Hesitant fuzzy semigroups with a frontier

1 Introduction

As a useful generalization of the fuzzy set, Torra [7] introduced the hesitant fuzzy set which permits the membership degree of an element to a set to be represented by a set of possible values between 0 and 1 (see [7, 8]). The hesitant fuzzy set provides a more accurate representation of peoples hesitancy in stating their preferences over objects than the fuzzy set or its classical extensions. Hesitant fuzzy sets are a new approach to handle uncertainty. Many authors are keenly working in this area all over the world. Hesitant fuzzy set theory has been applied to several practical problems, primarily in the area of decision making (see [6, 8–13]). Liao et al. [5] investigated different types of distance and similarity measures for the hesitant fuzzy linguistic term sets. After giving the basic axioms for distance and similarity measures, they developed a family of distance and similarity measures for the hesitant fuzzy linguistic term sets based on the well known Hamming distance, the Euclidean distance, the Hausdorff distance and their generalizations. Jun et al. applied the notion of hesitant fuzzy sets to semigroups, MTL-algebras and EQ-algebras (see [1]–[4]).

In this paper, we introduce the notion of hesitant fuzzy semigroups with a frontier, and investigate several properties. We consider characterizations of a hesitant fuzzy semigroups with a frontier, and provide a condition for a special set to be a subsemigroup. We show that the hesitant intersection of two hesitant fuzzy semigroups with a frontier is a hesitant fuzzy semigroup with a frontier. We discuss the hesitant fuzzy pre-image and hesitant fuzzy image of a hesitant fuzzy semigroup with a frontier under the homomorphism.

2 Preliminaries

Let S be a semigroup. Let A and B be subsets of S. Then the multiplication of A and B is defined as follows: AB = { ab ∈ S ∣ a∈ A and b∈ B } . A nonempty subset A of S is called a subsemigroup of S if AA ⊆ A, that is, ab ∈ A for all a, b ∈ A, Torra [7] defined hesitant fuzzy sets in terms of a function that returns a set of membership values for each element in the domain. We display the basic notions on hesitant fuzzy sets. For more details, we refer to references.

Let S be a reference set. Then we define hesitant fuzzy set on S in terms of a function H that when applied to X returns a subset of [0, 1].

For a hesitant fuzzy set H on S and x, y, z ∈ S, we use the notations H x : = H ( x ), H x y : = H ( x ) ∩ H ( y ), H x ( ɛ ) : = H ( x ) ∩ ɛ and H x y ( ɛ ) : = H ( x ) ∩ H ( y ) ∩ ɛ where ɛ ∈ 𝓅 ([0, 1]). It is clear that H x y = H y x, H x y ( ɛ ) ⊆ H x ( ɛ ) and H x = H y ⇔ H x ⊆ H y , H y ⊆ H x for all x, y ∈ S.

Let H and G be two hesitant fuzzy sets on S. The hesitant union H ⊔ G and hesitant intersection H ⊓ G of H and G are defined to be hesitant fuzzy sets on S as follows: H ⊔ G : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ H x ∪ G x _(1) and H ⊓ G : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ H x ∩ G x ,(2) respectively.

For any hesitant fuzzy sets H and G on S, we define

H ⊑ G if H x ⊆ G x for all x ∈ S.

For a hesitant fuzzy set H on S and a subset ɛ of [0, 1], the set S ( H ; ɛ ) : = { x ∈ S ∣ ɛ ⊆ H x } , is called the hesitant level set of H.

In what follows, we take a semigroup S as a reference set unless otherwise specified.

Definition 3.1. [2] A hesitant fuzzy set H on a semigroup S is called a hesitant fuzzy semigroup on S if H x y ⊆ H xy for all x, y ∈ S.

Definition 3.2. Let ɛ ∈𝓅^* ([0, 1]) : =𝓅 ([0, 1]) \ {∅}. If a hesitant fuzzy set H on S satisfies H x y ( ɛ ) ⊆ H xy for all x, y ∈ S, then we say that ɛ is a frontier of H and H is a hesitant fuzzy semigroup with a frontier ɛ (briefly, ɛ-hesitant fuzzy semigroup) on S.

Obviously, every hesitant fuzzy semigroup is an ɛ-hesitant fuzzy semigroup for all ɛ ∈ 𝓅^* ([0, 1]). Also, if ɛ ∈ 𝓅 ([0, 1]) satisfies H x ⊆ ɛ for all x ∈ S then every ɛ-hesitant fuzzy semigroup H on S is a hesitant fuzzy semigroup on S.

For a hesitant fuzzy set H on S, we know that there exists ɛ ∈ 𝓅^* ([0, 1]) such that H is an ɛ-hesitant fuzzy semigroup, but not a hesitant fuzzy semigroup as seen in the following example.

Example 3.3. Let S = {e, a, b} be a semigroup with the following Cayley table: · e a b e e a b a a b e b b e a Let H be a hesitant fuzzy set on S defined as follows: H : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { [ 0.4 , 0.8 ) if x = e , ( 0.4 , 0.8 ) if x = a , [ 0.6 , 0.8 ) if x = b Then H is an ɛ-hesitant fuzzy semigroup on S with ɛ = [0.6, 0.8). But it is not a hesitant fuzzy semigroup on S since H aa = H b = [ 0.6 , 0.8 ) ⊉ ( 0.4 , 0.8 ) = H a a.

Example 3.4. Let S = {a, b, c, d} be a semigroup with the following Cayley table:

· a b c d a a a c c b b b d d c a a c c d b b d d Let H be a hesitant fuzzy set on S defined as follows: H : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { ( 0 , 1 ] if x = a , { 0.2 , 0.4 , 0.6 , 0.8 , 1 } if x = b { 0.1 , 0.2 , 0.3 , 0.4 , 0.5 } if x = c { 0.2 , 0.8 } if x = d Then H is not a hesitant fuzzy semigroup on S since H bc = H d = { 0.2 , 0.8 } ⊉ { 0.2 , 0.4 } = H b c . But it is an ɛ-hesitant fuzzy semigroup on S with ɛ = {0.2, 0.6, 1}.

Theorem 3.5. Let ɛ ∈ 𝓅^* ([0, 1]) . Then a hesitant fuzzy set H on S is an ɛ-hesitant fuzzy semigroup on S if and only if the set S ( H ; δ ) : = { x ∈ S ∣ δ ⊆ H x } , which is called the hesitant level set of H, is a subsemigroup of S for all δ ∈ 𝓅 ([0, 1]) with S ( H ; δ ) ≠ ∅ and δ ⊆ ɛ.

Proof. Assume that H is an ɛ-hesitant fuzzy subsemigroup on S. Let x , y ∈ S ( H ; δ ) where δ ∈ 𝓅 ([0, 1]) with δ ⊆ ɛ . Then H x ⊇ δ and H y ⊇ δ. It follows that H xy ⊇ H x y ( ɛ ) ⊇ δ ∩ ɛ = δ . Hence xy ∈ S ( H ; δ ), and S ( H ; δ ) is a subsemigroup of S for all δ ∈ 𝓅 ([0, 1]) with S ( H ; δ ) ≠ ∅ and δ ⊆ ɛ.

Conversely, suppose that the nonempty hesitant level set S ( H ; δ ) of H is a subsemigroup of S for all δ ∈ 𝓅 ([0, 1]) with δ ⊆ ɛ. Let x, y ∈ S be such that H x = δ x and H y = δ y . Take δ = ɛ ∩ δ _x  ∩ δ _y . Then x , y ∈ S ( H ; δ ), and so xy ∈ S ( H ; δ ). Hence H xy ⊇ δ = ɛ ∩ δ x ∩ δ y = H x y ( ɛ ) . Therefore H is an ɛ-hesitant fuzzy semigroup on S .hesitant fuzzy ill□

Corollary 3.6. For ɛ ∈ 𝓅^* ([0, 1]) , if a hesitant fuzzy set H on S is a ɛ-hesitant fuzzy semigroup on S, then the nonempty δ ∩ ɛ-hesitant level set S ( H ; δ ∩ ɛ ) of H is a subsemigroup of S for all δ ∈ 𝓅 ([0, 1]).

Proposition 3.7. Let H be an ɛ-hesitant fuzzy semigroup over U . Then ( ∀ x , y ∈ S ) ( H x ⊇ ɛ , H y ⊇ ɛ ⇒ H xy ⊇ ɛ ) .(3)

Proof. Straightforward.hesitant fuzzy ill□

The following example shows that there exists a hesitant fuzzy set H on S such that

The condition (3) is valid.

Example 3.8. Let S = {a, b, c, d} be a semigroup with the following Cayley table: · a b c d a a a a a b a b a b c c c c c d a b c d Let H be a hesitant fuzzy set on S defined as follows: H : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.2 , 0.5 , 1 } if x = a { 0.1 , 0.2 , 0.3 , 0.4 , 0.5 } if x = b { 0.2 , 0.4 , 0.8 } if x = c ( 0 , 1 ] if x = d . If we take ɛ = {0.3, 0.4} , then the condition (3) is valid. But H is not an ɛ-hesitant fuzzy semigroup on S since H bc = H a = { 0.2 , 0.5 , 1 } ⊉ { 0.4 } = H b c ( ɛ ) .

Theorem 3.9. If H is an ɛ-hesitant fuzzy semigroup on S, then the set S a : = { x ∈ S ∣ H x ⊇ H a ( ɛ ) }

is a subsemigroup of S for all a ∈ S .

Proof. Obviously a ∈ S _a for all a ∈ S . Let x, y ∈ S _a . Then H x ⊇ H a ( ɛ ) and H y ⊇ H a ( ɛ ). It follows that H xy ⊇ H x y ( ɛ ) ⊇ H a ( ɛ ). Thus xy ∈ S _a , and S _a is a subsemigroup of S for all a ∈ S.□

The following example shows that there exist a ∈ S and a hesitant fuzzy set H on S such that

The set S _a is a subsemigroup of S.

Example 3.10. Let S = {e, a, b} be a semigroup with the following Cayley table:

· e a b e e e e a e a e b e e b Let H be a hesitant fuzzy set on S defined by H : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.1 , 0.3 , 0.5 } if x = e { 0 , 0.1 , 0.4 } if x = a { 0.3 , 0.4 } if x = b . If ɛ = {0.1, 0.4}, then S _e  = {e, a} is a subsemigroup of S. But H is not an ɛ-hesitant fuzzy semigroup on S since H ab = H e = { 0.1 , 0.3 , 0.5 } ⊉ { 0.4 } = H a b ( ɛ ) .

Theorem 3.11. The hesitant intersection of two ɛ-hesitant fuzzy semigroups on S is an ɛ-hesitant fuzzy semigroup on S.

Proof. Let H and G be ɛ-hesitant fuzzy semigroups on S. For any x, y ∈ S, we have ( H ⊓ G ) xy = H xy ∩ G xy ⊇ ɛ ∩ H xy ∩ G xy ⊇ ɛ ∩ ( H x y ( ɛ ) ∩ G x y ( ɛ ) ) = ɛ ∩ H x y ∩ G x y = ɛ ∩ ( H x ∩ G x ) ∩ ( H y ∩ G y ) = ( H ⊓ G ) x y ( ɛ ) . Therefore H ⊓ G is an ɛ-hesitant fuzzy semigroup on S.□

The following example shows that the hesitant union of two ɛ-hesitant fuzzy semigroup over U is not an ɛ-hesitant fuzzy semigroup on S in general.

Example 3.12. Let S = {a, b, c} be a semigroup with the following Cayley table: · a b c a a a a b a b a c c c c Let H and G be hesitant fuzzy sets on S defined as follows: H : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.6 } if x = a , { 0.2 , 0.7 } if x = b , { 0.1 , 0.3 , 0.5 } if x = c , and G : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.2 , 0.6 , 1 } if x = a , { 0.3 , 0.6 } if x = b , { 0.7 , 0.9 } if x = c , respectively. Then the hesitant union H ⊔ G of H and G is given by H ⊔ G : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.2 , 0.6 , 1 } if x = a , { 0.2 , 0.3 , 0.6 , 0.7 } if x = b , { 0.1 , 0.3 , 0.5 , 0.7 , 0.9 } if x = c . If we take ɛ = {0.7, 1}, then H and G are ɛ-hesitant fuzzy semigroups on S, but the hesitant union H ⊔ G is not an ɛ-hesitant fuzzy semigroup on S since ( H ⊔ G ) bc = ( H ⊔ G ) a = { 0.2 , 0.6 , 1 } ⊉ { 0.7 } = ( H ⊔ G ) b c ( ɛ ) .

Theorem 3.13. For a subset A of S and ɛ ∈ 𝓅^* ([0, 1]), define a hesitant fuzzy set [ H A ] on S as follows: [ H A ] : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { ɛ ∩ ϱ if x ∈ A , τ otherwise where ϱ, τ ∈ 𝓅 ([0, 1]) with τ ⊊ ɛ ∩ ϱ . Then [ H A ] is an ɛ-hesitant fuzzy semigroup on S if and only if A is a subsemigroup of S.

Proof. Assume that [ H A ] is an ɛ-hesitant fuzzy semigroup on S. Let x, y ∈ A . Then [ H A ] xy ⊇ [ H A ] x y ( ɛ ) = ɛ ∩ ϱ , and so xy ∈ A . Thus A is a subsemigroup of S .

Conversely, suppose that A is a subsemigroup of S . Let x, y ∈ S . If x, y ∈ A, then xy ∈ A . Hence [ H A ] xy = ɛ ∩ ϱ = [ H A ] x y . If x ∉ A or y ∉ A, then [ H A ] x = τ or [ H A ] y = τ . Hence [ H A ] xy ⊇ τ = [ H A ] x y . Therefore [ H A ] is a hesitant fuzzy semigroup on S, and so an ɛ-hesitant fuzzy semigroup on S.□

The following example illustrate Theorem 3.13.

Example 3.14. Let S = {a, b, c} be a semigroup with the following Cayley table: · a b c a a a a b a b a c c c c Note that A = {a, b} is a subsemigroup of S . If we take ɛ = {0.1, 0.2, 0.6, 1} , ϱ = {0.2, 0.4, 0.6, 0.8, 1} and τ = {0.2, 1}, then the hesitant fuzzy set [ H A ] on S which is given by [ H A ] : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.2 , 0.6 , 1 } if x ∈ { a , b } , { 0.2 , 1 } if x = c , is an ɛ-hesitant fuzzy semigroup on S .

For any semigroups S and T, let μ : S → T be a function and H and G be hesitant fuzzy sets on S and T, respectively. The hesitant fuzzy pre-image of G under μ is denoted by μ - 1 G and is defined to be the hesitant fuzzy set on T given by μ - 1 ( G ) x = G μ ( x ). The hesitant fuzzy image of H under μ is denoted by μ H and is defined to be the hesitant fuzzy set on S given by μ ( H ) y = { ⋃ x ∈ μ - 1 ( y ) H x if μ - 1 ( y ) ≠ ∅ , ∅ otherwise .

Theorem 3.15. Let μ : S → T be a homomorphism of semigroups and G a hesitant fuzzy set on T . If G is an ɛ-hesitant fuzzy semigroup on T, then the hesitant fuzzy pre-image μ - 1 G of G under μ is an ɛ-hesitant fuzzy semigroup on S .

Proof. For any x₁, x₂ ∈ S, we have μ - 1 ( G ) x 1 x 2 = G μ ( x 1 x 2 ) = G μ ( x 1 ) μ ( x 2 ) ⊇ G μ ( x 1 ) μ ( x 2 ) ( ɛ ) = μ - 1 ( G ) x 1 x 2 ( ɛ ) Hence μ - 1 G is an ɛ-hesitant fuzzy semigroup on S .□

Theorem 3.16. Let μ : S → T be a homomorphism of semigroups and H a hesitant fuzzy set on S . If H is an ɛ-hesitant fuzzy semigroup on S, then the hesitant fuzzy image μ H of H under μ is an ɛ-hesitant fuzzy semigroup on T .

Proof. Let y₁, y₂ ∈ T . If at least one of μ^-1 (y₁) and μ^-1 (y₂) is empty, then the inclusion μ ( H ) y 1 y 2 ( ɛ ) ⊆ μ ( H ) y 1 y 2 is clear. Assume that μ^-1 (y₁)≠ ∅ and μ^-1 (y₂) ≠ ∅ . Then μ^-1 (y₁y₂)≠ ∅, and so μ ( H ) y 1 y 2 = ⋃ x ∈ μ - 1 ( y 1 y 2 ) H x ⊇ ∪ x 1 ∈ μ - 1 ( y 1 ) x 2 ∈ μ - 1 ( y 2 ) ⋃ H x 1 x 2 ⊇ ∪ x 1 ∈ μ - 1 ( y 1 ) x 2 ∈ μ - 1 ( y 2 ) ⋃ H x 1 x 2 ( ɛ ) = μ ( H ) y 1 y 2 ( ɛ ) . Therefore μ H is an ɛ-hesitant fuzzy semigroup on T.□

Theorem 3.15 is illustrated by the following example.

Example 3.17. Consider two semigroups S = {a₁, a₂, a₃} and T = {b₁, b₂, b₃, b₄} with the following Cayley tables, respectively. · a 1 a 2 a 3 a 1 a 1 a 1 a 1 a 2 a 1 a 2 a 3 a 3 a 3 a 3 a 3 · b 1 b 2 b 3 b 4 b 1 b 1 b 1 b 1 b 1 b 2 b 1 b 2 b 2 b 4 b 3 b 1 b 3 b 3 b 4 b 4 b 4 b 4 b 4 b 4 The function μ : S → T , x ↦ { b 1 if x = a 1 , b 3 if x = a 2 , b 4 if x = a 3 , is a homomorphism. Let G be a hesitant fuzzy set on T which is given by G : T → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.4 , 0.8 } if x = b 1 , { 0.2 , 0.4 , 0.7 } if x = b 2 , { 0.1 , 0.7 , 1 } if x = b 3 , [ 0 , 1 ] if x = b 4 . Then G is an ɛ-hesitant fuzzy semigroup on T with ɛ = {0.4, 0.7, 1}, and the hesitant fuzzy pre-image μ - 1 G of G under μ is described as follows: μ - 1 ( G ) : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.4 , 0.8 } if x = a 1 , { 0.1 , 0.7 , 1 } if x = a 2 , [ 0 , 1 ] if x = a 3 , and it is an ɛ-hesitant fuzzy semigroup on S with ɛ = {0.4, 0.7, 1}.

The following example shows that the converse of Theorem 3.15 may not be true.

Example 3.18. Consider two semigroups S = {a₁, a₂, a₃} and T = {b₁, b₂, b₃, b₄} with the following Cayley tables, respectively.

· a 1 a 2 a 3 a 1 a 1 a 1 a 1 a 2 a 2 a 2 a 2 a 3 a 3 a 3 a 3 · b 1 b 2 b 3 b 4 b 1 b 2 b 2 b 2 b 3 b 2 b 2 b 2 b 2 b 2 b 3 b 3 b 3 b 3 b 3 b 4 b 4 b 4 b 4 b 4

The function μ : S → T , x ↦ { b 2 if x = a 1 , b 3 if x = a 2 , b 4 if x = a 3 , is a homomorphism. Let G be a hesitant fuzzy set on T which is given by G : T → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.2 , 0.7 , 1 } if x = b 1 , { 0.2 , 0.5 } if x = b 2 , ( 0 , 1 ] if x = b 3 , { 0.3 , 0.5 , 0.7 , 1 } if x = b 4 . Then G is not an ɛ-hesitant fuzzy semigroup on T with ɛ = {0.2, 0.5, 1} since G b 1 b 1 = G b 2 = { 0.2 , 0.5 } ⊉ { 0.2 , 1 } = G b 1 b 1 ( ɛ ) . Now the hesitant fuzzy pre-image μ - 1 G of G under μ is described as follows: μ - 1 ( G ) : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.2 , 0.5 } if x = a 1 , ( 0 , 1 ] if x = a 2 , { 0.3 , 0.5 , 0.7 , 1 } if x = a 3 , and it is an ɛ-hesitant fuzzy semigroup on S with ɛ = {0.2, 0.5, 1}.

Theorem 3.16 is illustrated by the following example.

Example 3.19. Consider two semigroups S = {a₁, a₂, a₃} and T = {b₁, b₂, b₃} with the following Cayley tables, respectively: · a 1 a 2 a 3 a 1 a 2 a 2 a 2 a 2 a 2 a 2 a 2 a 3 a 2 a 2 a 3 · b 1 b 2 b 3 b 1 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 3 b 3 b 3 b 3 The function μ : S → T , x ↦ { b 1 if x = a 1 , b 2 if x ∈ { a 2 , a 3 } , is a homomorphism. Let H be a hesitant fuzzy set on S which is given by H : S → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.7 } if x = a 1 , { 0.2 , 0.7 } if x = a 2 , { 0.1 , 0.2 , 0.8 , 1 } if x = a 3 . Then H is an ɛ-hesitant fuzzy semigroup on S with ɛ = {0.7, 1} . Now the hesitant fuzzy image μ H of H under μ is described as follows: μ ( H ) : T → 𝓅 ( [ 0 , 1 ] ) , x ↦ { { 0.7 } if y = b 1 , { 0.1 , 0.2 , 0.7 , 0.8 , 1 } if y = b 2 , ∅ if y = b 3 , and it is an ɛ-hesitant fuzzy semigroup on S with ɛ = {0.7, 1} .

We have introduced the notion of hesitant fuzzy semigroups with a frontier, and investigated several properties. We have considered characterizations of a hesitant fuzzy semigroups with a frontier, and provided a condition for a special set to be a subsemigroup. We have shown that the hesitant intersection of two hesitant fuzzy semigroups with a frontier is a hesitant fuzzy semigroup with a frontier. We have provided an example to show that the hesitant union of two hesitant fuzzy semigroups with a frontier may not be a hesitant fuzzy semigroup with a frontier. We have discussed the hesitant fuzzy pre-image and hesitant fuzzy image of a hesitant fuzzy semigroup with a frontier under the homomorphism, and displayed examples.

In the future, we hope that our proposed method and results can be applied in some fields such as hesitant soft and rough set theory in algebraic structures, decision making, pattern recognition, and image processing.