New Properties of Anti Fuzzy Ideals of Regular Semigroups

In this article, we study some properties of anti-fuzzy sub-semigroup, anti fuzzy left (right, two sided) ideal, anti fuzzy ideal, anti fuzzy generalized bi-ideal, anti fuzzy interior ideals and anti fuzzy two sided ideal of regular semigroup. Also, we characterized regular LA-semigroup in terms of their anti fuzzy ideal.


Introduction and Basic Concept
Fuzzy sub-semigroup and fuzzy interior ideal in semigroup was introduced by Hong, et al., in [1].And the concept of fuzzy ideal and fuzzy bi-ideals in semigroups was studied by Nobuaki Kuroki in (1981)", [2].The concept of the product of two fuzzy subset and anti product of two fuzzy subset was introduced by Shabir and Nawaz [3].The concept of characterizations of semigroups by their anti fuzzy ideals was studied by Khan and Asif in [4].The concept of intra-regular (left almost semigroup denoted by LA-semigroups) characterized by their anti fuzzy ideals by Khan and Faisal in [5].Many other authors interested studied of fuzzy ideal, for example see [6][7][8][9].Through out of this paper we are denoted of a regular semigroup byℵ  .Definition 1 [1].

Definition 5 [3].
Let ζ and φ be any fuzzy subsets of a semigroup ℵ then the product ζ ∘ φ is defined by

Definition 8 [4].
A fuzzy subset ζ of a semigroup ℵ is said to be anti fuzzy left (right

Definition 9 [4].
A fuzzy subset ζ of a semigroup ℵ is said to be anti fuzzy ideal of ℵ if it is both anti fuzzy left ideal and anti fuzzy right ideal.

New Properties of Anti Fuzzy Ideals of a Regular Semigroup
In this section we introduce some properties anti fuzzy ideal

Definition 17
ℵ is said to be a regular semigroup if w=wzw, whenever w, z ∈ ℵ or equivalently w ∈ wℵw.

Theorem 18
Every fuzzy interior ideal in ℵ  is idempotent.

Theorem 19
Let ζ be a fuzzy subset in ℵ  then it is an anti fuzzy two sided ideal of ℵ iff it is an anti fuzzy interior ideal of ℵ.Proof ⟹Since ζ be anti fuzzy two sided ideal of ℵ, then obviously, ζ is an anti fuzzy interior ideal of ℵ. ⟸ Suppose that ζ is an anti fuzzy interior ideal of ℵ.Let w, z ∈ ℵ, by by hypotheses so ∃ s, r ∈ ℵ, s.

Example 20
Let ℵ ={s, r, t, v} be a set with operation as follows: .
Then clearly, ζ is anti fuzzy interior ideal of ℵ but it is not an anti fuzzy two sided ideal of ℵ, since {s, r} is not a two sided ideal of ℵ.

Proposition 21
In regular semigroup ℵ, then i-Every anti fuzzy right ideal is idempotent.ii-Every anti fuzzy interior ideal is idempotent.Proof

Example 23
Consider the semigroup ℵ= {s, r, t , v} with the operation as follows: .

Proposition 26
A fuzzy subset ζ of ℵ  , then ζ is anti fuzzy bi-ideal of ℵ iff it is an anti fuzzy generalized bi-ideal of ℵ.Proof ⟹ Suppose that ζ be any anti fuzzy bi-ideal of ℵ, the obviously, ζ is an anti fuzzy generalized bi-ideal of ℵ. ⟸ Suppose that ζ be any anti fuzzy generalized bi-ideal of ℵ, since ℵ is a regular of a semigroup, so whenever w ∈ ℵ, ∃ z ∈ ℵ s.

Theorem 29
For every anti fuzzy left ideal α, every anti fuzzy generalized bi-ideal μ, and every anti fuzzy interior ideal
5. An LA-semigroup with right identity becomes a commutative semigroup with identity.if an LA-semigroup contains left identity, the following law holds w (r j) = r (w j); whenever w, r, j ∈ ℵ.

Proposition 30
A fuzzy subset ζ of ℵ  is a fuzzy right ideal iff it is a fuzzy left ideal.Proof ⟹ Suppose that ζ is a fuzzy right ideal of ℵ, since ℵ is a regular so whenever w∈ ℵ, ∃ z ∈ ℵ, s.t w=wzw, so by using ( 1

Theorem 32
For a fuzzy subset ζ of a regular LA-semigroup ℵ, with left identity then ζ is a fuzzy two sided ideal of ℵ iff it is a fuzzy interior ideal of ℵ.Proof ⟹ Suppose that ζ be a fuzzy two sided ideal of ℵ, then obviously, ζ is a fuzzy interior ideal of ℵ. ⟸ Suppose that ζ be a fuzzy interior ideal of ℵ, and w, r∈ ℵ, then since ℵ is a regular of ALsemigroup, so ∃ z, y ∈ ℵ s.

Conclusion
From the research / the evidence we conclude that 1.Let ζ be a fuzzy subset in ℵ  then it is an anti fuzzy two sided ideal of ℵ iff is an anti fuzzy interior ideal of ℵ. 2. In a regular semigroup ℵ, then the following are satisfy the following i) Every anti fuzzy right ideal is idempotent.ii) Every anti fuzzy interior ideal is idempotent.

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019
Then we can easily see that (ℵ,.) is not a regular semigroup.Define the fuzzy subset ζ of ℵ as ζ The ideals of ℵ are {s}, {s, r}, {s, r, t} and {s, r, t, v} Let us define two fuzzy subsets ζ and μ Example 25Let ℵ ={s, r, t} be a semigroup with the following table: Define a fuzzy subset ζ of ℵ by ζ(s)=0.6,ζ(r)=0.5, ζ(t)=0.4.By routine calculation, we can check that ζ is an anti fuzzy ideal, anti fuzzy interior ideal and anti fuzzy bi-ideal of ℵ  .Now, we give other fuzzy characterizations of a regular semigroup.