On Fuzzy Corsini′s Hyperoperations

Definition 1.1 (see [8].)Let 〈H, R〉 be a a pair of sets where H is a nonempty set and R is a binary relation on H. Corsini′s hyperoperation (briefly, C-hyperoperation) *_R associated with R is defined in the following way:

where P(H) denotes the family of all the subsets of H.

Definition 2.1. Let 〈H, R〉 be a pair of sets where H is a non-empty set and R is a fuzzy relation on H. We define a fuzzy hyperoperation *_R : H × H → F(H), for any x, y, z ∈ H, as follows:

*_R is called a fuzzy Corsini's hyperoperation (briefly, F-C-hyperoperation) associated with R. The fuzzy hyperstructure 〈H, *_R〉 is called a partial F-C-hypergroupoid.

Remark 2.2. It is obvious that the concept of F-C-hyperoperation is a generalization of the concept of C-hyperoperation.

Example 2.3. Letting H = {a, b} be a non-empty set, R is a fuzzy relation on H as described in Table 1.

Definition 2.4. Supposing R, S are two fuzzy relations on a non-empty set H, the composition of R and S is a fuzzy relation on H and is defined by (R∘S)(x, y)≐ ⋁_z∈H(R(x, z)∧S(z, y)), for all x, y ∈ H.

Proposition 2.5. A partial F-C-hypergroupoid 〈H, *_R〉 is a F-C-hypergroupoid if and only if supp (R∘R) = H × H, where supp (R∘R) = {(x, y)∣(R∘R)(x, y) ≠ 0}.

Proof. Suppose that 〈H, *_R〉 is a hypergroupoid. For any x, y ∈ H, there exists at least one z ∈ H, such that (x *_R y)(z) ≠ 0 holds.

So (R∘R)(x, y) = ⋁_z∈H(R(x, z)∧R(z, y)) ≠ 0. Thus (x, y) ∈ supp (R∘R). And we conclude that H × H⊆supp (R∘R).

supp (R∘R)⊆H × H is obvious. And so supp (R∘R) = H × H.

Conversely, if supp (R∘R) = H × H, then for any x, y ∈ H, (x, y) ∈ H × H = supp (R∘R). So (R∘R)(x, y) = ⋁_z∈H(R(x, z)∧R(z, y)) ≠ 0. That is, there exists at least one z ∈ H such that (x *_R y)(z) ≠ 0 holds. And so 〈H, *_R〉 is a hypergroupoid.

Thus we complete the proof.

Definition 2.6. Letting H be a non-empty set, * is a fuzzy hyperoperation of H, the hyperoperation *_p is defined by x *_p y = (x*y)_p, for all x, y ∈ H, p ∈ [0,1]. *_p is called the p-cut of *.

Definition 2.7. Letting R be a fuzzy relation on a non-empty set H, we define a binary relation R_p on H, for all p ∈ (0,1], as follows:

R_p is called the p-cut of the fuzzy relation R.

Proposition 2.8. Let 〈H, *_R〉 be a partial F-C-hypergroupoid. Then $$ is a C-hyperoperation associated with R_p, for all 0 < p ≤ 1.

Proof. For any 0 < p ≤ 1 and for any x, y ∈ H, we have

From the definition of C-hyperoperation, we conclude that (*_R) _p is a C-hyperoperation associated with R_p.

Thus we complete the proof.

Proposition 2.9. Let H be a non-empty set and let * be a fuzzy hyperoperation of H, then the fuzzy hyperoperation * is an F-C-hyperoperation associated with a fuzzy relation R on H if and only if *_p is a C-hyperoperation associated with R_p, for any 0 < p ≤ 1.

Proposition 3.1. Let 〈H, *_R〉 be a partial or nonpartial F-C-hypergroupoid defined on H ≠ ∅. Then, for all x, y, a, b ∈ H, we have

Proof. For any x, y, a, b, z ∈ H, we have that (x *_R y∩a *_R b)(z) = (x *_R y)(z)∧(a *_R b)(z) = R(x, z)∧R(z, y)∧R(a, z)∧R(z, b) = R(x, z)∧R(z, b)∧R(a, z)∧R(z, y) = (x *_R b∩a *_R y)(z).

So

for all x, y, a, b ∈ H.

Proposition 3.2. Let 〈H, *_R〉 be a partial F-C-hypergroupoid and x, y ∈ H, x *_R y = ∅. Then,

• (1)

  x *_R H∩H *_R y = ∅;

• (2)

  If H = x *_R H then H *_R y = ∅;

• (3)

  If H = H *_R x then y *_R H = ∅.

Proof. (1) Supposing x *_R H∩H *_R y ≠ ∅, then there exist a, b ∈ H, such that x *_R a∩b *_R y ≠ ∅. So from the previous proposition, we have x*_Ry∩b*_Ra ≠ ∅. This is a contradiction.

(2) From H = x *_R H and x *_R H∩H *_R y = ∅, we have that H∩H *_R y = ∅, and so, H *_R y = ∅.

(3) is proved similar to (2).

Proposition 3.3. Letting *_R be the F-C-hyperoperation defined on the non-empty set H, p ∈ (0,1], then the following are equivalent:

• (1)

  for some a ∈ H, $$;

• (2)

  for all x, y ∈ H, $$.

Proof. Let $$. Then, for all x, y ∈ H, we have that (a *_R a)(x) ≥ p, (a *_R a)(y) ≥ p, that is R(a, x) ≥ p, R(x, a) ≥ p, R(a, y) ≥ p, R(y, a) ≥ p and so R(x, a)∧R(a, y) ≥ p. Thus $$, for all x, y ∈ H.

Conversely, let a ∈ (x *_R y) _p, for all x, y ∈ H. Specially, we have a ∈ (a *_R x) _p and a ∈ (x *_R a) _p. Thus, R(a, x) ≥ p and R(x, a) ≥ p. And so x ∈ (a*_Ra) _p.

Proposition 3.4. Let 〈H, *_R〉 be a partial or nonpartial F-C-hypergroupoid defined on H ≠ ∅. Then, for all a, b ∈ H, p ∈ (0,1], we have

Proof. For any a, b ∈ H, we have that

The remaining part can be proved similarly.

Definition 4.1. A fuzzy relation R on a non-empty set H is called p-fuzzy reflexive if for any x ∈ H,

Example 4.2. The fuzzy relation R introduced in Example 2.3 is 0.1-fuzzy reflexive. Of course, it is p-fuzzy reflexive, where 0 ≤ p ≤ 0.1.

Proposition 4.3. Letting 〈H, *_R〉 be a partial F-C-hypergroupoid defined on H ≠ ∅, R is p-fuzzy reflexive. Then, for all a, b ∈ H, p ∈ (0,1], the following are equivalent:

• (1)

  R(a, b) ≥ p;

• (2)

  $$;

• (3)

  $$.

Proof. “(1)⇒(2)”

From R(a, a) ≥ p and R(a, b) ≥ p we have that R(a, a)∧R(a, b) ≥ p which shows that $$.

“(2)⇒(3)”

From $$ we have that R(a, b) ≥ p. Since R(b, b) ≥ p, so R(a, b)∧R(b, b) ≥ p which implies that $$.

“(3)⇒(1)”

It is obvious.

Proposition 4.4. Letting 〈H, *_R〉 be a partial F-C-hypergroupoid defined on H ≠ ∅, R is p-fuzzy reflexive. Then, for any a ∈ H, we have that

Proof. From R(a, a) ≥ p we have R(a, a)∧R(a, a) ≥ p. That is $$.

Proposition 4.5. Letting 〈H, *_R〉 be a partial F-C-hypergroupoid defined on H ≠ ∅, R is p-fuzzy reflexive. Then, for any a, b ∈ H, p ∈ (0,1], we have that

Proof. From $$ we have that R(a, b)∧R(b, a) ≥ p. So R(a, b) ≥ p and R(b, a) ≥ p. Thus R(a, a)∧R(a, b) ≥ p and R(b, a)∧R(a, a) ≥ p. That is (a *_R b)(a) ≥ p and (b *_R a)(a) ≥ p. So (a *_R b∩b *_Ra )(a) ≥ p. Thus $$.

Conversely, suppose that $$. Then (a *_R b)(a)∧(b *_R a)(a) ≥ p. Thus R(a, a)∧R(a, b)∧R(b, a)∧R(a, a) ≥ p. So R(a, b)∧R(b, a) ≥ p. That is $$.

Corollary 4.6. Letting 〈H, *_R〉 be a partial F-C-hypergroupoid defined on H ≠ ∅, R is p-fuzzy reflexive. Then, for any a, b ∈ H, p ∈ (0,1], we have that

Proposition 4.7. Letting 〈H, *_R〉 be a partial F-C-hypergroupoid defined on H ≠ ∅, R is p-fuzzy reflexive. Then, for any a, b ∈ H, we have that

Proof. If $$, then R(a, c) ≥ p and R(c, b) ≥ p. Thus $$ and $$. So $$.

Conversely, if $$, then (a *_R c)(c)∧(c *_R b)(c) ≥ p. Thus R(a, c)∧R(c, c)∧R(c, c)∧R(c, b) ≥ p. And so R(a, c)∧R(c, b) ≥ p. Thus $$.

Proposition 4.8. Letting 〈H, *_R〉 be a partial F-C-hypergroupoid defined on H ≠ ∅, R is p-fuzzy reflexive. Then, for any a, b, c ∈ H, p ∈ (0,1], the following are equivalent:

• (1)

  $$;

• (2)

  $$ and $$;

• (3)

  $$ and $$.

Proof. “(1)⇒(2)”

Suppose that $$. Then R(a, c) ≥ p and R(c, b) ≥ p. So R(a, a)∧R(a, c) ≥ p and R(c, b)∧R(b, b) ≥ p. Thus $$ and $$.

“(2)⇒(3)”

Suppose that $$. Then R(c, b) ≥ p. Thus R(c, c)∧R(c, b) ≥ p. And so $$.

“(3)⇒(1)”

From $$ and $$, we have that R(a, c) ≥ p and R(c, b) ≥ p. Thus R(a, c)∧R(c, b) ≥ p. So $$.

Definition 5.1. A fuzzy binary relation R on a non-empty set H is called p-fuzzy symmetric if for any x, y ∈ H,

Example 5.2. The fuzzy relation R introduced in Example 2.3 is 0.2-fuzzy symmetric. Of course, it is p-fuzzy reflexive, where 0 ≤ p ≤ 0.2.

Proposition 5.3. Letting 〈H, *_R〉 be a partial F-C-hypergroupoid defined on H ≠ ∅, R is p-fuzzy symmetric relation. Then, for all a, b ∈ H, we have that

Proof. For all a, b ∈ H, two cases are possible.

• (1)

  If $$, then $$.

• (2)

  If $$, let $$. Then R(a, x) ≥ p and R(x, b) ≥ p.

Since R is p-fuzzy symmetric, so R(x, a) ≥ p and R(b, x) ≥ p. Thus (b *_R a)(x) = R(b, x)∧R(x, a) ≥ p. So $$. And in this case, we also have that $$.

The remaining part can be proved by exchanging a and b.

Proposition 5.4. Let 〈H, *_R〉 be a partial F-C-hypergroupoid defined on H ≠ ∅, p ∈ (0,1], if

• (1)

  for all a, b ∈ H, $$,

• (2)

  for any x ∈ H, there exists a y ∈ H, such that R(x, y) ≥ p.

Then R is a p-fuzzy symmetric binary relation on H.

Proof. For all a, b ∈ H, suppose that R(a, b) ≥ p. We need to show that R(b, a) ≥ p.

Since for b ∈ H, there exists a x ∈ H, such that R(b, x) ≥ p. So R(a, b)∧R(b, x) ≥ p. That is, $$. And so R(x, b)∧R(b, a) ≥ p. And finally we have that R(b, a) ≥ p.

Definition 6.1. A fuzzy binary relation R on a non-empty set H is called p-fuzzy transitive if for any x, y, z ∈ H,

Example 6.2. The fuzzy relation R introduced in Example 2.3 is 0.1-fuzzy transitive. Of course, it is p-fuzzy transitive, where 0 ≤ p ≤ 0.1.

Proposition 6.3. Letting 〈H, *_R〉 be a partial F-C-hypergroupoid defined on H ≠ ∅, R is a p-fuzzy transitive relation on H, p ∈ (0,1]. Then for all x, y ∈ H, we have that

Proof. (1) If $$, then obviously $$.

Supposing that $$, then for any $$, we have that R(x, w)∧R(w, x) ≥ p, that is, R(x, w) ≥ p and R(w, x) ≥ p. From R(w, x) ≥ p and R(x, y) ≥ p we have that R(w, y) ≥ p. From R(x, w) ≥ p and R(w, y) ≥ p we conclude that $$.

So $$.

(2) If $$, then obviously $$.

Supposing that $$, then for any $$, we have that R(y, w)∧R(w, y) ≥ p, that is, R(y, w) ≥ p and R(w, y) ≥ p. From R(y, w) ≥ p and R(x, y) ≥ p we have that R(x, w) ≥ p. From R(x, w) ≥ p and R(w, y) ≥ p we conclude that $$.

So $$.

Proposition 6.4. Letting 〈H, *_R〉 be a partial F-C-hypergroupoid defined on H ≠ ∅, R is a p-fuzzy transitive binary relation. For any a, b, c ∈ H, we have that

• (1)

  $$;

• (2)

  $$.

Proof. (1) If $$, then it is obvious that $$.

Suppose that $$. Then for any $$, there exists a $$ such that $$. That is R(a, w₁) ≥ p, R(w₁, b) ≥ p, R(w₁, w) ≥ p and R(w, c) ≥ p. From R(a, w₁) ≥ p and R(w₁, w) ≥ p, we have that R(a, w) ≥ p. Thus R(a, w)∧R(w, c) ≥ p∧p = p. That is, $$. So $$.

(2) Can be proved similarly.