ a → y (v ∈ y → a) there exists u ∈ a → x (u ∈ x → a) such that ρ (u, v) ≥ ρ (x, y), for all x, y, a ∈ A. Now, fuzzy congruence relation ρ on A is called a fuzzy regular relation on A, if ρ ( x , y ) ≥ min ( sup a ∈ x → y ρ ( a , 1 ) , sup b ∈ y → x ρ ( b , 1 ) ) .

Theorem 4.2. If ρ is a fuzzy regular relation on A, then for all t ∈ [0, 1], ρ _t ≠ ∅ is a regular relation on A. Conversely, if fuzzy relation ρ on A satisfies in the sup-property condition and for all t ∈ [0, 1], ρ _t ≠ ∅ is a regular relation on A, then ρ is a fuzzy regular relation on A.

Proof. (⇒) The proof is straightforward.

(←) It is clear that ρ is a fuzzy congruence relation on A. Now, let r = min ( sup a ∈ x → y ρ ( a , 1 ) , sup b ∈ y → x ρ ( b , 1 ) ) .

Since ρ satisfies the sup-property, there exist u ∈ x → y and v ∈ y → x such that, ρ ( u , 1 ) = sup a ∈ x → y ρ ( a , 1 ) ≥ r , and ρ ( v , 1 ) = sup b ∈ y → x ρ ( b , 1 ) ≥ r .

Hence, x → yρ _r  {1} and y → xρ _r  {1}. Since, ρ _r is a regular relation, then xρ _r y and so ρ ( x , y ) ≥ r = min ( sup a ∈ x → y ρ ( a , 1 ) , sup b ∈ y → x ρ ( b , 1 ) ) . Therefore, ρ is a fuzzy regular relation on A.

Note. For any fuzzy relation ρ on A, we define the fuzzy subset μ x ρ on A by, μ x ρ ( y ) = ρ ( y , x ), for all y ∈ A. Moreover, we let ϑ = sup u , w ∈ A ρ ( u , w ).

Proposition 4.3. Let ρ be a fuzzy equivalence relation on hyper hoop A. Then,

for all x, y ∈ A, μ x ρ = μ y ρ if and only if ρ (x, y) = ϑ,s

Proof. (i) Let μ x ρ = μ y ρ, for x, y ∈ A. Since, ρ is a fuzzy reflexive relation, then, ρ ( x , y ) = μ y ρ ( x ) = μ x ρ ( x ) = ρ ( x , x ) = ϑ . Conversely, let ρ (x, y) = ϑ, for x, y ∈ A. Since, ρ is a fuzzy symmetric and fuzzy transitive relation, then for all z ∈ A, μ x ρ ( z ) = ρ ( z , x ) = ρ ( x , z ) ≥ min ( ρ ( x , y ) , ρ ( y , z ) ) = min ( ϑ , ρ ( y , z ) ) = ρ ( y , z ) = ρ ( z , y ) = μ y ρ ( z ) .

By the similar way, we can show that μ y ρ ( z ) ≥ μ x ρ ( z ). Hence, for all z ∈ A, μ x ρ ( z ) = μ y ρ ( z ) and so μ x ρ = μ y ρ. (ii) Let x ∈ [1]  _{ρ t} . Then xρ _t 1 and so μ₁ (x) = ρ (x, 1) ≥ t. Hence, x ∈ (μ₁)  _t . Conversely, if x ∈ (μ₁)  _t then μ₁ (x) ≥ t and so ρ (x, 1) ≥ t. Then xρ _t 1 and so x ∈ [1]  _{ρ t} .

Theorem 4.4. Let ρ be a fuzzy regular relation on A and define, A ρ = { μ x ρ | x ∈ A } .

Then we define hyperoperations ⊗ and ↪ on A ρ as follows, μ x ρ ⊗ μ y ρ = μ x ⊙ y = { μ z ρ | z ∈ x ⊙ y } , μ x ρ ↪ μ y ρ = μ x → y = { μ z ρ | z ∈ x → y } . Then ( A ρ , ⊗ , ↪ , μ 1 ) is a hyper hoop.

Proof. First, we show that hyperoperations ⊗ and ↪ on A ρ are well-defined. Let μ x ρ = μ x ′ ρ, and μ y ρ = μ y ′ ρ, where x, x′, y, y′ ∈ A. Then, by Proposition 4.3 (i), ρ (x, x′) = ρ (y, y′) = ϑ and so xρ _ϑ x′ and yρ _ϑ y′. By Theorem 4.2 ρ _ϑ is a regular relation on A and then x ⊙ y ρ ϑ ¯ x ′ ⊙ y ′. Now, let μ z ρ ∈ μ x ρ ⊗ μ y ρ. Then there exists z′ ∈ x ⊙ y such that μ z ρ = μ z ′ ρ and so zρ _ϑ z′. Since z′ ∈ x ⊙ y and x ⊙ y ρ ϑ ¯ x ′ ⊙ y ′, there exists w ∈ x′ ⊙ y′ such that z′ρ _ϑ w and so since by Theorem 4.2 ρ _ϑ is a transitive relation, we get zρ _ϑ w and so by Proposition 4.3 (i) μ z ρ = μ w ρ. Since, w ∈ x′ ⊙ y′, we get μ z ρ = μ w ρ ∈ μ x ′ ρ ⊗ μ y ′ ρ and so μ x ρ ⊗ μ y ρ ⊆ μ x ′ ρ ⊗ μ y ′ ρ. Similarly, we can show that μ x ′ ρ ⊗ μ y ′ ρ ⊆ μ x ρ ⊗ μ y ρ. Then μ x ρ ⊗ μ y ρ = μ x ′ ρ ⊗ μ y ′ ρ and so hyperoperation ⊗ is well-defined. Similarly, we can show that the hyperoperation ↪ is well-defined.

Now, we prove that ( A ρ , ⊗ , ↪ , μ 1 ) is a hyper hoop. For all x, y, z ∈ A, we have,

(F-HHA1): Let μ v ρ ∈ ( μ x ρ ⊙ μ y ρ ) ⊙ μ z ρ. Then there exist u ∈ x ⊙ y and w ∈ u ⊙ z such that μ v ρ = μ w ρ. We have w ∈ u ⊙ z ⊆ (x ⊙ y) ⊙ z = x ⊙ (y ⊙ z) and so μ w ρ = μ v ρ ∈ μ x ρ ⊗ ( μ y ρ ⊗ μ z ρ ).

Then ( μ x ρ ⊗ μ y ρ ) ⊗ μ z ρ ⊆ μ x ρ ⊗ ( μ y ρ ⊗ μ z ρ ). By the similar way, we can show that μ x ρ ⊗ ( μ y ρ ⊗ μ z ρ ) ⊆ ( μ x ρ ⊗ μ y ρ ) ⊗ μ z ρ. Hence, μ x ρ ⊗ ( μ y ρ ⊗ μ z ρ ) = ( μ x ρ ⊗ μ y ρ ) ⊙ μ z ρ and ⊗ is associative on A ρ. Since ⊙ is commutative on A, we get ⊗ is commutative on A ρ. Since 1 is a unit of hyper hoop A, we get μ₁ is a unit of A ρ.

(F-HHA2): By condition (HHA2) on A, we get (F-HHA2) on A ρ.

(F-HHA3): Let μ v ρ ∈ ( μ x ρ ↪ μ y ρ ) ⊗ μ x ρ. Then there exist u ∈ (x → y) ⊙ x such that μ v ρ = μ u ρ. By (HHA3), u ∈ (x → y) ⊙ x = (y → x) ⊙ y. Then μ v ρ = μ u ρ ∈ ( μ y ρ ↪ μ x ρ ) ⊗ μ y ρ. This implies that ( μ x ρ ↪ μ y ρ ) ⊗ μ x ρ ⊆ ( μ y ρ ↪ μ x ρ ) ⊗ μ y ρ. By the similar way, we can show that ( μ y ρ ↪ μ x ρ ) ⊗ μ y ρ ⊆ ( μ x ρ ↪ μ y ρ ) ⊗ μ x ρ. Therefore, ( μ x ρ ↪ μ y ρ ) ⊗ μ x ρ = ( μ y ρ ↪ μ x ρ ) ⊗ μ y ρ.

(F-HHA4): The proof is similar to the proof of (F-HHA3).

(F-HHA5): By (HHA5), 1 ∈ x → 1, then μ 1 ∈ μ x ρ → μ 1.

(F-HHA6): Let μ 1 ∈ μ x ρ ↪ μ y ρ and μ 1 ∈ μ y ρ ↪ μ x ρ. Then there exist u ∈ x → y and v ∈ y → x such that μ 1 = μ u ρ = μ v ρ. Then by Proposition 4.3, we get ρ (u, 1) = ρ (v, 1) = ϑ. Hence, uρ _ϑ 1 and vρ _ϑ 1. Since u ∈ x → y and v ∈ y → x we get x → yρ _ϑ  {1} and y → xρ _ϑ  {1}. By Theorem 4.2 we get ρ _ϑ is regular and so xρ _ϑ y. Hence, ρ (x, y) = ϑ and then by Proposition 4.3, μ x ρ = μ y ρ.

(F-HHA7): Let μ 1 ∈ μ x ρ ↪ μ y ρ and μ 1 ∈ μ y ρ ↪ μ z ρ. Then there exist u ∈ x → y and v ∈ y → z such that μ 1 = μ u ρ = μ v ρ. Then by Proposition 4.3 we get ρ (u, 1) = ϑ and ρ (v, 1) = ϑ. Hence, uρ _ϑ 1 and vρ _ϑ 1. Since u ∈ x → y and v ∈ y → z we get x → yρ _ϑ  {1} and y → zρ _ϑ  {1}. By Theorem 4.2, ρ _ϑ is a regular relation on A and so by Proposition 2.3, x → zρ _ϑ  {1}. Hence, there exists w ∈ x → z such that wρ _ϑ  {1} and then ρ (w, 1) = ϑ. So by Proposition 4.3 (i), we get μ w ρ = μ 1. Since w ∈ x → z, we get μ 1 = μ w ρ ∈ μ x ρ ↪ μ z ρ.

Therefore, ( A ρ , ⊗ , ↪ , μ 1 ) is a hyper hoop.

Theorem 4.5. Let ρ be a fuzzy regular relation on A. Then there is a regular relation θ on A ρ such that, A ρ θ ≅ A ρ ϑ

Proof. Since ρ is a fuzzy reflexive relation on A, then ρ (1, 1) = ϑ and so (1, 1) ∈ ρ _ϑ . Hence, ρ _ϑ ≠ ∅ and so by Theorem 4.2, ρ _ϑ is a regular relation on A and then by Theorem 2.4, (A ρ ϑ , ⊗ , ↪ , [ 1 ] ρ ϑ ) is a hyper hoop. Now, let φ : A ρ → A ρ ϑ be defined by φ ( μ x ρ ) = [ x ] ρ ϑ. Now, we show that φ is well-defined. Let μ x ρ = μ y ρ, for x, y ∈ A. Then, by Proposition 4.3 (i), ρ (x, y) = ϑ. Hence, xρ _ϑ y and so [x]  _{ρ ϑ}  = [y]  _{ρ ϑ} i.e. φ ( μ x ρ ) = φ ( μ y ρ ). Hence, φ is well-defined. For all μ x ρ , μ y ρ ∈ A ρ, we have, φ ( μ x ρ ⊗ μ y ρ ) = φ { μ z ρ | z ∈ x ⊙ y } = { φ ( μ z ρ ) | z ∈ x ⊙ y } = { [ z ] ρ ϑ | z ∈ x ⊙ y } = [ x ] ρ ϑ ⊗ [ y ] ρ ϑ = φ ( μ x ρ ) ⊗ φ ( μ y ρ ) .

By the similar way, we can show that φ ( μ x ρ ↪ μ y ρ ) = φ ( μ x ρ ) ↪ φ ( μ y ρ ). Moreover, φ ( μ 1 ) = [ 1 ] ρ ϑ = 1 A ρ ϑ. Hence, φ is a homomorphism of hyper hoops. Then we define an equivalence relation θ on A ρ as follows, for all x, y ∈ A, μ x ρ θ μ y ρ ⇔ x ρ ϑ y .

Since, ρ _ϑ is a regular relation on A, we get θ is a regular relation on A ρ. Now, we show that kerφ = [μ₁]  _θ  . μ x ρ ∈ Ker φ ⇔ φ ( μ x ρ ) = [ 1 ] ρ ϑ ⇔ [ x ] ρ ϑ = [ 1 ] ρ ϑ ⇔ x ∈ [ 1 ] ρ ϑ ⇔ x ρ ϑ 1 ⇔ μ x ρ θ μ 1 ⇔ μ x ρ ∈ [ μ 1 ] θ .

It is clear that φ is onto. Hence, by Theorem 2.5, we get A ρ θ ≅ A ρ ϑ.

Theorem 4.6. (First isomorphism theorem) Let ρ be a fuzzy regular relation on A and f : A → A′ be an epimorphism of hyper hoops such that ker ( f ) = { x ∈ A | ρ ( x , 1 ) = ρ ( 1 , 1 ) } . Then A ρ ≅ A ′.

Proof. Let φ : A ρ → A ′ be defined by φ ( μ x ρ ) = f ( x ), for all x ∈ A. First, we show that φ is well-defined. Let μ x ρ = μ y ρ, for x, y ∈ A. Then by Proposition 4.3 (i), ρ (x, y) = ϑ and so xρ _ϑ y. Since, by Theorem 4.2, ρ _ϑ is a regular relation on A, we get x → y ρ ϑ ¯ y → y and x → x ρ ϑ ¯ y → x. Since, 1 ∈ x → x, there exists a ∈ x → y such that aρ _ϑ 1 and so ρ (a, 1) = ϑ. Since, ρ is fuzzy reflexive, ρ (1, 1) = ϑ = ρ (a, 1) and so a ∈ ker (f). Hence, 1_A′ = f (a) ∈ f (x → y) = f (x) → f (y) and so f (x) ≤ f (y). Similarly, we can show that f (y) ≤ f (x). Thus, f (x) = f (y) and so φ is well-defined. Let μ x ρ , μ y ρ ∈ A ρ. Then, φ ( μ x ρ ⊗ μ y ρ ) = φ { μ z ρ | z ∈ x ⊙ y } = { f ( z ) | z ∈ x ⊙ y } = f ( x ⊙ y ) = f ( x ) ⊙ f ( y ) = φ ( μ x ρ ) ⊙ φ ( μ y ρ ) .

Similarly, we can show that φ ( μ x ρ ↪ μ y ρ ) = φ ( μ x ρ ) → φ ( μ y ρ ) and it is clear that φ (μ₁) =1_A′. So φ is a homomorphism of hyper hoops. Since f is onto, we get φ is onto. Now, Let μ z ρ ∈ ker (φ). Then f ( z ) = φ ( μ z ρ ) = 1 A ′ and so z ∈ ker (f) = {x ∈ A|ρ (x, 1) = ρ (1, 1)}. Then ρ (z, 1) = ρ (1, 1) = ϑ. Hence, by Proposition 4.3 (i), μ z ρ = μ 1 and so kerφ = {μ₁}. Hence φ is one to one. Therefore, φ is an isomorphism and so A ρ ≅ A ′.

Lemma 4.7. Let ρ and σ be two fuzzy regular relations on hyper hoop A such that μ y ρ ( x ) = σ ( x , y ), μ₁ (x) = σ (x, 1), for all x, y ∈ A and ρ satisfies the sup property. Then ρ σ is a fuzzy regular relation on A σ, where ρ σ ( μ x ρ , μ y ρ ) = ρ ( x , y ).

Proof. Since, σ is a fuzzy regular relation on A, by Theorem 4.4, A σ is a hyper hoop. Moreover, since ρ is a fuzzy regular relation on A, clearly ρ σ is a fuzzy congruence relation on A. Now, let s = min ( sup μ a ρ ∈ μ x ρ → μ y ρ ρ σ ( μ a ρ , μ 1 ) , sup μ b ρ ∈ μ y ρ → μ x ρ ρ σ ( μ b ρ , μ 1 ) ) .

Since ρ satisfies the sup property, we get ρ σ satisfies the sup property, too. Thus, there exist u ∈ x → y and v ∈ y → x such that, ρ ( u , 1 ) = ρ σ ( μ u ρ , μ 1 ) = sup μ a ρ ∈ μ x ρ → μ y ρ ρ σ ( μ a ρ , μ 1 ) ≥ s and ρ ( v , 1 ) = ρ σ ( μ v ρ , μ 1 ) = sup μ b ρ ∈ μ y ρ → μ x ρ ρ σ ( μ b ρ , μ 1 ) ≥ s .

Since, ρ is a fuzzy regular relation on A, we get, ρ σ ( μ x ρ , μ y ρ ) = ρ ( x , y ) ≥ min ( sup a ∈ x → y ρ ( a , 1 ) , sup b ∈ y → x ρ ( b , 1 ) ) ≥ min ( ρ ( u , 1 ) , ρ ( v , 1 ) ) ≥ s = min ( sup μ a ρ ∈ μ x ρ → μ y ρ ρ σ ( μ a ρ , μ 1 ) , sup μ b ρ ∈ μ y ρ → μ x ρ ρ σ ( μ b ρ , μ 1 ) ) .

Hence, ρ σ is a fuzzy regular relation on A σ.

Theorem 4.8. (Second isomorphism theorem) Let ρ and σ be two fuzzy regular relations on hyper hoop A such that σ ⊆ ρ, ρ satisfies the sup property, and there exists a ∈ A such that σ (a, a) =1. Let fuzzy subsets η _y and μ y ρ on A are defined by η _y  (x) = ρ (x, y) and μ y ρ ( x ) = σ ( x , y ), for all x, y ∈ A and we define A ρ : = { η x | x ∈ A } and A σ : = { μ x ρ | x ∈ A }. Then, A σ ρ σ ≅ A ρ .

Proof. By Lemma 4.7, ρ σ is a fuzzy regular relation on A σ such that ρ σ ( μ x ρ , μ y ρ ) = ρ ( x , y ). Then by Theorem 4.4, A σ ρ σ is a hyper hoop. Now, let φ : A σ → A ρ be defined by φ ( μ x ρ ) = η x, for all x ∈ A. First, We show that φ is well-defined. Let μ x ρ = μ y ρ, for x, y ∈ A. Then, by Proposition 4.3 (i), σ ( x , y ) = sup x , y ∈ A σ ( x , y ) = 1. Since, σ ⊆ ρ, then ρ (x, y) ≥ σ (x, y) =1 and so ρ ( x , y ) = sup z , w ∈ A ρ ( z , w ). Hence, by Proposition 4.3 (i), η _x  = η _y , i.e. φ is well-defined. It is easy to check that φ is an epimorphism. Now, ker φ = { μ x ρ ∈ A σ | η x = φ ( μ x ρ ) = η 1 } = { μ x ρ ∈ A σ | ρ ( x , 1 ) = ρ ( 1 , 1 ) } = { μ x ρ ∈ A σ | ρ σ ( μ x ρ , μ 1 ) = ρ σ ( μ 1 , μ 1 ) } .

Hence, by Theorem 4.6, we get A σ ρ σ ≅ A ρ .