Bipolar Fuzzy Almost Quasi-Ideals in Semigroups

. The aim of paper, we give the concept bipolar fuzzy, almost quasi-ideal in semigroups. We present the properties of bipolar fuzzy, almost quasi-ideals in semigroups. Moreover, we prove the relationship between almost quasi-ideals and bipolar fuzzy quasi-ideals in semigroups


Introduction
The theory fuzzy sets are a kind of proper mathematical structure to represent a collection of objects whose boundary is vague, which was studied by Zaden in 1965 [10].In 1994 Zhang [11] extended the concept of the fuzzy set to bipolar fuzzy sets, which is an extension of fuzzy sets whose membership degree range is [−1, 0] ∪ [0, 1].A bipolar fuzzy set is the membership degree of an element means that the element is irrelevant to the related property, the membership degree of an element indicates that the element somewhat satisfies the property, and the membership degree of an element indicates that the element somewhat satisfies the implicit counter-property.The ideals, introduced by her are still central concepts in ring theory, and the notion of a one-sided ideal of any algebraic structure is a generalization of the notion of an ideal.The almost ideal theory in semigroups was studied by Grosek and Satko in 1980 [2].In 1981, Bogdanvic, [3] established definitions of almost bi-ideals in semigroups and studied properties of almost bi-ideals in semigroups.
Later, Chinram gives definition definitions of the types of almost ideals in semigroups such that almost quasi-ideal [9], almost i-ideal, (m, n)-almost ideal.In 2019, Murugadas et al. [7] studied fuzzy almost quasi-ideals in semigroup.They proved the basic properties of almost quasi-ideals and fuzzy almost quasi-ideals in semigroups.In 2020, Chinram et al. [4] introduced almost interior and weakly almost interior ideals in semigroups and studied the properties of its.Recently N. Sarasit et al. [8] studied almost ternary subsemirings.
In this paper, we define bipolar fuzzy almost quasi-ideals in semigroup, which it is developed to study on bipolar fuzzy sets.We investigate the fundamental properties of bipolar fuzzy almost quasi-ideals in semigroups.

Preliminaries
In this section we give the concepts and results, which will be helpful in later sections.A . By an ideal of a semigroup E, we mean a non-empty set of E which is both a left and a right ideal of Theorem 2.1.[7] Every quasi-ideal of a semigroup E is an almost quasi-ideal of E.

For any
We see that for any h, r ∈ [0, 1], we have h ∨ r = max{h, r} and h ∧ r = min{h, r}.
A fuzzy set (fuzzy subset) of a non-empty set E is a function For any two fuzzy sets ϑ and ξ of a non-empty set E, define the symbol as follows: (1 Definition 2.1.[6] A bipolar fuzzy set (BF set) ϑ on a non-empty set E is an object having the form Remark 2.1.For the sake of simplicity we shall use the symbol ϑ = (E; ϑ p , ϑ n ) for the BF set ϑ = {(h, ϑ p (h), ϑ n (h)) | h ∈ E}.
The following example of a BF set.
Definition 2.2.[6] A non-empty set K of a semigroup E. A positive characteristic function and a negative characteristic function are respectively defined by and For the sake of simplicity we shall use the symbol Lemma 2.1.[5] Let K and M be non-empty subsets of a semigroup E. Then the following holds. (1) The following example of a BF subsemigroup.
Example 2.2.Let E be a semigroup defined by the following table:

Main Results
In this section, we give define the bipolar fuzzy almost quasi-ideal in semigroups and we investigate properties of bipolar fuzzy almost qausi-ideal in semigroups.
Theorem 3.3.Let K be a nonempty subset of a semigroup E. Then K is an almost quasi-ideal of E if and Proof.Suppose that K is an almost quasi-ideal of a semigroup E. Then (Ks ∩ sK) ∩ K ∅ for all s ∈ E. Thus there exists r ∈ E such that c ∈ (Ks ∩ sK) ∩ K. Let x ∈ E and t ∈ (0, 1] and s ∈ [−1, 0).Then (x So ϑ p (z) 0 and ϑ n (z) 0, there exists w 1 , w 2 ∈ E such that z = w 1 w 2 and ϑ p (w 1 ) 0, ϑ p (w 2 ) 0 and ϑ n (w 1 ) 0, ϑ p (w 2 ) 0. Thus, (x Therefore, ≥ s ∈ E, we have (x Then there exists w 1 , w 2 ∈ supp(ϑ) such that c = w 1 w 2 .Thus ϑ p (w 1 ) 0, ϑ p (w 2 ) 0 and ϑ n (w 1 ) 0, Next, we investigate minimal BF almost quasi-ideals in semigroups and study relationships between minimal almost quasi-ideals and minimal BF almost quasi-ideals of semigroups.
Proof.Assume that K is a minimal almost quasi-ideal of E. Then K is an almost quasi-ideal of E.
Thus by Theorem 3.3, We give definition of prime (resp., semiprime, strongly prime) almost quasi-ideal and prime (resp., semiprime strongly prime) BF almost quasi-ideal.We study the relationships between prime (resp., semiprime strongly prime) almost quasi-ideals and their bioplar fuzzification of semigroups Definition 3.4.Let K be an almost quasi-ideal of semigroup E. Then we called (1) K is a prime if for any almost quasi-ideals M and L of E such that ML ⊆ K implies that M ⊆ K or L ⊆ K.
(2) K is a semiprime if for any almost quasi-ideal M of E such that M 2 ⊆ K implies that M ⊆ K.
(3) K is a strongly prime if for any almost quasi-ideals M and L of E such that ML ∩ LM ⊆ K implies that M ⊆ K or L ⊆ K. Definition 3.5.A BF almost quasi-ideal ϑ = (E; ϑ p , ϑ n ) on a semigroup E. Then we called (1) ϑ is a prime if for any two BF almost quasi-ideals ξ = (E; ξ p , ξ n) and ν = (E; ν p , ν n ) of E such taht ξ p • ν p leqϑ p and ξ n • ν n ≥ ϑ n implies that ξ p ≤ ϑ p and ξ n ≥ ϑ n or ν p ≤ ϑ p and ν n ≥ ϑ n .
(3) ϑ is a strongly prime if for any two BF almost quasi-ideals ξ = (E; ξ p , ξ n ) and ν = (E; Proof.Suppose that K is a prime almost quasi-ideal of E. Then K is an almost quasi-ideal of E.

Conclusion
In this paper, we give the concept of BF almost quasi-ideals in semigroup and we study properties of BF almost quasi-deal in semigroups.Moreover, we prove relationship between BF almost quasiideals and almost quasi-ideals.In the future we extend study other kinds of almost quasi-ideals and interval valued fuzzy set or class of kinds fuzzy sets.

p
supp(ϑ) is a BF almost quasi-ideal of E. By Theorem 3.3, supp(ϑ) is an almost quasi-ideal of E. Conversely, suppose that supp(ϑ) is an almost quasi-ideal of E. By Theorem 3.3, ≥ p supp(ϑ) is a BF almost quasi-ideal of E. Then for any BF points x p t , x n

Definition 3 . 2 .Definition 3 . 3 .Theorem 3 . 5 .
An almost quasi-ideal K of a semigroup E is said minimal if for any almost quasi-ideal M of E if whenever M ⊆ K, then M = K.A BF almost quasi-ideal ϑ = (E; ϑ p , ϑ n ) of a semigroup E is said minimal if for any BF almost quasi-ideal ξ = (E; ξ p , ξ n ) of E if whenever ξ ⊆ ϑ, then supp(ξ) = supp(ϑ).Let K be a nonempty subset of a semigroup E. Then K is a minimal almost quasi-ideal of E if and only if

Corollary 3 . 1 .
Let E be a semigroup.Then E has no proper almost quasi-ideal if and only if supp(ϑ) = E for every BF almost quasi-ideal ϑ = (E; ϑ p , ϑ n ) of E.Proof.Suppose that E has no proper almost quasi-ideal and let ϑ = (E; ϑ p , ϑ n ) be a BF almost quasi-ideal of E. Then by Theorem 3.4, supp(ϑ) is an almost quasi-ideal of E. By assumption, supp(ϑ) = E.Conversely, suppose that supp(ϑ) = E and M is a proper almost quasi-ideal of E. Then by Theorem 3.3,≥ M = (E; ≥ p M , ≥ n K ) is a BF almost quasi-ideal of E. Thus supp(≥ M ) = M E.It is a contradiction.Hence E has no proper almost quasi-ideal.

pKK
and ϑ n ≥ λ n K or ξ p ≤ λ p K and ξ n ≥ λ n K .Therefore λ K = (E;λ p K , λ n K ) is a BF strongly prime almost quasi-ideal of E. Conversely, suppose that λ K = (E; λ p K , λ n K ) is a BF strongly prime almost quasi-ideal of E. Then λ K = (E; λ p K , λ n K ) is a BF almost quasi-ideal of E. Thus by Theorem 3.3, K is an almost quasi-ideal of E. Let M and L be almost quasi-ideals of E such that ML ∩ LM ≤ K. Then λ M = (E; λ p M , λ n K ) and λ L = (E; λ p L , λ n L ) are BF almost quasi-ideals of E. By Lemma 2and (λ n M • λ n L ) ∨ (λ n L • λ n M ) = λ n ML ∨ λ n LM = λ n ML∪LM ≥ λ n K .By assumption, λ p M ≤ λ p K and λ n M ≥ λ n K or λ p L ≤ λ p K and λ n L ≥ λ n K .Thus M ⊆ K or L ⊆ K.
Let ϑ = (E; ϑ p , ϑ n ) and ξ = (E; ξ p , ξ n ) be a BF subset of a semigroup E such that ϑ ⊆ ξ.If ϑ is a BF almost quasi-ideal of E, then ξ is also a BF almost quasi-ideal of E.Proof.Supppose that ϑ is a BF almost quasi-ideal of E and let x for any BF point x p t , x n s ∈ E. Theorem 3.1.p t , x n s be BF points of E. By assumption, that ξ p ⊆ ϑ p and ξ n ≥ ϑ n or ν p ≤ ϑ p and ν n ≥ ϑ n .