Hybrid ideals in semigroups

Abstract: The notions of hybrid subsemigroups and hybrid left (resp., right) ideals in semigroups are introduced, and several properties are investigated. Using these notions, characterizations of subsemigroups and left (resp., right) ideals are discussed. The concept of hybrid product is also introduced, and characterizations of hybrid subsemigroups and hybrid left (resp., right) ideals are considered by using the notion of hybrid product. Relations between hybrid intersection and hybrid product are displayed.


Introduction
The study of the fuzzy algebraic structures has started with the introduction of the concepts of fuzzy (subgroupoids) subgroups and fuzzy (left, fight) ideals in the pioneering paper of Rosenfeld (1971). Since then, several authors applied fuzzy set theory to semigroups, and Mursaleen, Srivastava, and Sunil (2016) studied certain new spaces of statistically convergent and strongly summable sequences of fuzzy numbers. As a parallel circuit of fuzzy sets and soft sets (or, hesitant fuzzy set), Jun, Song, and Muhiuddin (in press) introduced the notion of hybrid structure in a set of parameters over an initial universe set, and applied it to BCK / BCI-algebras and linear spaces.
As a new mathematical tool for dealing with uncertainties, Molodtsov (1999) introduced the soft set theory. Torra introduced the concept of a hesitant fuzzy set (Torra, 2010;Torra & Narukawa, ABOUT THE AUTHORS Saima Anis and Madad Khan had worked in semigroups and left almost semigroups. Jun introduced the notion of hybrid structure in a set of parameters over an initial universe set. In this paper, we applied hybrid structure to semigroups.

PUBLIC INTEREST STATEMENT
A semigroup is a nonempty set with a binary operation which is associative. An ideal of a semigroup S is a subset A of S such that AS or SA is contain in S. We introduced new kind of ideals in semigroups called as hybrid ideals and have investigated various properties in it. Using these notions, we have considered characterizations of subsemigroups and left (right) ideals. We also have introduced the concept of hybrid product, and have discussed characterizations of hybrid subsemigroups and hybrid left (resp., right) ideals by using the notion of hybrid product. We have provided relations between hybrid intersection and hybrid product.
Using the notions and results in this paper, we will study the hybrid structures in related algebraic structures and decision making problems etc.
2009) which is a generalization of Zadeh's fuzzy set (Zadeh, 1965). The hesitant fuzzy set is very useful to express peoples hesitancy in daily life, and it is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers.
In this paper, we apply the notion of hybrid structure to semigroups. We introduce the notions of hybrid subsemigroups and hybrid left (resp., right) ideals in semigroups, and investigate several properties. Using these notions, we consider characterizations of subsemigroups and left (resp., right) ideals. We also introduce the concept of hybrid product, and discuss characterizations of hybrid subsemigroups and hybrid left (resp., right) ideals by using the notion of hybrid product. We provide relations between hybrid intersection and hybrid product.

Fundamentals on semigroups
Let L be a semigroup. Let A and B be subsets of L. Then the multiplication of A and B is defined as follows: A semigroup L is said to be regular if for every x ∈ L there exists a ∈ L such that xax = x.
• a two-sided ideal of L if it is both a left and a right ideal of L.

Fundamentals on hybrid structures
In what follows, let I be the unit interval, L a set of parameters and (U) denote the power set of an initial universe set U. where f⊆g means that f (x) ⊆g(x) and ⪰ means that (x) ≥ (x) for all x ∈ L. Note that (H(L), ≪) is a poset.

Hybrid subsemigroups and ideals
Definition 3.1 Let L be a semigroup. A hybrid structure f in L over U is called a hybrid subsemigroup of L over U if the following assertions are valid: Example 3.2 Let L = {0, 1, 2, 3, 4, 5} be a semigroup with the following Cayley table: Let f be a hybrid structure in L over U = Z which is given by Table 1.  It is easy to verify that f is a hybrid subsemigroup of L over U = Z.
Also the hybrid structure g in L over U = Z which is given by Table 2 is a hybrid subsemigroup of L over U = Z. Definition 3.3 Let L be a semigroup. A hybrid structure f in L over U is called a hybrid left (resp., right) ideal of L over U if the following assertions are valid: If a hybrid structure f in L over U is both a hybrid left ideal and a hybrid right ideal of L over U, we say that f is a hybrid two-sided ideal of L over U. Then the hybrid structure f in L over an initial universe set U = {u 1 , u 2 , u 3 , u 4 , u 5 } which is given by Table 3 is a hybrid two-sided ideal of L over U.   Table 3. Tabular representation of the hybrid structure f Obviously, every hybrid left (resp., right) ideal is a hybrid subsemigroup, but the converse is not true in general. In fact, the hybrid subsemigroup f in Example 3.2 is not a hybrid left ideal of L over For a nonempty subset A of L and , ∈ (U) with ⊋ , and s, t where and which is called the -characteristic (resp., identity) hybrid structure in L over U with = U, = �, t = 0 and s = 1 is called the characteristic (resp., identity) hybrid structure in L over U, and is denoted by Theorem 3.5 For any nonempty subset A of a semigroup L, the following are equivalent: (i) A is a left (resp., right) ideal of L.
(ii) The characteristic hybrid structure A (f ) in L over U is a hybrid left (resp., right) ideal of L over U.
Proof Assume that A is a left ideal of L. For any x, y ∈ L, if y ∉ A then A (f )(xy) ⊇ � = A (f )(y) and A ( )(xy) ≤ 1 = A ( )(y). If y ∈ A, then xy ∈ A and so A (f )(xy) = U = A (f )(y) and A ( )(xy) = 0 = A ( )(y). Therefore A (f ) is a hybrid left ideal of L over U. Similarly, A (f ) is a hybrid right ideal of L over U when A is a right ideal of L.
Conversely, suppose that A (f ) is a hybrid left ideal of L over U. Let x ∈ L and y ∈ A. Then A (f )(y) = U and A ( )(y) = 0, and so A (f )(xy) ⊇ A (f )(y) = U and A ( )(xy) ≤ 0 = A ( )(y). Hence xy ∈ A and therefore A is a left ideal of L. Similarly, we can show that if A (f ) is a hybrid right ideal of L over U, then A is a right ideal of L. ✷ Corollary 3.6 For any nonempty subset A of a semigroup L, the following are equivalent: (i) A is a two-sided ideal of L.
(ii) The characteristic hybrid structure A (f ) in L over U is a hybrid two-sided ideal of L over U.

Theorem 3.7 A hybrid structure f in L over U is a hybrid subsemigroup of L over U if and only if the nonempty sets
are subsemigroups of L for all ( , t) ∈ (U) × I.
Proof Suppose that a hybrid structure f in L over U is a hybrid subsemigroup of L over U. Assume that L̃f ≠ ∅ ≠ L t for all ( , t) ∈ (U) × I. Let x, y ∈ L̃f ∩ L t . Then f (x) ⊇ , f (y) ⊇ , (x) ≤ t and (y) ≤ t. It follows from (3.1) that Hence xy ∈ L̃f ∩ L t , and so L̃f and L t are subsemigroups of L.
Conversely, assume that the nonempty sets L̃f and L t are subsemigroups of L for all ( , t) ∈ (U) × I For any x, y ∈ L, let f (x) = x and f (y) = y . If we put : = x ∩ y , then x, y ∈ L̃f and so f (xy) ⊇ = x ∩ y =f (x) ∩f (y). Now, for any a, b ∈ L, let (a) = t a and (b) = t b . Taking t: Therefore f is a hybrid subsemigroup of L over U.
✷ Note that f [ , t] = L̃f ∩ L t for all ( , t) ∈ (U) × I. Hence we have the following corollary. Proof It is the same as the proof of Theorem 3.7. ✷ For any hybrid structures f and g in L over U, the hybrid product of f and g is defined to be a hybrid structure f ⊙g = f•g , • in L over U where and for all x ∈ L.
Proof Let x ∈ L. If x is not expressed as x = yz for y, z ∈ L, then clearly Assume that x = yz for some y, z ∈ L. Then (3.4) Therefore f 1 ⊙f 2 ≪g 1 ⊙g 2 . ✷

Lemma 3.11 For subsets A and B of L, let A (f ) and B (f ) be characteristic hybrid structures in L over U. Then
It follows that Hence (i) is valid.
(ii) For any x ∈ L, if x ∈ AB then x = ab for some a ∈ A and b ∈ B. Thus and and so . Suppose x ∉ AB. Then x ≠ ab for all a ∈ A and b ∈ B. If x = yz for some y, z ∈ L, then y ∉ A or z ∉ B. Thus If x ≠ yz for all y, z ∈ L, then . . .

In any case, we have
This completes the proof. ✷ Theorem 3.12 A hybrid structure f in a semigroup L over U is a hybrid subsemigroup of L over U if and only if f ⊙f ≪f .
Proof Assume that f is a hybrid subsemigroup of L over U. Then for all x ∈ L with x = yz. Hence for all x ∈ L. Thus f•f⊆f and • ⪰ , and so f ⊙f ≪f .
Conversely suppose that f ⊙f ≪f and let x, y ∈ L. Then and so f is a hybrid subsemigroup of L over U. ✷ Theorem 3.13 For a hybrid structure g and the identity hybrid structure f in a semigroup L over U, the following assertions are equivalent.
(i) g is a hybrid left ideal of L over U.
Proof Assume that g is a hybrid left ideal of L over U. Let x ∈ L. If x = yz for some y, z ∈ L, then Otherwise, we have f•g (x) = � ⊆g(x) and • (x) = 1 ≥ (x). Hence f•g⊆g and • ⪰ . Therefore f ⊙g ≪g .
Conversely, suppose f ⊙g ≪g . For any x, y ∈ L, we have Consequently, g is a hybrid left ideal of L over U. ✷ Similarly, we have the following theorem. Theorem 3.14 For a hybrid structure g and the identity hybrid structure f in a semigroup L over U, the following assertions are equivalent.
(i) g is a hybrid right ideal of L over U.
Theorem 3.15 If f and g are hybrid subsemigroups of a semigroup L over U, then so is the hybrid intersection f ⋒g .
Proof For any x, y ∈ L, we have and Hence f ⋒g is a hybrid subsemigroup of a semigroup L over U. ✷ By the similar way, we can prove the following theorem.
Theorem 3.16 If f and g are hybrid left (resp., right) ideals of a semigroup L over U, then so is the hybrid intersection f ⋒g .
Theorem 3.17 Let f and g be hybrid structures in a semigroup L over U. If f is a hybrid left ideal of L over U, then so is the hybrid product f ⊙g .
Proof Assume that f is a hybrid left ideal of L over U and let x, y ∈ L. If there exist a, b ∈ L such that y = ab, then xy = x(ab) = (xa)b and If y is not expressible as y = ab for a, b ∈ L, then f•g (y) = � ⊆ f•g (xy) and ( • )(y) = 1 ≥ ( • )(xy). Hence for all x, y ∈ L. Therefore f ⊙g is a hybrid left ideal of L over U. ✷ Similarly, we have the following theorem.
f∩g (xy) =f (xy) ∩g(xy) Proposition 3.22 Let L be a regular semigroup. If f is a hybrid right ideal of L over U, then f ⋒g ≪f ⊙g for every hybrid structure g in L over U.
Proof Let x ∈ L. Then there exists a ∈ L such that xax = x since L is regular. Hence On the other hand, we have since f is a hybrid right ideal of L over U. Since xax = x, we get It follows from (3.5) and (3.6) that f∩g (x) ⊆ f•g (x) and ( ∨ )(x) ≥ ( • )(x) for all x ∈ L. Therefore f ⋒g ≪f ⊙g . ✷ In a similar way, we obtain the following.
Proposition 3.23 Let L be a regular semigroup. If g is a hybrid left ideal of L over U, then f ⋒g ≪f ⊙g for every hybrid structure f in L over U.
Combining Propositions 3.22 and 3.23, we have the following theorem.
Theorem 3.24 If a semigroup L is regular, then f ⋒g =f ⊙g for every hybrid right ideal f and hybrid left ideal g of L over U.

Conclusion
We have introduced the notions of hybrid subsemigroups and hybrid left (resp., right) ideals in semigroups, and have investigated several properties. Using these notions, we have considered characterizations of subsemigroups and left (resp., right) ideals. We also have introduced the concept of hybrid product, and have discussed characterizations of hybrid subsemigroups and hybrid left (resp., right) ideals by using the notion of hybrid product. We have provided relations between hybrid intersection and hybrid product. Using the notions and results in this paper, we will study the hybrid structures in related algebraic structures and decision making problems etc.