Fuzzy hyperrings and fundamental relation

1 Introduction

The study of fuzzy hyperstructures is an interesting research topic of fuzzy sets. A hyperoperation assigns to every pair of elements of H a non-empty subset of H, while a fuzzy hyperoperation assigns to every pair of elements of H a nonzero fuzzy set on H. This idea was introduced by Corsini and Tofan [5] and then studied by Kehagias, for the interesting properties obtained in connections with an important hyperstructure, called join space. Using the fuzzy hypersemigroup notion, introduced in [12], Leoreanu and Davvaz defined and studid the fuzzy hyperring notion and connections with hyperrings. This approach followed by some researchers and they extended it to fuzzy hypermodules. The fundamental relations are one of the most important and interesting concepts in fuzzy hyperstructures that ordinary algebraic structures are derived from fuzzy hyperstructures by them. Fundamental relation α^* on fuzzy hypersemigroups is studied in [1]. This relation is the smallest equivalence relation on a given fuzzy hypersemigroup such that it’s quotient structure is a semigroup. Now in this paper after review some definitions and basic results about fuzzy hyperstructures in next section, we follow the results obtained by Leoreanu and Davvaz about fuzzy hyperrings (see [9]) to introduce and study the fundamental relation of fuzzy hyperrings. We will proceed to introduce and study the basic properties of complete parts of fuzzy hyperrings and investigate relationship between complete parts and fundamental relation of fuzzy hyperrings. Then the commutative fundamental relation is introduced such that its quotient structure is a commutative ring. Finally some necessary and sufficient conditions such that the commutative fundamental relation be transitive, are determined.

2 Preliminary

First of all, we recall some definitions and theorems from [2, 12], that we need them in our paper.

Definition 2.1. Let S be a non-empty set and F^* (S) denotes the set of all nonzero fuzzy subset of S. A fuzzy hyperoperation on S is a map ∘ : S × S ⟶ F^* (S), which associates a nonzero fuzzy subset a ∘ b or ab to any pair (a, b) of elements of S. The couple (S, ∘) is called a fuzzy hypergroupoid.

Definition 2.2. A fuzzy hypergropoid (S, ∘) is called a fuzzy hypersemigroup if for all a, b, c ∈ S, (a ∘ b) ∘ c = a ∘ (b ∘ c), where for any fuzzy subset μ of S ( μ ∘ a ) ( r ) = { ⋁ t ∈ S ( μ ( t ) ∧ ( t ∘ a ) ( r ) ) , if μ ≠ 0 0 otherwise ( a ∘ μ ) ( r ) = { ⋁ t ∈ S ( ( a ∘ t ) ( r ) ∧ μ ( t ) ) , if μ ≠ 0 0 , otherwise for all r ∈ S.

Let μ, ν be two fuzzy subsets of a fuzzy hypergropoid (S, ∘), then we define μ ∘ ν by ( μ ∘ ν ) ( t ) = ⋁ p , q ∈ S ( μ ( p ) ∧ ( p ∘ q ) ( t ) ∧ ν ( q ) ) , for all t ∈ S.

Definition 2.3. A fuzzy hypersemigroup (S, ∘) is called a fuzzy hypergroup if x ∘ S = S ∘ x = χ _S , for all x ∈ S.

Definition 2.4. Let R be a non-empty set and ⊕, ⊙ be two fuzzy hyperoperations on R. The triple (R, ⊕ , ⊙) is called a fuzzy hyperring if following axioms hold:

a ⊕ (b ⊕ c) = (a ⊕ b) ⊕ c for all a, b, c ∈ R;

The fuzzy hyperring (R, ⊕ , ⊙) is called commutative if for all x, y ∈ R, x ⊙ y = y ⊙ x .

Definition 2.5. Let ρ be an equivalence relation on a fuzzy hypersemigroup (S, ∘) and μ, ν two fuzzy subsets on (S, ∘). If μ (a) >0 implies that there exists b ∈ S such that ν (b) >0 and aρb and also if ν (x) >0 implies that there exists y ∈ S, such that μ (y) >0 and xρy, then we say that μ ρ ¯ ν . If for all x ∈ S such that μ (x) >0 and for all y ∈ S such that ν (y) >0, xρy we say that μ ρ ¯ ¯ ν .

Definition 2.6. An equivalence relation ρ on a fuzzy hypersemigroup (S, ∘) is said to be a fuzzy (strongly) regular relation on (S, ∘) if aρb, a′ρb′ implies ( a ∘ a ′ ) ρ ¯ ( b ∘ b ′ ) ( ( a ∘ a ′ ) ρ ¯ ¯ ( b ∘ b ′ ) ).

Definition 2.7. An equivalence relation ρ on a fuzzy hyperring (R, ⊕ , ⊙) is called fuzzy regular (strongly regular) relation on (R, ⊕ , ⊙) if it is a fuzzy regular (strongly regular) relation on both (R, ⊕) and (R, ⊙).

Let R be a fuzzy hyperring. For every map f : R →  R′ where R′ is a ring, we define f ( a ⊕ b ) = { f ( t ) : ( a ⊕ b ) ( t ) > 0 } and f ( a ⊙ b ) = { f ( t ) : ( a ⊙ b ) ( t ) > 0 } .

Theorem 2.8. Let (R, ⊕ , ⊙) be a fuzzy hyperring and ρ be an equivalence relation on R, then we consider the following hyperoperations on the quotient set R/ρ as follows: for every aρ, bρ ∈ R/ρ a ρ ⊕ ˆ b ρ = { c ρ : ( a ′ ⊕ b ′ ) ( c ) > 0 , a ρ a ′ , b ρ b ′ } , a ρ ⊙ ˆ b ρ = { c ρ : ( a ′ ⊙ b ′ ) ( c ) > 0 , a ρ a ′ , b ρ b ′ } Then

the relation ρ is a fuzzy regular relation on (R, ⊕ , ⊙) iff ( R / ρ , ⊕ ˆ , ⊙ ˆ ) be a hyperring,

Proof. Straightforward.

3 Fundamental relation

Definition 3.1. Let (R, ⊕ , ⊙) be a fuzzy hyperring. A fundamental relation on (R, ⊕ , ⊙) is the smallest equivalence relation ρ on R such that the quotient structure ( R / ρ , oplus ˆ , ⨀ ˆ ) is a ring.

R/ρ is called fundamental ring.

If we denote the set of all finite hyrersums of finite hyperproducts of elements of R by R, then the relation γ^* is the transitive closure of the relation γ, defind as follows: x γ y ⇔ ∃ u ∈ R : u ( x ) > 0 , u ( y ) > 0 .

We can consider this relation by the following Definition:

Definition 3.2. Let (R, ⊕ , ⊙) be a fuzzy hyperring. We define the relation γ on R in the following way: aγb if and only if ∃ n ∈ ℕ , ∃ z i 1 , . . . , z ik i ∈ R k i , i = 1 , . . . , n : ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( a ) > 0 , ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( b ) > 0 .

It is clear that γ is symetric. Define for any a ∈ R, a (a) = (χ _a ) (a) =1, thus the relation γ is also reflexive. We take γ^* the transitive closure of γ, then γ^* is an equivalence relation on R.

Theorem 3.3. Let (R, ⊕ , ⊙) be a fuzzy hyperring. Then γ^* is the fundamental relation on R .

Proof. we define ⊕ ˆ and ⊙ ˆ on R/γ^* in the usual manner: γ * ( a ) ⊕ ˆ γ * ( b ) = { γ * ( c ) ∣ ( a ′ ⊕ b ′ ) ( c ) > 0 , a γ * a ′ , b γ * b ′ } γ * ( a ) ⊙ ˆ γ * ( b ) = { γ * ( d ) ∣ ( a ′ ⊙ b ′ ) ( d ) > 0 , a γ * a ′ , b γ * b ′ } let a′γ^*a and b′γ^*b. Then we have:

a′γ^*a ⇔ ∃ a₁, . . . , a_p+1 ∈ R, a₁ = a′, a_p+1 = a such that a _r γa_r+1 (r = 1, . . . , p)

b′γ^*b ⇔ ∃ b₁, . . . , b_q+1 ∈ R, b₁ = b′, b_q+1 = b such that b _s γb_s+1 (s = 1, . . . , q)

and so a r γ a r + 1 ⇔ ∃ n r ∈ ℕ , ∃ ( k r 1 , . . . , k rn r ) ∈ ℕ n r , ∃ ( x ri 1 , . . . , x rik ri ) ∈ R k ri ( i = 1 , . . . , n r ) such that

( ∑ i = 1 n r ∏ j = 1 k ri x rij ) ( a r ) > 0 { and { ( ∑ i = 1 n r ∏ j = 1 k ri x rij ) ( a r + 1 ) > 0 and

b s γ b s + 1 ⇔ ∃ m s ∈ ℕ , ∃ ( t s 1 , . . . , t sm s ) ∈ ℕ m s , ∃ ( y si 1 , . . . , y sit si ) ∈ R t si ( i = 1 , . . . , m s ) such that

( ∑ i = 1 m s ∏ j = 1 t si y sij ) ( b s ) > 0 { and { ( ∑ i = 1 m s ∏ j = 1 t si y sij ) ( b s + 1 ) > 0 therefore, we obtain ( ∑ i = 1 n r ∏ j = 1 k ri x rij + ∑ i = 1 m 1 ∏ j = 1 t 1 i y 1 ij ) ( a r + b 1 ) > 0 ( ∑ i = 1 n r ∏ j = 1 k ri x rij + ∑ i = 1 m 1 ∏ j = 1 t 1 i y 1 ij ) ( a r + 1 + b 1 ) > 0 , and ( ∑ i = 1 n r ∏ j = 1 k ri x rij + ∑ i = 1 m 1 ∏ j = 1 t 1 i y 1 ij ) ( a p + 1 + b s ) > 0 , ( ∑ i = 1 n r ∏ j = 1 k ri x rij + ∑ i = 1 m 1 ∏ j = 1 t 1 i y 1 ij ) ( a p + 1 + b s + 1 ) > 0 .

Now, we choose the elements c₁, . . . , c_p+q such that ( a r ⊕ b 1 ) ( c r ) > 0 ( r = 1 , . . . , p ) , ( a p + 1 ⊕ b s + 1 ) ( c p + s ) > 0 ( s = 1 , . . . , q ) , using the above in clusions we get c _r γc_r+1. Therefore, (a₁ ⊕ b₁) (c₁) > 0 and (a_p+1 ⊕ b_q+1) (c_p+q) > 0 and (a ⊕ b) (c_p+q) > 0 thus γ * ( a ) ⊕ ˆ γ * ( b ) = γ * ( c ) , ∀ c ∈ R ; ( a ′ ⊕ b ′ ) ( c ) > 0 , a γ * a ′ , b γ * b ′ .

In a similar way, it is proved that γ * ( a ) ⊙ ˆ γ * ( b ) = γ * ( d ) , ∀ d ∈ R ; ( a ′ ⊙ b ′ ) ( d ) > 0 , a γ * a ′ , b γ * b ′ .

The associativity and distributivity on R guarantee that the associativity and distributivity are valid for R/γ^*. Let θ be an equivalence relation such that R/θ is ring and let φ : R ⟶ R/θ be the canonical projection. If xγy then there exist n ∈ ℕ , ( k 1 , . . . , k n ) ∈ ℕ n and (x _{ik 1} , . . . , x _{ik i} ) ∈ R ^{k _i}  (i = 1, . . . , n) such that, ( ⊕ ∑ i = 1 n ⊙ ∏ j = 1 k i x ij ) ( x ) > 0 and ( ⊕ ∑ i = 1 n ⊙ ∏ j = 1 k i x ij ) ( y ) > 0 ; since θ is fuzzy regular relation we have xθy and since θ is transivitive, we obtain x ∈ γ * ( y ) ⇒ x ∈ θ ( y ) { therefore γ * ⊆ θ .

Definition 3.4. Let M be a non-empty subset of a fuzzy hyperring R. We say that M is a fuzzy complete part of R if the following implication holds:

∀ n ∈ ℕ , ∀ i = 1 , . . . , n , k i ∈ ℕ , ∀ ( z i 1 , . . . , z ik ) ∈ R k i, if ∃a ∈ M such that ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( a ) > 0 then ∀ y ∈ R \ M : ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( y ) = 0.

Example 3.5. Let ρ be a fuzzy strongly regular relation on a fuzzy hyperring (R, ⊕ , ⊙), z ∈ R and ρ (z) be the class of z modulo ρ. Then ρ (z) is fuzzy complete part.

Definition 3.6. Let M be a non-empty subset of fuzzy hyperring (R, ⊕ , ⊙). The intersection of all fuzzy complete parts of R and contain M is called the fuzzy complete closure of M in R, it will be denoted by ∂ (M).

Theorem 3.7. Let M be a non-empty subset of a fuzzy hyperring (R, ⊕ , ⊙) and [ z ] k 1 , . . . , k n n = { ( z i 1 , . . . , z i k i ) ∈ R k i ∣ i = 1 , . . . , n } ,

K₁ (M) = M,

K n + 1 ( M ) = { Z ∈ R ∣ ∃ ( z i 1 , . . . , z i k i ) ∈ [ z ] k 1 , . . . , k n n , ∃ y ∈ K n ( M ) : ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( z ) > 0 , ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( y ) > 0 }

If K (M) = ⋃ _n≥1K _n  (M), then ∂ (M) = K (M).

Proof. We prove the following:

K (M) is a fuzzy complete part,

For (i), if ∃x ∈ K (M) such that ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( x ) > 0, Then ∃ p ∈ ℕ, such that x ∈ K _p  (M) and ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( x ) > 0. Now if ∃y ∈ R \ K (M) such that ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( y ) > 0 then y ∈ K_p+1 (M). It contradicts with y ∈ R \ K (M).

For (ii), since M = K₁ (M) then N ⊃ K₁ (M). Suppose N ⊃ K _n  (M) and we prove N ⊃ K_n+1 (M). If z ∈ K_n+1 (M), therefore exists ( z i 1 , . . . , z i k i ) ∈ [ z ] k 1 , . . . , k n n such that ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( z ) > 0 and ∃k ∈ R such that k ∈ K _n  (M) and ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( k ) > 0. Since N ⊃ K _n  (M) thus k ∈ N and since N is fuzzy complete part, therefore ∀ y ∈ R \ N : ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( y ) = 0, thus ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( z ) > 0, implies that z ∈ N .

Lemma 3.8. Let R be a fuzzy hyperring.

∀n ≥ 2, ∀ x ∈ R, K _n  (K₂ (x)) = K_n+1 (x) ,

Proof.

(1) K 2 ( K 2 ( x ) ) = { z ∣ ∃ ( z i 1 , . . . , z i k i ) ∈ [ z ] k 1 , . . . , k n p , ∃ y ∈ K 2 ( x ) : ( ∑ i = 1 p ∏ j = 1 k i z ij ) ( z ) > 0 , ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( y ) > 0 } = K 3 ( x ).

We now proceed by induction, suppose that K_n-1 (K₂ (x)) = K _n  (x), then

K n ( K 2 ( x ) ) = { z ∣ ∃ ( z i 1 , . . . , z i k i ) ∈ [ z ] k 1 , . . . , k n p , ∃ t ∈ K n - 1 ( K 2 ( x ) ) s . t ( ∑ i = 1 p ∏ j = 1 k i z ij ) ( z ) > 0 , ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( t ) > 0 } = { z ∣ ∃ ( z i 1 , . . . , z i k i ) ∈ [ z ] k 1 , . . . , k n p , ∃ t ∈ K n ( ( x ) ) s . t ( ∑ i = 1 p ∏ j = 1 k i z ij ) ( z ) > 0, ( ∑ i = 1 p ∏ j = 1 k i z ij ) ( t ) > 0 } = k n + 1 ( x ) .

(2) We also prove this by induction. It is clear that x ∈ K₂ (y) ⇔ y ∈ K₂ (x). Suppose x ∈ K_n-1 (y) ⇔ y ∈ K_n-1 (x). Let x ∈ K _n  (y), then ∃ ( z i 1 , . . . , z i k i ) ∈ [ z ] k 1 , . . . , k n p , ∃ s ∈ K n - 1 ( y ) such that ( ∑ i = 1 p ∏ j = 1 k i z ij ) ( x ) > 0 , ( ∑ i = 1 p ∏ j = 1 k i z ij ) ( s ) > 0 thus s ∈ K₂ (x).

From s ∈ K_n-1 (y) we have y ∈ K_n-1 (K₂ (x)) = K _n  (x) .

Theorem 3.9. The relation xKy ⇔ x ∈ ∂ ({y}) is an equivalence.

Proof. K is clearly reflexive. Now let xKy and yKz . Since y ∈ ∂ ({z}) then ∂ ({y}) ⊂ ∂ ({z}), therefore x ∈ ∂ ({z}) and xKz. The simmetricity of K follows in a direct way from the previous lemma.

In the next theorem, we see the relationship between fundamental relation and fuzzy complete parts in fuzzy hyperrings.

Theorem 3.10. For every (x, y) ∈ R², We have   xKy ⇔ xγ^*y.

Proof. From xγy, ∃ ( z i 1 , . . . , z i k i ) ∈ [ z ] k 1 , . . . , k n p such that ( ∑ i = 1 p ∏ j = 1 k i z ij ) ( x ) > 0 , ( ∑ i = 1 p ∏ j = 1 k i z ij ) ( y ) > 0 thus x ∈ K₂ (y), therefore x ∈ K (y), from which γ ⊂ K, and since K is a equivalence relation thus γ^* ⊂ K. On the converse if xKy, n ∈ ℕ exists such thatx ∈ K_n+1 (y), therefore exists ( z i 1 , . . . , z i k i ) ∈ [ z ] k 1 , . . . , k n p such that ( ∑ i = 1 p ∏ j = 1 k i z ij ) ( x ) > 0 and exists x₁ ∈ K _n  (y) such that ( ∑ i = 1 p ∏ j = 1 k i z ij ) ( x 1 ) > 0 thus xγx₁, x₁ ∈ K _n  (y). So as a consequence ∃x₂, . . . , x _n  ∈ R such that x _i  ∈ K_n-i+1 (y), x_i-1γx _i and i = 2, . . . , n, now we have x _n  ∈ k₁ (y) = y and thus x _n  = y and xγx₁γ . . . γx _n  = y. Therefore K ⊂ γ^* .

4 Commutative fundamental relation

Some times we need that the fundamental ring be commutative with respect to both sum and product, that is, the fundamental ring is an ordinary commutative ring. Thus the following definition is presented:

Definition 4.1. Let (R, ⊕ , ⊙) be a fuzzy hyperring. We define the relation α on R in the following way: aαb if and only if

∃ n ∈ ℕ , ∃ ( k 1 , . . . , k n ) ∈ ℕ n , ∃ σ ∈ S n and ∃ (z_i1, . . . , z _{ik i} ) ∈ R ^{k _i} , ∃ σ _i  ∈ S _{k i}  (i = 1, . . . , n) such that ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( a ) > 0 , ( ∑ i = 1 n A σ ( i ) ) ( b ) > 0 where A i = ∏ j = 1 k i z i σ i ( j ).

It is clear that α is symetric. Define for any a ∈ R, a (a) = (χ _a ) (a) =1, thus the relation α is also reflexive. We take α^* to be the transitive closure of α, then α^* is an equivalence relation on R.

Lemma 4.2. The relation α^* is a fuzzy strongly regular relation on both (R, ⊕) and (R, ⊙).

Proof. It is enough to show that x α y ⇒ { x ⊕ a α ¯ ¯ y ⊕ a , a ⊕ x α ¯ ¯ a ⊕ y x ⊙ a α ¯ ¯ y ⊙ a , a ⊙ x α ¯ ¯ a ⊙ y for every a ∈ R. If xαy, then ∃ n ∈ ℕ , ∃ ( k 1 , . . . , k n ) ∈ ℕ n , ∃ σ ∈ S n and ∃ (z_i1, . . . , z _{ik i} ) ∈ R ^{k _i} , ∃ σ _i  ∈ S _{k i}  (i = 1, . . . , n) such that ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( x ) > 0 , ( ∑ i = 1 n ∏ j = 1 k σ ( i ) z σ ( i ) σ σ ( i ) ( j ) ) ( y ) > 0 and for every t, s ∈ R such that (x ⊕ a) (t) >0 and (y ⊕ a) (s) >0 ( ( ∑ i = 1 n ∏ j = 1 k i z ij ) ⊕ a ) ( t ) > 0 , ( ( ∑ i = 1 n ∏ j = 1 k σ ( i ) z σ ( i ) σ σ ( i ) ( j ) ) ⊕ a ) ( s ) > 0 .

Now, let k_n+1 = 1, z_n+11 = a, σ_n+1 = id and τ ∈ S_n+1 such that τ ( i ) = { σ ( i ) , i = 1 , . . . , n , n + 1 , i = n + 1 .

Hence for every t, s ∈ R such that (x ⊕ a) (t) >0 and (y ⊕ a) (s) >0 ( ( ∑ i = 1 n + 1 ∏ j = 1 k i z ij ) ) ( t ) > 0 , ( ( ∑ i = 1 n + 1 ∏ j = 1 k τ ( i ) z σ ( i ) σ σ ( i ) ( j ) ) ) ( s ) > 0 . Therefore for all t, s ∈ R such that (x ⊕ a) (t) >0 and (y ⊕ a) (s) >0, tαs. This means that x ⊕ a α ¯ ¯ y ⊕ a. In the similar way we see a ⊕ x α ¯ ¯ a ⊕ y. Since α^* is transitive, it is routine to show that x ⊕ a α * ¯ ¯ y ⊕ a and a ⊕ x α * ¯ ¯ a ⊕ y.

Now, for all t, s ∈ R such that (x ⊙ a) (t) >0 and (y ⊙ a) (s) >0, ( ( ∑ i = 1 n ∏ j = 1 k i z ij ) ⊙ a ) ( t ) > 0 , ( ( ∑ i = 1 n ∏ j = 1 k σ ( i ) z σ ( i ) σ σ ( i ) ( j ) ) ⊙ a ) ( s ) > 0 .

Thus ( ∑ i = 1 n ( ∏ j = 1 k i z ij ⊙ a ) ) ( t ) > 0 , ( ∑ i = 1 n ( ∏ j = 1 k σ ( i ) z σ ( i ) σ σ ( i ) ( j ) ⊙ a ) ) ( s ) > 0 .

Now, let k i ′ = k i + 1 , z ik i ′ = a, and define τ i ( r ) = { σ i ( r ) , r = 1 , . . . , k i , k i + 1 , r = k i + 1 .

Therefore τ i ∈ S k i ′ ( i = 1 , . . . , n ) , and for all t, s ∈ R such that (x ⊙ a) (t) >0 and (y ⊙ a) (s) >0, ( ∑ i = 1 n ( ∏ j = 1 k i ′ z ij ) ) ( t ) > 0 , ( ∑ i = 1 n ( ∏ j = 1 k σ ( i ) ′ z σ ( i ) τ σ ( i ) ( j ) ) ) ( s ) > 0 .

Hence, for all t, s ∈ R such that (x ⊙ a) (t) >0 and (y ⊙ a) (s) >0, tαs. This means that x ⊙ a α * ¯ ¯ y ⊙ a , and a ⊙ x α * ¯ ¯ a ⊙ y.

Theorem 4.3. The quotient R/α^* is a commutative ring.

Proof. By Theorem 2.8 and Lemma 4.2, R/α^* with two operations ⊕ ˆ and ⊙ ˆ α * ( a ) ⊕ ˆ α * ( b ) = α * ( c ) : ( a ′ ⊕ b ′ ) ( c ) > 0 , a α * a ′ , b α * b ′ α * ( a ) ⊙ ˆ α * ( b ) = α * ( c ) : ( a ′ ⊙ b ′ ) ( c ) > 0 , a α * a ′ , b α * b ′ is a ring. It is enough to show that the operation ⊙ ˆ is commutative. Suppose that σ is the permutation of S₂ such that σ (1) =2. For every a, b ∈ R such that (x₁ ⊙ x₂) (a) >0 and (x_σ(1) ⊙ x_σ(2)) (b) >0, we see aαb . Thus aα^*b and α * ( x 1 ) ⊙ ˆ α * ( x 2 ) = α * ( a ) = α * ( b ) = α * ( x 2 ) ⊙ ˆ α * ( x 1 )

Therefore R/α^* is a commutative ring.

In the following we will determine some necessary and sufficient conditions so that the relation α is transitive.

Definition 4.4. Let R is a fuzzy hyperring and M be a non-empty subset of R. We say M is a α- part of R, if for ( ∀ n ∈ ℕ , i = 1 , . . . , n , ∀ k i ∈ ℕ, (z_i1, z _{i 2} , . . . , z _{ik i} ) ∈ R ^{k _i} , ∀ σ ∈ S _n , ∀ σ _i  ∈ S _{k i} ), there exists z ∈ M such that ( ∑ i = 1 n ∏ j = 1 k i z ij ) ( z ) > 0, then for every w ∈ R \ M we have ( ∑ i = 1 n A σ ( i ) ) ( w ) = 0, where A i = ∏ j = 1 k i z i σ i ( j ).

Lemma 4.5. Let R is a fuzzy hyperring and M be a non-empty subset of R. The following conditions are equivalent:

M is a α- part of R,

Proof. 1 → 2) Let (x, y) ∈ R² be a pair such that x ∈ M and xαy, then there exists n ∈ ℕ , ∃ ( k 1 , . . . , k n ) ∈ ℕ n , ∃ σ ∈ S n, ∃ (z _{i 1} , . . . , z _{ik i} ) ∈ R ^{k _i} and ∃σ _i  ∈ S _{k i} , (i = 1, . . . , n) such that ( ∑ i = 1 n ( ∏ j = 1 k i z ij ) ) ( x ) > 0 and ( ∑ i = 1 n A σ ( i ) ) ( y ) > 0, where A i = ∏ j = 1 k i z i σ i ( j ). Since M is a α- part of R, ( ∑ i = 1 n A σ ( i ) ) ( y ) > 0 implies that y ∈ M.

2 → 3) Let (x, y) ∈ R² be a pair such that x ∈ M and xα^*y, then there exist m ∈ ℕ and (u₀ = x, u₁, . . . , u _m  = y) ∈ R^m+1 such that xαu₁γu₂α . . . αu_m-1αy. Since x ∈ M, by applying (2) m times, we see y ∈ M.

3 → 1) Let x ∈ M and (z_i1, . . . , z _{ik i} ) ∈ R ^{k _i} such that( ∑ i = 1 n ( ∏ j = 1 k i z ij ) ) ( x ) > 0. Then for every σ ∈ S _n and every σ _i  ∈ S _{k i}  (i = 1, . . . , n), and every y ∈ R such that ( ∑ i = 1 n A σ ( i ) ) ( y ) > 0, where A i = ∏ j = 1 k i z i σ i ( j ), we have xαy, therefore xα^*y and x ∈ M. By (3) we obtain y ∈ M. Whence M is a α- part of R.

For next results, first we consider the following notions: Let R be a fuzzy hyperring and x ∈ R, we set;

[ x ] k 1 , . . . , k n n = { ( x i 1 , x i 2 , . . . , x ik i ) ∈ R k i ∣ i = 1 , . . . , n }

T n ( x ) = { ( x i 1 , x i 2 , . . . , x ik i ) ∈ [ x ] k 1 , . . . , k n n ∣ ( ∑ i = 1 n ( ∏ j = 1 k i x ij ) ) ( x ) > 0 }

P n ( x ) = ⋁ { ∑ i = 1 n ( ∏ j = 1 k i x σ ( i ) σ σ ( i ) ( j ) ) ∣ σ ∈ S n , σ i ∈ S k i , ( x i 1 , x i 2 , . . . , x ik i ) ∈ [ x ] k 1 , . . . , k n n ∈ T n ( x ) }

P _σ  (x) = ⋁ _n≥1P _n  (x)

From the above notations and definitions we have following results:

Lemma 4.6. For every x, y ∈ H, x α y ⇔ P σ ( x ) ( y ) > 0

Proof. x α y ⇔ ( ∃ n ∈ ℕ * , ∃ ( k 1 , . . . , k n ) ∈ ℕ n , ∃ σ ∈ S n [ ∃ ( z i 1 , . . . , z ik i ) ∈ R k i , ∃ σ i ∈ S k i ( i = 1 , . . . , n ) ] ) : ( ∑ i = 1 n ( ∏ j = 1 k i z ij ) ) ( x ) > 0 , ( ∑ i = 1 n A σ ( i ) ) ( y ) > 0, where A i = ∏ j = 1 k i z i σ i ( j ) . ⇔ ∃ n ∈ ℕ * , P n ( x ) ( y ) > 0 ⇔ P σ ( x ) ( y ) > 0

Theorem 4.7. If R be a fuzzy hyperring, then the following conditions are equivalent:

α is transitive,

Proof. 1 → 2) For every (x, y) ∈ R², by previous Lemma we have: y ∈ α * ( x ) ⇔ x α * y ⇔ x α y ⇔ P σ ( x ) ( y ) > 0 . 2 → 3) By Lemma 4.5, if φ ≠ M ⊆ R, then M is a α- part of R if and only if M is the union of equivalence classes modulo α^*. Particularly, α^* (x) is a α- part. 3 → 1) If xαy and yαz, then there exist n , m ∈ ℕ *, ( x i 1 , . . . , x ik i ) ∈ [ x ] k 1 , . . . , k n n ∈ T n ( x ), ( y i 1 , . . . , y ik i ′ ) ∈ [ y ] k 1 ′ , . . . , k n ′ m ∈ T m ( y ), ∃σ ∈ S _n and ∃τ ∈ S _m , ∃ σ _i  ∈ S _{k i} , (i = 1, . . . , n) and ∃ τ ∈ S k i ′ , ( i = 1 , . . . , m ) such that ( ∑ i = 1 n ( ∏ j = 1 k i x ij ) ) ( x ) > 0 , ( ∑ i = 1 n A σ ( i ) ) ( y ) > 0, where A i = ∏ j = 1 k i x i σ i ( j ) and ( ∑ i = 1 m ( ∏ j = 1 k i ′ y ij ) ) ( y ) > 0 , ( ∑ i = 1 m B τ ( i ) ) ( z ) > 0, where B i = ∏ j = 1 k i ′ y i τ i ( j ). Since {y ∈ R : P _σ  (x) (y) >0} is a α- part and P _σ  (x) (x) >0, ( ∑ i = 1 n ( ∏ j = 1 k i x ij ) ) ( x ) > 0 implies that P _σ  (x) (y) >0. We have ( ∑ i = 1 m ( ∏ j = 1 k i ′ y ij ) ) ( y ) > 0 , and {y ∈ S : P _σ  (x) (y) >0} is a α- part, therefore ( ∑ i = 1 m B τ ( i ) ) ( z ) > 0 implies that P _σ  (x) (z) >0. So there exists k ∈ ℕ such that P _k  (x) (z) >0, this means that zαx.

Remark 4.8. If R is a commutative fuzzy hyperring, then the relation γ is equal to the relation α.