New Types of Intuitionistic Fuzzy Ideals in Γ-Semigroups

Thiti Gaketem; Pannawit Khamrot

doi:10.5391/IJFIS.2022.22.2.135

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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 135-143

Published online June 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.2.135

New Types of Intuitionistic Fuzzy Ideals in Γ-Semigroups

Thiti Gaketem¹ and Pannawit Khamrot²

¹Department of Mathematics, School of Science, University of Phayao, Phayao, Thailand
²Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Phitsanulok, Phitsanulok, Thailand

Correspondence to :
Thiti Gaketem (josemour25@yahoo.com)

Received: July 1, 2021; Revised: February 14, 2022; Accepted: March 29, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Go to

Abstract
1. Introduction
2. Preliminaries
3. New Types of Intuitionistic Ideals
4. New Types of Intuitionistic Almost Ideals
5. Conclusion
Conflict of Interest
References
Biographies

In this study, we define new types of intuitionistic fuzzy ideals and intuitionistic fuzzy almost ideals in Γ-semigroups, which can be applied to minimal ideals. We discuss the properties of intuitionistic fuzzy ideals and almost ideals in Γ-semigroups. Moreover, minimal ideals and almost minimal ideals are investigated with respect to their properties.

Keywords: Intuitionistic fuzzy ideals, Intuitionistic fuzzy almost ideals, Minimal intuitionistic fuzzy ideal, Minimal intuitionistic almost ideal

1. Introduction

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Abstract
1. Introduction
2. Preliminaries
3. New Types of Intuitionistic Ideals
4. New Types of Intuitionistic Almost Ideals
5. Conclusion
Conflict of Interest
References
Biographies

The theory for relation to the theory of fuzzy sets was first proposed by Zadeh [1]. This theory has been applied in many fields, including medical science, robotics, computer science, information science, control engineering, measure theory, logic, set theory, and topology. In 2000, the concept of intuitionistic fuzzy sets was developed by Atanassov as a generalization of the fuzzy sets, and it was used for the study of vagueness. In 2011, Sarar et al. [2] studied intuitionistic fuzzy sets in Γ-semigroups and investigated the properties of intuitionistic fuzzy ideals in Γ-semigroups. The structure of the ideal theory is important in the study of semigroups, and many researchers use the knowledge of ideals in studies related to Gamma-semigroups in fuzzy semigroups. For instance, Chinram [3] studied almost quasi-Γ-ideal and fuzzy almost quasi-Γ-ideals in Γ-semigroups; Marapureddy and Doradla [8] studied weak interior ideals of Γ-semigroups; and Majumder and Mandal [3] studied fuzzy generalized bi-ideal in Γ-semigroups. In regards to the concept of intuitionistic fuzzy ideals, several researchers expanded on this idea [4–7]. Recently, in 2021, Simuen et al. [8], Simuen et al. studied the concept of ideals and fuzzy ideals of Γ-semigroups.

In this study, we extend the concept of new fuzzy ideals to intuitionistic fuzzy ideals of Γ-semigroups and investigate the properties of the new types intuitionistic fuzzy ideals.

2. Preliminaries

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Abstract
1. Introduction
2. Preliminaries
3. New Types of Intuitionistic Ideals
4. New Types of Intuitionistic Almost Ideals
5. Conclusion
Conflict of Interest
References
Biographies

With respect to the content of this paper, some basic definitions are provided below, which are important for the proper understanding of the theory presented in the next section.

A sub-Γ-semigroup of a Γ-semigroup E is a non-empty set J of E such that JΓJ ⊆ J. A left (right) ideal of a Γ-semigroup E is a non-empty set J of E such that EΓJ ⊆ J(JΓE ⊆ J). An ideal of a Γ-semigroup E represents both the left and right ideal of E. A quasi-ideal of a Γ-semigroup E is a non-empty set J of S such that JΓE ∩ EΓJ ⊆ J. A sub-Γ-semigroup K of a Γ-semigroup E is called a bi-ideal of E if JΓEΓJ ⊆ J.

Definition 2.1 [8]

Let E be a Γ-semigroup and J be a nonempty subset of E, for all e ∈ E and α, β ∈ Γ. Thus, the following definitions are applicable to J.

(1) A left (right) almost ideal of Γ-semigroup E is a non-empty set J such that (eΓJ) ∩ J ≠ ∅︀ ((JΓe) ∩ J ≠ ∅︀)
(2) An almost bi-ideal of Γ-semigroup E is a non-empty set J such that (JΓeΓJ) ∩ J ≠ ∅︀.
(3) An almost quasi-ideal of Γ-semigroup E is a non-empty set J such that (eΓJ ∩ JΓe) ∩ J ≠ ∅︀.
(4) A left α-ideal of Γ-semigroup J is a non-empty set J such that EαJ ⊆ J. A right α-ideal of a Γ-semigroup E is a non-empty set J such that JβE ⊆ J.
(5) An (α, β)-ideal of Γ-semigroup E is a non-empty set J such that it is both a left α-ideal and right β-ideal of E.

For any m_i ∈ [0, 1] and , we define:

∨i∈Ami:supi∈A{mi} and ∧i∈Ami:=infi∈A{mi}.

For any m, n ∈ [0, 1], we have

m∨n=max{m,n} and m∧n=min{m,n}.

Definition 2.2 [1]

A fuzzy set ξ of a non-empty set J is a function ξ : J → [0, 1].

For any two fuzzy sets ξ and ς of a non-empty set J, we define ≥, =, ∧, and ∨ as follows:

(1) ξ ≥ ς ⇔ ξ(j) ≥ ς(j) for all j ∈ J.
(2) ξ = ς ⇔ ξ ≥ ς and ς ≥ ξ.
(3) (ξ ∧ ς)(j) = ξ(j) ∧ ς(j) for all j ∈ J.
(4) (ξ ∨ ς)(j) = ξ(k) ∨ ς(j) for all j ∈ J.

For any two fuzzy sets of ξ and ς of a non-empty of J, we define

ξ⊆ς if ξ(u)≤ς(u),(ξ∪ς)(u)=ξ(u)∧ς(u),and (ξ∩ς)(u)=ξ(u)∨ς(u) for all u∈J.

For any element j in a semigroup E, we define the set F_j as F_j := {(m, n) ∈ E × E | j = mn}.

For two fuzzy sets ξ and ς on a semigroup E, we define the product ξ ○ ς for all j ∈ E as follows:

(ξ∘ς)(j)={⋁(m,n)∈Fj{ξ(m)∧ς(n)∣(m,n)∈Fj},if Fj≠∅,0if Fj=∅.

The following conditions define the types of fuzzy almost ideal on semigroups.

Definition 2.3 [8]

A fuzzy set ξ of a semigroup J is considered to be:

(1) a fuzzy almost left (right) ideal of S if ℰ ○ ξ ∩ ξ ≠ 0 (ξ ○ ℰ ∩ξ ≠ 0).
(2) a fuzzy almost ideal of E if it is both a fuzzy almost left and fuzzy almost right ideal of E,
(3) a fuzzy almost bi-ideal of E if ξ ○ ℰ ○ ξ ∩ ξ ≠ 0.
(4) a fuzzy almost quasi-ideal of E if (ℰ ○ ξ ∩ ξ ○ ℰ) ∩ ξ ≠ 0.

The following conditions define the types of fuzzy sub-semigroups on Γ-semigroups.

Definition 2.4 [9]

A fuzzy set ξ of a Γ-semigroup E is said to be

(1) A fuzzy sub-semigroup of E if ξ(uγv) ≥ ξ(u) ∧ ξ(v) for all u, v ∈ E and γ ∈ Γ.
(2) A fuzzy left (right) ideal of E if ξ(uγv) ≥ ξ(v) (ξ(uγv) ≥ ξ(u)) for all u, v ∈ E and γ ∈ Γ.
(3) A fuzzy ideal of E if it is both a fuzzy left and fuzzy right ideal of E.
(4) A fuzzy bi-ideal of E if ξ is a fuzzy sub-semigroup of E and ξ(uγvβw) ≥ ξ(u) ∧ ξ(w) for all u, v,w ∈ E and γ, β ∈ Γ.

Now, we review the definition of the intuitionistic fuzzy sets and their basic properties, which are discussed in the following section.

Definition 2.5 [2]

Let μ and γ be fuzzy sets. An intuitionistic fuzzy (IF ) set on a non-empty set E is a function

G:={(j,μ(j),γ(j))|j∈E}

which satisfies 0 ≤ μ(j) + γ(j) ≤ 1.

For IF sets J₁ = (μ_J_₁, γ_J_₁) and J₂ = (μ_J_₂, γ_J_₂), we define the following relations:

(1) J₁ ⊆ J₂ if and only if μ_J_₁(j) ≤ μ_J_₂(j) and γ_J_₁(j) ≥ γ_J_₂(j).
(2) J₁ = J₂ if and only if J₁ ⊆ J₂ and J₂ ⊆ J₁.
(3) J₁ ∩ J₂ = {(μ_J_₁∩ μ_J_₂)(j), (γ_J_₁∪ γ_J_₂)(j) | j ∈ E}.
(4) J₁ ∪ J₂ = {(μ_J_₁∪ μ_J_₂)(j), (γ_J_₁∩ γ_J_₂)(j) | j ∈ E}.

For k ∈ E, we define F_k = {(y, z) ∈ E × E | k = yz}.

For IF sets J₁ = (μ_J_₁, γ_J_₁) and J₂ = (μ_J_₂, γ_J_₂), we define the product J₁ ○ J₂ = (μ_J_₁○_J_₂, γ_J_₁○_J_₂) as follows:

(μJ1∘μJ2)(j)={∨(m,n)∈Fj{μJ1(m)∧μJ2(n)},if Fk≠∅,0,if otherwise.

and

(γJ1∘γJ2)(j)={∧(m,n)∈Fk{γJ1(m)∨γJ2(n)},if Fk≠∅,0,if otherwise.

For any element j in a Γ-semigroup E, we define the set F_j_{_α}as F_j_{_α}:= {(m, α, n) ∈ E × Γ × E | j = mαn}.

For the IF sets J₁ = (μ_J_₁, γ_J_₁) and J₂ = (μ_J_₂, γ_J_₂), we define the product ξ ○_α ς for all k ∈ S as follows:

(μJ1∘αμJ2)(j)={⋁(m,α,n)∈Fjα{μJ1(m)∧μJ2(n)},if Fjα≠∅,0,if otherwise.

and

(γJ1∘αγJ2)(j)={⋀(m,α,n)∈Fjα{γJ1(m)∨γJ2(n)},if Fjα≠∅,0,if otherwise.

Definition 2.6

Let J be a non-empty set of a semigroup E. An intuitionistic characteristic function is defined as χ_J = (μ_χ_{_J}, γ_χ_{_J}), where

μχJ(j):={1,j∈J,0,j∉j,

and

γχJ(j):={0,j∈J,1,j∉J.

Remark 2.7

For the sake of simplicity, we use the symbol χ_E = (μ_χ_{_E}, γ_χ_{_E}) for the IF set χ_E := {(u, μ_χ_{_E}(u), γ_χ_{_E}(u)) | u ∈ E}.

Definition 2.8 [2]

An IF set J = (μ_J, γ_J ) on a Γ-semigroup E is defined as an:

(1) IF subsemigroup on E if it satisfies μ_J (uαv) ≥ μ_J (u) ∧ μ_J (v) and γ_J (uαv) ≤ γ_J (u) ∨ γ_J (v)
(2) IF left ideal on E if it satisfies μ_J (uαv) ≥ μ_J (v) and γ_J (uαv) ≤ γ_J (v)
(3) IF right ideal on E if it satisfies μ_J (uαv) ≥ μ_J (u) and γ_J (uαv) ≤ γ_J (u)
(4) IF ideal on E if it is both intuitionsitic fuzzy left ideal and right ideal of E

for all u, v ∈ E and α ∈ Γ.

3. New Types of Intuitionistic Ideals

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Abstract
1. Introduction
2. Preliminaries
3. New Types of Intuitionistic Ideals
4. New Types of Intuitionistic Almost Ideals
5. Conclusion
Conflict of Interest
References
Biographies

In this section, we define the bipolar fuzzy (α, β)-ideal and study its basic properties.

Definition 3.1

Let J = (μ_J, γ_J ) be an IF set on a Γ-semigroup E and α, β ∈ Γ. Then J = (μ_J, γ_J ) is defined as:

(1) An IF left α-ideal of E if μ_J (uαv) ≥ μ_J (v) and γ_J (uαv) ≤ γ_J (v) for all u, v ∈ E.
(2) An IF right β-ideal of E if μ_J (uβv) ≥ μ_J (v) and γ_J (uβv) ≤ γ_J (v) for all u, v ∈ E.
(3) An IF (α, β)-ideal of E if it is both an IF left α-ideal and an IF right β-ideal of E.
(4) An IF α-ideal of E if it is an IF (α, α)-ideal of E.

Theorem 3.2

Let J be a non-empty subset of Γ-semigroup E. Then, J is a left α-ideal (right β-ideal, (α, β)-ideal) of E if and only if χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF left α-ideal (right β-ideal, (α, β)-ideal) of E.

Proof

Let us suppose that J is a left α-ideal of E and u, v ∈ E.

If v ∈ J, then uαv ∈ J. Thus, μ_χ_{_J}(v) = μ_χ_{_J}(uαv) = 1 and

γ_χ_{_J}(v) = γ_χ_{_J}(uαv) = 0. Hence, μ_χ_{_J}(uαv) ≥ μ_χ_{_J}(v) and γ_χ_{_J}(uαv) ≤ γ_χ_{_J}(v).

If v ∉ J, then uαv ∈ J. Thus, μ_χ_{_J}(v) = 0, γ_χ_{_J}(v) = 1 and μ_χ_{_J}(uαv) = 1, γ_χ_{_J}(uαv) = 0. Hence, μ_χ_{_J}(uαv) ≥ μ_χ_{_J}(v) and γ_χ_{_J}(uαv) ≤ γ_χ_{_J}(v). Therefore, χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF left α-ideal of S.

Conversely, assuming that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF left α-ideal of E and u, v ∈ E with v ∈ J. Then, μ_χ_{_J}(v) = μ_χ_{_J}(uαv) = 1 and γ_χ_{_J}(v) = γ_χ_{_J}(uαv) = 0. By assumption, μ_χ_{_J}(uαv) ≥ μ_χ_{_J}(v) and γ_χ_{_J}(uαv) ≤ γ_χ_{_J}(v). Thus, uαv ∈ J. Hence, J is a left α-ideal of E.

Theorem 3.3

The intersection and union of any two IF left α-ideals (right β-ideals, (α, β)-ideals) of a Γ-semigroup E is an IF left α-ideal (right β-ideal, (α, β)-ideal) of E.

Proof

Let J₁ = (μ_J_₁, γ_J_₁) and J₂ = (μ_J_₂, γ_J_₂) be IF left α-ideals of E and let u, v ∈ E. Then,

(μJ1∪μJ2)(uαv)=μJ1(uαv)∧μJ2(uαv)≥μJ1(v)∧μJ2(v),

and

(γJ1∩γJ2)(uαv)=γJ1(uαv)∨γJ2(uαv)≤γJ1(v)∨γJ2(v).

Thus, J₁ ∩ J₂ is an IF left α-ideal of E. Similarly,

(μJ1∪μJ2)(uαv)=μJ1(uαv)∨μJ2(uαv)≥μJ1(v)∨μJ2(v),

and

(γJ1∪γJ2)(uαv)=γJ1(uαv)∧γJ2(uαv)≤γJ1(v)∧γJ2(v).

Thus, J₁ ∪ J₂ is an IF left α-ideal of E.

Next, we provide the definitions for IF (α, β)-bi-ideals and study their basic properties.

Definition 3.4

Let J = (μ_J, γ_J ) be an IF set of a Γ-semigroup E and α, β ∈ Γ. Then, J = (μ_J, γ_J ) is called an IF (α, β)-bi-ideal of E if μ_J ○_α μ_χ_{_E}○_β μ_J ≥ μ_J and γ_J ○_α γ_χ_{_E}○_β γ_J ≤ γ_J, where χ_E = (μ_χ_{_E}, γ_χ_{_E}) is an IF set mapping every element of E to 1 and 0.

Theorem 3.5

Let J be a non-empty subset of Γ-semigroup E. Then, J is an (α, β)-bi-ideal of E if and only if the characteristic function χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-bi-ideal of E.

Proof

Let us suppose that J is an (α, β)-bi-ideal of E and JαEβJ ⊆ J.

If u ∈ JαEβJ, then μ_J (u) = (μ_J ○_α μ_χ_{_J}○_β μ_J )(u) = 1 and γ_J (u) = (γ_J ○_α γ_χ_{_J}○_β γ_J )(u) = 0. Hence, (μ_J ○_α μ_χ_{_J}○_β μ_J )(u) ≥ μ_J (u) and (γ_J ○_α γ_χ_{_J}○_β γ_J )(u) ≤ γ_J (u).

If u ∉ JαEβJ, then μ_J (u) = 0, (μ_J ○_α μ_χ_{_J}○_β μ_J )(u) = 1 and γ_J (u) = 1, (γ_J ○_α γ_χ_{_J}○_β γ_J )(u) = 0. Hence, (μ_J ○_α μ_χ_{_J}○_β μ_J )(u) ≥ μ_J (u) and (γ_J ○_α γ_χ_{_J}○_β γ_J )(u) ≤ γ_J (u).

Therefore, χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-bi-ideal of E.

Conversely, assuming that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-bi-ideal of E and u ∈ JαEβJ. Then, (μ_J○_αμ_χ_{_J}○_βμ_J )(u) = 1 and (γ_J ○_α γ_χ_{_J}○_β γ_J )(u) = 0. By assumption, (μ_J ○_α μ_χ_{_J}○_β μ_J )(u) ≥ μ_J (u) and (γ_J ○_α γ_χ_{_J}○_β γ_J )(u) ≤ γ_J (u). Thus, u ∈ J. Hence, J is an (α, β)-bi-ideal of E.

Theorem 3.6

The intersection and union of any two IF (α, β)-bi-ideals of a Γ-semigroup E is an (α, β)-bi-ideal of E.

Proof

Let J₁ = (μ_J_₁, γ_J_₁) and J₂ = (μ_J_₂, γ_J_₂) be (α, β)-bi-ideals of E and u ∈ E. Then,

((μJ1∩μJ2)∘αμχE∘β (μJ1∩μJ2))(u)≥(μJ1∘αμχE∘βμJ1)(u)∧(μJ2∘αμχJ∘βμJ2)(u)≥(μJ1∩μJ2)(u),

and

((γJ1∩γJ2)∘αγχE∘β (γJ1∩γJ2))(u)≤γJ1∘α(γχE∘βγJ1)(u)∨(γJ2γJ2γJ2)(u)≤(γJ1∩γJ2)(u).

Thus, J₁ ∩ J₂ is an IF α-bi-ideal of S.

Next, we define conditions for IF (α, β)-quasi-ideal and study its basic properties.

Definition 3.7

Let J = (μ_J, γ_J ) be an IF set of a Γ-semigroup E and α, β ∈ Γ. Then, J = (μ_J, γ_J ) is called an IF (α, β)-quasi-ideal of E if μ_χ_{_E}○_α μ_J ∩ μ_J ○_β μ_χ_{_E}⊆ μ_J and γ_χ_{_E}○_α γ_J ∪ γ_J ○_β γ_χ_{_E}⊇ γ_J.

Theorem 3.8

If L = (μ_L, γ_L) and R = (μ_R, γ_R) are IF left and right α-ideals of a Γ-semigroup E, respectively, then L ∩ R is an IF α-quasi-ideal of E.

Proof

Let L = (μ_L, γ_L) and R = (μ_R, γ_R) be IF left and right α-ideals of E, respectively. Then, μ_L○_αμ_R ⊆ μ_χ_{_E}○_αμ_L ⊆ μ_L and μ_L ○_α μ_R ⊆ μ_R ○_α μ_χ_{_E}⊆ μ_R. Similarly, γ_L ○_α γ_R ⊆ γ_χ_{_E}○_α γ_L ⊆ γ_L and γ_L ○_α γ_R ⊆ γ_R ○_α γ_χ_{_E}⊆ γ_R Thus, μ_L ○_α μ_R ⊆ μ_L ∩ μ_R and γ_L ○_α γ_R ⊇ γ_L ∪ γ_R. Hence,

μχE∘α (μL∩μR)∩(μL∩μR)∘αμχE⊆μχE∘α (μL∩μR)∘αμχE⊆μL∩μR,

and

γχE∘α (γL∩γR)∩(γL∩γR)∘αγχE⊇γχE∘α (γL∪γR)∘αγχE⊇γL∪γR.

Thus, L ∩ R is an IF α-quasi-ideal of S.

Theorem 3.9

Every IF (α, β)-quasi-ideal of a Γ-semigroup E is an intersection of an IF left α-ideal and IF right β-ideal of S

Proof

Let Q = (μ_Q, γ_Q) be an IF (α, β)-quasi-ideal of E. Considering μ_L = μ_Q ∪ (μ_χ_{_E}○_α μ_Q) and γ_L = γ_Q ∪ (γ_χ_{_E}○_α γ_Q), where L = (μ_L, γ_L) μ_R = μ_Q ∪ (μ_Q ○_β μ_χ_{_E}) and γ_R = γ_Q ∪ (γ_Q ○_β γ_χ_{_E}), where R = (μ_R, γ_R), we obtain:

μχE∘αμL=μχE∘α(μQ∪(μχE∘αμQ))=(μχE∘αμQ)∪(μχE∘α(μχE∘αμQ))=(μχE∘αμQ)∪((μχE∘αμχE)∘αμQ)=(μχE∘αμQ)∪(μχE∘αμQ)⊆μQ∪(μχE∘αμQ)=μL.

Additionally, we obtain:

μR∘βμχE=(μQ∪(μQ∘βμχE))∘βμχE=(μQ∘αμχE)∪((μQ∘βμχE)∘αμχE)=(μQ∘αμχE)∪(μQ∘β(μχE∘αμχE))=(μQ∘αμχE)∪(μQ∘βμχE)⊆μQ∪(μQ∘βμχE)=μR.

Similarly, we can show that: γ_χ_{_E}○_α γ_L ⊆ γ_L and γ_R ○_β γ_χ_{_E}⊆ γ_R. Thus, L = (μ_L, γ_L) and R = (μ_R, γ_R) are IF left and right β-ideals of E, respectively. Furthermore, we know that

μQ⊆(μQ∪(μχE∘αμQ))∩(μQ∪(μQ∘βμχE))=μL∩μR,

and

μL∩μR=(μQ∪(μχE∘αμQ))∩(μQ∪(μQ∘βμχE))=μQ∩((μχE∘αμQ)∪(μQ∘βμχE))⊆μQ∩μQ=μQ.

Hence, μ_Q = μ_L ∩ μ_R. Similarly, we can show that γ_Q = γ_L ∩ γ_R.

Theorem 3.10

Let J be a non-empty subset of a Γ-semigroup E. Then, J is an (α, β)-quasi-ideal of E if and only if the characteristic function χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-quasi-ideal of S.

Proof

Let us suppose that J is an (α, β)-quasi-ideal of S and u ∈ E.

If u ∈ (EαJ) ∩ (JβE), then u ∈ J. Thus, μ_χ_{_J}(u) = 1 and γ_χ_{_J}(u) = 0. Hence, ((μ_χ_{_J}○_α μ_χ_{_E}) ∩ (μ_χ_{_J}○_β μ_χ_{_E}))(u) ≤ μ_χ_{_J}(u) and ((γ_χ_{_J}○_α γ_χ_{_E}) ∪ (γ_χ_{_E}○_β γ_χ_{_J}))(u) ≥ γ_χ_{_J}(u).

If u ∉ (EαJ) ∩ (JβE), then μ_χ_{_J}(u) = 0 and γ_χ_{_J}(u) = 1. Hence, ((μ_χ_{_J}○_α μ_χ_{_E}) ∩ (μ_χ_{_J}○_β μ_χ_{_E}))(u) ≤ μ_χ_{_J}(u) and ((γ_χ_{_J}○_α γ_χ_{_E}) ∪ (γ_χ_{_E}○_β γ_χ_{_J}))(u) ≥ γ_χ_{_J}(u).

Therefore, χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-quasi-ideal of E.

Conversely, assuming that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-quasi-ideal of S and u ∈ (EαJ)∩(JβE), we obtain: ((μ_χ_{_J}○_α μ_χ_{_E}) ∩ (μ_χ_{_J}○_β μ_χ_{_E}))(u) = 1, and ((γ_χ_{_J}○_α γ_χ_{_E}) ∪ (γ_χ_{_E}○_β γ_χ_{_J}))(u) = 0. By assumption, ((μ_χ_{_J}○_α μ_χ_{_E}) ∩ (μ_χ_{_J}○_β μ_χ_{_E}))(u) ≤ μ_χ_{_J}(u), and ((γ_χ_{_J}○_α γ_χ_{_E})∪(γ_χ_{_E}○_β γ_χ_{_J}))(u) ≥ γ_χ_{_J}(u). Thus, u ∈ J. Hence, J is an (α, β)-quasi-ideal of E.

4. New Types of Intuitionistic Almost Ideals

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Abstract
1. Introduction
2. Preliminaries
3. New Types of Intuitionistic Ideals
4. New Types of Intuitionistic Almost Ideals
5. Conclusion
Conflict of Interest
References
Biographies

Definition 4.1

Let J = (μ_J, γ_J ) be an IF set of a Γ-semigroup E and α, β ∈ Γ, then is said to be

(1) An IF almost left α-ideal of E if (μ_χ_{_e}○_α μ_J ) ∧ μ_J ≠ 0 and (γ_χ_{_e}○_α γ_J ) ∨ γ_J ≠ 1 for all e ∈ E.
(2) An IF almost right β-ideal of E if (μ_χ_{_e}○_β μ_J ) ∧ μ_J ≠ 0 and (γ_χ_{_e}○_β γ_J ) ∨ γ_J ≠ 1 for all e ∈ E.
(3) An IF almost (α, β)-ideal of E if it is both an IF almost left α-ideal and an IF almost right β-ideal of E.

Here, χ_e = (μ_χ_{_e}, γ_χ_{_e}) is an IF set of E mapping every element of E to e.

Theorem 4.2

If J = (μ_J, γ_J ) is an IF almost left α-ideal (right β-ideal, (α, β)-ideal) of a Γ-semigroup E and K = (μ_K, γ_K) is an IF set of E such that J ⊆ K, then K = (μ_K, γ_K) is an IF almost left α-ideal (right β-ideal, (α, β)-ideal) of E.

Proof

Let us suppose that J = (μ_J, γ_J ) is an IF almost left α-ideal of E and K = (μ_K, γ_K) is an IF set of E such that J ⊆ K. Then, (μ_χ_{_e}○_αμ_J )∧μ_J ≠ 1and (γ_χ_{_e}○_α γ_J )∨γ_J ≠ 0. Thus (μ_χ_{_e}○_α μ_J ) ∧ μ_J ⊆ (μ_χ_{_e}○_α μ_K) ∧ μ_K ≠ 0 and (γ_χ_{_e}○_α γ_J ) ∨ γ_J ⊆ (γ_χ_{_e}○_α γ_K) ∨ γ_K ≠ 1. Hence, K = (μ_K, γ_K) is an IF almost left α-ideal of E.

Theorem 4.3

Let J be a non-empty subset of Γ-semigroup E. Then, J is an almost left α-ideal (right β-ideal, (α, β)-ideal) of E if and only if the characteristic function χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF almost left α-ideal (right β-ideal, (α, β)-ideal) of E.

Proof

Let us suppose that J is an almost left α-ideal of E. Then, uαJ ∩ J ≠ ∅︀ for all u ∈ E. Thus, there exist v ∈ uαJ and v ∈ J such thato (μ_χ_{_E}○_α μ_χ_{_J})(v) = μ_χ_{_J}(v) = 1 and (γ_χ_{_e}○_αγ_χ_{_J})(v) = γ_χ_{_J}(v) = 0. Hence, (μ_χ_{_e}○_αμ_χ_{_J})∧μ_χ_{_J}≠ 0 and (γ_χ_{_E}○_α γ_χ_{_J}) ∨ γ_χ_{_J}≠ 1. Therefore, χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF almost left α-ideal of S.

Conversely, assuming that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF almost left α-ideal of S and u ∈ S, we obtain: (μ_χ_{_e}○_αμ_χ_{_J})∧μ_χ_{_J}≠ 0 and (γ_χ_{_e}○_α γ_χ_{_J}) ∨ γ_χ_{_J}≠ 1. Thus, there exists an r ∈ S such that ((μ_χ_{_e}○_αμ_χ_{_J})∧μ_χ_{_J})(r) ≠ 0 and ((γ_χ_{_e}○_αγ_χ_{_J})∨γ_χ_{_J})(r) ≠ 1. Hence, r ∈ uαJ ∩ J Implies that uαJ ∩ J ≠ ∅︀. Therefore, J is an almost left α-ideal of E.

Next, we review the definition of supp(J) and study the relation between supp(J) and IF almost left α-ideal (right β-ideal, (α, β)-ideal) of Γ-semigroups.

Let J = (μ_J, γ_J ) be an IF set of a non-empty of E. Then the support of J is determined rather than supp(J) = {u ∈ E | J(u) ≠ 0}, where μ_J (u) ≠ 0 and γ_J (u) ≠ 1 for all u ∈ E.

Theorem 4.4

Let J = (μ_J, γ_J ) be an IF set of a non-empty Γ-semigroup E. Then, J = (μ_J, γ_J ) is an IF almost left α-ideal (right β-ideal, (α, β)-ideal) of S if and only if supp(J) is an almost left α-ideal (right β-ideal, (α, β)-ideal) of E.

Proof

Let J = (μ_J, γ_J ) be an IF almost left α-ideal of S and u ∈ E. Then, (μ_χ_{_e}○_αμ_χ_{_J})∧μ_χ_{_J}≠ 0 and (γ_χ_{_e}○_αγ_χ_{_J})∨γ_χ_{_J}≠ 1. Thus, there exists an r ∈ E such that ((μ_χ_{_e}○_α μ_χ_{_J}) ∧ μ_χ_{_J})(r) ≠ 0 and ((γ_χ_{_e}○_α γ_χ_{_J}) ∨ γ_χ_{_J})(r) ≠ 1. Hence, there exists a k ∈ E such that r = uαJ, μ_χ_{_J}(r) ≠ 0, γ_χ_{_J}(r) ≠ 1, and μ_χ_{_J}(k) ≠ 0, γ_χ_{_J}(k) ≠ 1. This implies that r, k ∈ supp(J). Thus, (μ_χ_{_e}○_α μ_χ_{_supp₍_J₎})(r) ≠ 0, (γ_χ_{_e}○_α γ_χ_{_supp₍_J₎})(r) ≠ 1, and μ_χ_{_supp₍_J₎}≠ 0, γ_χ_{_supp₍_J₎}≠ 1. Hence, (μ_χ_{_e}○_α μ_χ_{_supp₍_J₎}) ∧ μ_χ_{_Jsupp₍_J₎}≠ 0 and (γ_χ_{_e}○_α γ_χ_{_supp₍_J₎})∨γ_χ_{_Jsupp₍_J₎}≠ 1. Therefore, χ_supp₍_J₎ is an IF almost left α-ideal of E. This shows that supp(J) is an almost left α-ideal of E.

Conversely, let supp(J) be an almost left α-ideal of E. Then, using Theorem 4.3, we can conclude that χ_J_{_supp₍_J₎}is an IF almost left α-ideal of E. Thus, (μ_χ_{_e}○_αμ_χ_{_supp₍_J₎})∧μ_χ_{_supp₍_J₎}≠ 0 and (γ_χ_{_e}○_α γ_χ_{_supp₍_J₎}) ∨ γ_χ_{_supp₍_J₎}≠ 1. Hence, there exists an r ∈ E such that ((μ_χ_{_e}○_α μ_χ_{_supp₍_J₎}) ∧ μ_χ_{_supp₍_J₎})(r) ≠ 0 and ((γ_χ_{_e}○_α γ_χ_{_supp₍_J₎}) ∨ γ_χ_{_supp₍_J₎})(r) ≠ 1. This implies that (μ_χ_{_e}○_α μ_χ_{_supp₍_J₎})(r) ≠ 0, (γ_χ_{_e}○_α γ_χ_{_supp₍_J₎})(r) ≠ 0 and μ_χ_{_J}(r) ≠ 0,γ_χ_{_J}(r) ≠ 1. Thus, there exists a k ∈ E such that r = uαJ, μ_χ_{_supp₍_J₎}(r) ≠ 0, γ_χ_{_supp₍_J₎}(r) ≠ 1 and μ_χ_{_supp₍_J₎}(k) ≠ 0, γ_χ_{_supp₍_J₎}(k) ≠ 1. Hence, (μ_χ_{_e}○_αμ_χ_{_supp₍_J₎})∧ μ_χ_{_supp₍_J₎}≠ 0 and (γ_χ_{_e}○_α γ_χ_{_Jsupp₍_J₎}) ∨ γ_χ_{_supp₍_J₎}≠ 1. Therefore, J = (μ_J, γ_J ) is an IF almost left α-ideal of E.

Definition 4.5

An ideal I of a Γ-semigroup S is called a minimal if for every ideal of J of S, for which J ⊆ I, we obtain J = I.

Definition 4.6

An IF almost left α-ideal (right β-ideal, (α, β)-ideal) J = (μ_J, γ_J ) of a Γ-semigroup E is minimal if for all BF almost left α-ideal (right β-ideal, (α, β)-ideal), there exists a K = (μ_K, γ_K) of E, where K ⊆ J, such that supp(K) = supp(J).

Theorem 4.7

Let J be a non-empty subset of a Γ-semigroup E. Then J is a minimal almost left α-ideal (right β-ideal, (α, β)-ideal) if and only if χ_J = (μ_χ_{_J}, γ_χ_{_J}) is a minimal IF almost left α-ideal (right β-ideal, (α, β)-ideal) of E.

Proof

Let us suppose that J is a minimal almost left α-ideal of E. Then, J is an almost left α-ideal of E. Thus, using Theorem 4.3, we can conclude that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF left α-ideal of E. Let K = (μ_K, γ_K) be an IF left α-ideal of E such that K ⊆ J. Then, using Theorem 4.4, it can be concluded that supp(Q) is an almost left α-ideal of E. Thus, supp(K) ⊆ supp(χ_J) = J. By assumption, supp(K) = K =supp(χ_J ). Thus, χ_J = (μ_χ_{_J}, γ_χ_{_J}) is a minimal IF almost left α-ideal of E.

Conversely, let us suppose that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is a minimal IF almost left α-ideal of E. Then, using Theorem 4.3, we can conclude that J is an almost left α-ideal of E. Additionally, let K be an almost left α-ideal of E such that K ⊆ J. Then, using Theorem 4.3, it can be concluded that χ_K = (μ_χ_{_K}, γ_χ_{_K}) is an IF left α-ideal of E such that χ_K ⊆ χ_J. Thus, K = supp(χ_K) = supp(χ_J) = J. Hence, J is a minimal almost left α-ideal of S.

Next, we provide the definitions for IF almost (α, β)-quasi-ideals and study their properties.

Definition 4.8

Let J = (μ_J, γ_J ) be an IF set of a Γ-semigroup E and α, β ∈ Γ, then is said to be an IF almost (α, β)-quasi-ideal of E if (J ○_α χ_e) ∩ (χ_e ○_β J) ≠ 0 and (J ○_α χ_e) ∪ (χ_e ○_β J) ≠ 1.

Theorem 4.9

If J = (μ_J, γ_J ) is an IF almost (α, β)-quasi-ideal of a Γ-semigroup E and K = (μ_K, γ_K) is an IF set of E such that J ⊆ K, then K = (μ_K, γ_K) is an IF (α, β)-quasi-ideal of E.

Proof

Let us suppose that J = (μ_J, γ_J ) is an IF almost (α, β)-quasi-ideal of E and K = (μ_K, γ_K) is an IF set of E such that J ⊆ K. Then, (μ_χ_{_E}○_α μ_J ) ∧ (μ_J ○_β μ_χ_{_e}) ≠ 0 and (γ_χ_{_e}○_αγ_J )∧(γ_J○_βγ_χ_{_e}) ≠ 1. Thus, (μ_χ_{_e}○_αμ_J )∧(μ_J○_βμ_χ_{_e}) ⊆ (μ_χ_{_e}○_αμ_J )∧(μ_K○_βμ_χ_{_e}) ≠ 0 and (γ_χ_{_e}○_αγ_J )∧(γ_J ○_β γ_χ_{_E}) ⊆ (γ_χ_{_e}○_α γ_K) ∧ (γ_K ○_β γ_χ_{_e}) ≠ 0 Hence, K = (μ_K, γ_K) is an IF (α, β)-quasi-ideal of E.

Theorem 4.10

Let J be a non-empty subset of a Γ-semigroup E. Then, J is an almost (α, β)-quasi-ideal of E if and only if the characteristic function χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF almost (α, β)-quasi-ideal of E.

Proof

Let us suppose that J is an almost (α, β)-quasi-ideal of E. Then, (Jαu) ∩ (uβJ) ∩ J ≠ ∅︀ for all u ∈ E. Thus, there exist v ∈ (Jαu)∩(uβJ)∩J and v ∈ J such that ((μ_χ_{_e}○_αμ_J )∧ (μ_J ○_β μ_χ_{_e}))(v) ≠ 0 and ((γ_χ_{_e}○_α γ_J ) ∨ (γ_J ○_β γ_χ_{_e}))(v) ≠ 1. Hence, (J○_αχ_e)∩(χ_E○_βJ) ≠ 0 and (J○_αχ_e)∪(χ_E○_βJ) ≠ 1. Therefore, χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF almost (α, β)-quasi-ideal of S.

Conversely, assuming that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF almost (α, β)-quasi-ideal of E and u ∈ E, we obtain (J ○_α χ_e) ∩ (χ_e ○_β J) ≠ 0 and (J ○_α χ_e) ∪ (χ_e ○_β J) ≠ 1. Thus there exists an r ∈ E such that ((μ_χ_{_e}○_α μ_J ) ∧ (μ_J ○_β μ_χ_{_e}))(r) ≠ 0 and ((γ_χ_{_e}○_α γ_J ) ∧ (γ_J ○_β γ_χ_{_e}))(r) ≠ 1. Hence, r ∈ (Jαu) ∩ (uβJ) ∩ J implies that (Jαu) ∩ (uβJ) ∩ J ≠ ∅︀. Therefore, J is an almost (α, β)-quasi-ideal of E.

In the following section, we study the properties and relationship between supp(J) and IF almost (α, β)-quasi-ideal of Γ-semigroups.

Theorem 4.11

Let J = (μ_J, γ_J ) be an IF set of a non-empty Γ-semigroup E. Then, J = (μ_J, γ_J ) is an IF almost (α, β)-quasi-ideal of E if and only if supp(J) is an almost (α, β)-quasi-ideal of E.

Proof

Let J = (μ_J, γ_J ) be an IF almost (α, β)-quasi-ideal of E and u ∈ E. Then, (J ○_α χ_E) ∩ (χ_E ○_β J) ≠ 0 and (J ○_α χ_e) ∪ (χ_e ○_β J) ≠ 1. Thus, there exists an r ∈ E such that ((μ_χ_{_e}○_α μ_J ) ∧ (μ_J ○_β μ_χ_{_E}))(r) ≠ 0 and ((γ_χ_{_e}○_α γ_J ) ∧ (γ_J ○_β γ_χ_{_e}))(r) ≠ 1. Therefore, there exist j₁, j₂ ∈ E such that r = j₁αu = uβj₂, μ_J (r) ≠ 0, γ_J (r) ≠ 1 and μ_J (j₁) ≠ 0, γ_J (j₂) ≠ 1. This implies that r, j₁, j₂ ∈ supp(J). Thus, ((μ_χ_{_supp₍_J₎}○_α μ_χ_{_e}) ∩ (μ_χ_{_e}○_β μ_χ_{_supp₍_J₎}))(r) ≠ 0 and μ_χ_{_Jsupp₍_J₎}(r) ≠ 0. Similarly, ((γ_χ_{_supp₍_J₎}○_α γ_χ_{_e}) ∩ (γ_χ_{_e}○_β γ_χ_{_supp₍_J₎}))(r) ≠ 1 and γ_χ_{_supp₍_J₎}(r) ≠ 1. Hence, (μ_χ_{_supp₍_J₎}○_α μ_χ_{_e})∩(μ_χ_{_e}○_β μ_χ_{_supp₍_J₎}) ≠ 0 and (γ_χ_{_supp₍_J₎}○_α γ_χ_{_e})∩(γ_χ_{_e}○_β γ_χ_{_supp₍_J₎}) ≠ 1. Therefore, χ_supp₍_J₎ is an IF almost (α, β)-quasi-ideal of S. This shows that supp(J) is an almost (α, β)-quasi-ideal of E.

Conversely, let supp(J) be an almost (α, β)-quasi-ideal of E. Then, using Theorem 4.10, we can infer that χ_supp₍_J₎ is an IF (α, β)-quasi-ideal of E. Thus, (μ_χ_{_supp₍_J₎}○_α μ_χ_{_e}) ∩ (μ_χ_{_e}○_β μ_χ_{_supp₍_J₎}) ≠ 0 and (γ_χ_{_supp₍_J₎}○_α γ_χ_{_e})∪(γ_χ_{_e}○_β γ_χ_{_supp₍_J₎}) ≠ 1. Hence, there exists an r ∈ E such that ((μ_χ_{_supp₍_J₎}○_α μ_χ_{_e}) ∧ (μ_χ_{_e}○_β μ_χ_{_supp₍_J₎}))(r) ≠ 0 and ((γ_χ_{_supp₍_J₎}○_α γ_χ_{_e}) ∨ (γ_χ_{_e}○_β γ_χ_{_supp₍_J₎}))(r) ≠ 1. This implies that (μ_χ_{_supp₍_J₎}○_α μ_χ_{_e})(r) ≠ 0, (μ_χ_{_e}○_β μ_χ_{_supp₍_J₎})(r) ≠ 0 and μ_χ_{_supp₍_J₎}(r) ≠ 0. Similarly, (γ_χ_{_supp₍_J₎}○_α γ_χ_{_e})(r) ≠ 1, (γ_χ_{_e}○_β γ_χ_{_supp₍_J₎})(r) ≠ 1 and γ_χ_{_supp₍_J₎}(r) ≠ 1. Thus, there exist j₁, j₂ ∈ E such that r = j₁αu = uβj₂, μ_J (r) ≠ 0, γ_J (r) ≠ 1 and μ_J (j₁) ≠ 0, γ_J (j₂) ≠ 1. Hence, (J ○_α χ_e) ∩ (χ_e ○_β J) ≠ 0 and (J ○_α χ_e) ∪ (χ_e ○_β J) ≠ 1. Therefore, J = (μ_J, γ_J ) is an IF almost (α, β)-quasi-ideal of E.

Definition 4.12

An almost ideal I of a Γ-semigroup S is called a minimal if for every almost ideal of J of S, where J ⊆ I, we have J = I.

Definition 4.13

An IF almost (α, β)-quasi-ideal J = (μ_J, γ_J ) of a Γ-semigroup E is minimal if for all IF almost (α, β)-quasi-ideal K = (μ_K, γ_K) of E, where K ⊆ J, supp(K) = supp(J).

Theorem 4.14

Let J be a non-empty subset of a Γ-semigroup E. Then J is a minimal almost (α, β)-quasi-ideal if and only if χ_J = (μ_χ_{_J}, γ_χ_{_J}) is a minimal IF almost (α, β)-quasi-ideal of E.

Proof

Let us suppose that J is a minimal almost left (α, β)-quasi-ideal of E. Then, J is an almost (α, β)-quasi-ideal of E. Thus, using Theorem 4.10, we can conclude that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-quasi-ideal of S. Let K = (μ_K, γ_K) be an IF (α, β)-quasi-ideal of E such that K ⊆ J. Then, using Theorem 4.9, we can conclude that supp(Q) is an almost (α, β)-quasi-ideal of E. Thus, supp(K) ⊆ supp(χ_J) = J. By assumption, supp(K) = K = supp(χ_J ). Thus, χ_J = (μ_χ_{_J}, γ_χ_{_J}) is a minimal IF almost (α, β)-quasi-ideal of E.

Conversely, let us suppose that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is a minimal IF almost (α, β)-quasi-ideal of E. Then, using Theorem 4.11, it can be shown that J is an IF almost α-ideal of E. Let K be an IF almost (α, β)-quasi-ideal of E such that K ⊆ J. Then, using Theorem 4.11, we can conclude that χ_K = (μ_χ_{_K}, γ_χ_{_K}) is an IF (α, β)-quasi-ideal of E such that χ_K ⊆ χ_J. Thus, K = supp(χ_K) = supp(χ_J) = J. Hence, J is a minimal almost (α, β)-quasi-ideal of E.

Next, we present the definitions of IF almost (α, β)-bi-ideals and study their properties.

Definition 4.15

Let J = (μ_J, γ_J ) be an IF set of a Γ-semigroup E and α, β ∈ Γ. is an IF almost (α, β)-bi-ideal of E if (μ_J ○_α μ_χ_{_e}○_β μ_J )∧μ_J ≠ 0 and (γ_J ○_α γ_χ_{_e}○_β γ_J )∨γ_J ≠ 1.

Theorem 4.16

If J = (μ_J, γ_J ) is an IF almost (α, β)-bi-ideal of a Γ-semigroup E and K = (μ_K, γ_K) is an IF set of E such that J ⊆ K, then K = (μ_K, γ_K) is an IF (α, β)-bi-ideal of E.

Proof

Let us suppose that J = (μ_J, γ_J ) is an IF almost (α, β)-bi-ideal of a Γ-semigroup E and K = (μ_K, γ_K) is an IF set of E such that J ⊆ K. Then, (μ_J ○_α μ_χ_{_E}○_β μ_J ) ∧ μ_J ≠ 0 and (γ_J ○_α γ_χ_{_e}○_β γ_J ) ∨ γ_J ≠ 1. Thus, (μ_J ○_α μ_χ_{_e}○_β μ_J ) ∧ μ_J ⊆ ((μ_K ○_α μ_χ_{_e}○_β μ_K) ∧ μ_J ≠ 0 and (γ_J ○_α γ_χ_{_E}○_β γ_J ) ∨ γ_J ⊆ (γ_K ○_α γ_χ_{_e}○_β γ_K) ∨ γ_K ≠ 1. Hence, K = (μ_K, γ_K) is an IF (α, β)-bi-ideal of S.

Theorem 4.17

Let J be a non-empty subset of Γ-semigroup E. Then, J is an almost (α, β)-bi-ideal of E if and only if the characteristic function χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF almost (α, β)-bi-ideal of E.

Proof

Let us suppose that J is an almost (α, β)-bi-ideal of E. Then, KαuβJ ∩ J ≠ ∅︀. for all u ∈ E. Thus, there exist v ∈ JαuβJ and v ∈ J such that ((μ_χ_{_J}○_αμ_χ_{_e}χ_e○_βμ_χ_{_J}))(v) = μ_χ_{_J}(v) = 1 and ((γ_χ_{_J}○_α γ_χ_{_e}○_β γ_χ_{_J}))(v) = γ_χ_{_J}(v) = 0. Hence, (μ_χ_{_J}○_α μ_χ_{_e}χ_e ○_β μ_χ_{_J}) ∧ μ_χ_{_J}≠ 0 and (γ_χ_{_J}○_α γ_χ_{_e}○_β γ_χ_{_J}) ∨ γ_χ_{_J}≠ 1. Therefore, χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF almost (α, β)-bi-ideal of E.

Conversely, assuming that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF almost (α, β)-bi-ideal of E and u ∈ E, we obtain: (μ_χ_{_J}○_α μ_χ_{_e}χ_e ○_β μ_χ_{_J})∧μ_χ_{_J}≠ 0 and (γ_χ_{_J}○_αγ_χ_{_e}○_β γ_χ_{_J})∨γ_χ_{_J}≠ 1. Thus, there exists an r ∈ E such that ((μ_χ_{_J}○_αμ_χ_{_e}χ_e○_βμ_χ_{_J})∧μ_χ_{_J})(r) ≠ 0 and ((γ_χ_{_J}○_αγ_χ_{_e}○_βγ_χ_{_J})∨γ_χ_{_J})(r) ≠ 1. Hence, r ∈ JαuβJ∩J implies that JαuβJ ∩ J ≠ ∅︀. Therefore, J is an almost (α, β)-bi-ideal of E.

In the next section, we study properties and relationship between supp(ξ) and IF almost (α, β)-bi-ideal of Γ-semigroups.

Theorem 4.18

Let J = (μ_J, γ_J ) be an IF of a non-empty oΓ-semigroup E. Then, J = (μ_J, γ_J ) is an IF almost (α, β)-bi-ideal of S if and only if supp(J) is an almost (α, β)-bi-ideal of E.

Proof

Let J = (μ_J, γ_J ) be an IF almost (α, β)-bi-ideal of E and u ∈ E. Then, (μ_χ_{_J}○_α μ_χ_{_e}χ_e ○_β μ_χ_{_J}) ∧ μ_χ_{_J}≠ 0 and (γ_χ_{_J}○_α γ_χ_{_e}○_β γ_χ_{_J}) ∨ γ_χ_{_J}≠ 1. Thus, there exists an r ∈ E such that ((μ_χ_{_J}○_α μ_χ_{_e}χ_e ○_β μ_χ_{_J}))(r) ≠ 0 and ((γ_χ_{_J}○_α γ_χ_{_e}○_β γ_χ_{_J}))(r) ≠ 1. Therefore, there exist j₁, j₂ ∈ E such that r = j₁αβj₂, μ_J (r) ≠ 0, γ_J (r) ≠ 1 and μ_J (k) ≠ 0, γ_J (k) ≠ 1. This implies that r, k₁, k₂ ∈ supp(J). Thus, (μ_χ_{_supp₍_J₎}○_αμ_χ_{_E}○_β μ_χ_{_supp₍_J₎})(r) ≠ 0 and μ_χ_{_supp₍_J₎}≠ 0. Similarly, (γ_χ_{_supp₍_J₎}○_αγ_χ_{_e}○_β γ_χ_{_supp₍_J₎})(r) ≠ 1and γ_χ_{_supp₍_J₎}≠ 1. Hence, ((μ_χ_{_supp₍_J₎}○_α μ_χ_{_e}○_β μ_χ_{_supp₍_J₎})) ∧ μ_χ_{_supp₍_J₎}≠ 0 and ((γ_χ_{_supp₍_J₎}○_αγ_χ_{_e}○_β γ_χ_{_supp₍_J₎}))∨γ_χ_{_supp₍_J₎}≠ 1. Thus, χ_supp₍_J₎ is an IF almost (α, β)-bi-ideal of E. This shows that supp(J) is an almost (α, β)-bi-ideal of E.

Conversely, let supp(ξ) be an almost (α, β)-bi-ideal of S. Then, using Theorem 4.17, we can conclude that χ_J_{_supp₍_J₎}is an IF almost (α, β)-bi-ideal of E. Thus, ((μ_χ_{_Jsupp₍_J₎}○_α μ_χ_{_E}○_βμ_χ_{_Jsupp₍_J₎}))∧μ_χ_{_Jsupp₍_J₎}≠ 0 and ((γ_χ_{_Jsupp₍_J₎}○_αγ_χ_{_E}○_β γ_χ_{_Jsupp₍_J₎})) ∨ γ_χ_{_Jsupp₍_J₎}≠ 1. Hence, there exists an r ∈ E such that ((μ_χ_{_Jsupp₍_J₎}○_αμ_χ_{_E}○_β μ_χ_{_Jsupp₍_J₎}))∧μ_χ_{_Jsupp₍_J₎}(r) ≠ 0 and ((γ_χ_{_Jsupp₍_J₎}○_α γ_χ_{_E}○_β γ_χ_{_Jsupp₍_J₎})) ∨ γ_χ_{_Jsupp₍_J₎}(r) ≠ 1 and γ_χ_{_Jsupp₍_J₎}≠ 1. This implies that (μ_χ_{_Jsupp₍_J₎}○_α μ_χ_{_E}○_β μ_χ_{_Jsupp₍_J₎})(r) ≠ 0 and μ_χ_{_Jsupp₍_J₎}≠ 0. Similarly, (γ_χ_{_Jsupp₍_J₎}○_αγ_χ_{_E}○_β γ_χ_{_Jsupp₍_J₎})(r) ≠ 1 and γ_χ_{_Jsupp₍_J₎}≠ 1. Therefore, there exists an r ∈ E such that ((μ_χ_{_J}○_α μ_χ_{_E}χ_E ○_β μ_χ_{_J}))(r) ≠ 0 and ((γ_χ_{_J}○_α γ_χ_{_E}○_β γ_χ_{_J}))(r) ≠ 1. Thus, there exist j₁, j₂ ∈ E such that r = j₁αβj₂, μ_J (r) ≠ 0, γ_J (r) ≠ 1 and μ_J (k) ≠ 0, γ_J (k) ≠ 1. Hence, (μ_χ_{_J}○_α μ_χ_{_E}χ_E ○_β μ_χ_{_J}) ∧ μ_χ_{_J}≠ 0 and (γ_χ_{_J}○_α γ_χ_{_E}○_β γ_χ_{_J}) ∨ γ_χ_{_J}≠ 1. Therefore, J is a BF almost (α, β)-bi-ideal of S.

Definition 4.19

An IF almost (α, β)-bi-ideal J = (μ_J, γ_J ) of a Γ-semigroup E is minimal if for all IF almost (α, β)-bi-ideal K = (μ_K, γ_K) of E, where K ⊆ J, supp(K) = supp(J).

Theorem 4.20

Let J be a non-empty subset of a Γ-semigroup E. Then, J is a minimal almost (α, β)-bi-ideal if and only if χ_J = (μ_χ_{_J}, γ_χ_{_J}) is a minimal IF almost (α, β)-quasi-ideal of E.

Proof

Let us suppose that J is a minimal almost left (α, β)-quasi-ideal of E. Then, J is an almost (α, β)-bi-ideal of E. Thus, using Theorem 4.17, it can be inferred that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-quasi-ideal of S. Let K = (μ_K, γ_K) be an IF (α, β)-bi-ideal of E such that K ⊆ J. Then, using Theorem 4.18, we can conclude that supp(Q) is an almost (α, β)-quasi-ideal of E. Thus, supp(K) ⊆ supp(χ_J) = J. By assumption, supp(K) = K = supp(χ_J ). Therefore, χ_J = (μ_χ_{_J}, γ_χ_{_J}) is a minimal IF almost (α, β)-bi-ideal of E.

Conversely, let us suppose that χ_J = (μ_χ_{_J}, γ_χ_{_J}) is a minimal IF almost (α, β)-bi-ideal of E. Then, using Theorem 4.18, we can conclude that J is an almost α bi-ideal of E. Let K be an IF almost (α, β)-bi-ideal of E such that K ⊆ J. Then, using Theorem 4.18, it can be deduced that χ_K = (μ_χ_{_K}, γ_χ_{_K}) is an IF (α, β)-bi-ideal of E such that χ_K ⊆ χ_J. Thus, K = supp(χ_K) = supp(χ_J) = J. Therefore, J is a minimal almost (α, β)-bi-ideal of S.

5. Conclusion

Go to

Abstract
1. Introduction
2. Preliminaries
3. New Types of Intuitionistic Ideals
4. New Types of Intuitionistic Almost Ideals
5. Conclusion
Conflict of Interest
References
Biographies

In this study, we present the concepts of intuitionistic fuzzy ideals and almost ideals in Γ-semigroups and apply the minimal condition to Γ-semigroups. In our future course of study, we plan to extend the concepts presented here to algebraic systems including, hyper semigroups, IUP-algebras, and UP-algebras.

Conflict of Interest

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Abstract
1. Introduction
2. Preliminaries
3. New Types of Intuitionistic Ideals
4. New Types of Intuitionistic Almost Ideals
5. Conclusion
Conflict of Interest
References
Biographies

No potential conflict of interest relevant to this study was reported.

References

Go to

Abstract
1. Introduction
2. Preliminaries
3. New Types of Intuitionistic Ideals
4. New Types of Intuitionistic Almost Ideals
5. Conclusion
Conflict of Interest
References
Biographies

Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Sardar, SK, Majumder, SK, and Mandal, M. (2011) . Atanassov’s intuitionistic fuzzy ideals of gamma semigroups. [Online]. Available: https://arxiv.org/abs/1101.3080
Chinram, R (2016). On quasi-gamma-ideals in gamma-semigroups. Science Asia. 32, 351-353. https://doi.org/10.2306/scienceasia1513-1874.2006.32.351
Jana, C, Pal, M, and Saied, B (2017). (∊, ∊ ∨ q)-bipolar fuzzy BCK/BCI-algebras. Missouri Journal of Mathematical Sciences. 29, 139-160. https://doi.org/10.35834/mjms/1513306827
Jana, C, and Pal, M (2017). On (∊, ∊ ∨ qβ)-fuzzy soft BCI-algebras. Missouri Journal of Mathematical Sciences. 29, 197-215. https://doi.org/10.35834/mjms/1513306831
Jana, C, and Pal, M (2017). Generalized intuitionistic fuzzy ideals of BCK/BCI-algebras based on 3-valued logic and its computational study. Fuzzy Information and Engineering. 9, 455-478. https://doi.org/10.1016/j.fiae.2017.05.002
Jana, C, Senapati, T, and Pal, M (2016). (∊, ∊ ∨ q)-intuitionistic fuzzy BCI-subalgebras of a BCI-algebra. Journal of Intelligent & Fuzzy Systems. 31, 613-621. https://doi.org/10.3233/IFS-162175
Simuen, A, Iampan, A, and Chinram, R (2021). A novel of ideals and fuzzy ideals of gamma-semigroups. Journal of Mathematics. 2021. article no. 6638299
Mordeson, JN, Malik, DS, and Kuroki, N (2003). Fuzzy Semigroups. Heidelberg, Germany: Springer

Biographies

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Abstract
1. Introduction
2. Preliminaries
3. New Types of Intuitionistic Ideals
4. New Types of Intuitionistic Almost Ideals
5. Conclusion
Conflict of Interest
References
Biographies

Thiti Gaketem is a lecturer at School of Science University of Phayao, Thailand. He received his B.S. and M.S. and Ph.D. degrees in mathematics from Naresuan University, Thailand. His areas of interest include the algebraic theory of semigroups and fuzzy algebraic structures.

E-mail: josemour25@yahoo.com

Pannawit Khamrot is a lecturer at Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Phitsanulok, Phitsanulok. He received his B.Ed. in mathematics from Nakhon Sawan Rajabhat University and M.S. degrees in Applied Statistics and his Ph.D. in mathematics from Naresuan University, Thailand. His areas of interest include the algebraic theory of semigroups and bipolar fuzzy algebraic structures.

E-mail: pk g@rmutl.ac.th

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 135-143

Published online June 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.2.135

New Types of Intuitionistic Fuzzy Ideals in Γ-Semigroups

Thiti Gaketem¹ and Pannawit Khamrot²

Correspondence to:Thiti Gaketem (josemour25@yahoo.com)

Received: July 1, 2021; Revised: February 14, 2022; Accepted: March 29, 2022

Abstract

Keywords: Intuitionistic fuzzy ideals, Intuitionistic fuzzy almost ideals, Minimal intuitionistic fuzzy ideal, Minimal intuitionistic almost ideal

1. Introduction

In this study, we extend the concept of new fuzzy ideals to intuitionistic fuzzy ideals of Γ-semigroups and investigate the properties of the new types intuitionistic fuzzy ideals.

2. Preliminaries

With respect to the content of this paper, some basic definitions are provided below, which are important for the proper understanding of the theory presented in the next section.

Definition 2.1 [8]

Let E be a Γ-semigroup and J be a nonempty subset of E, for all e ∈ E and α, β ∈ Γ. Thus, the following definitions are applicable to J.

(1) A left (right) almost ideal of Γ-semigroup E is a non-empty set J such that (eΓJ) ∩ J ≠ ∅︀ ((JΓe) ∩ J ≠ ∅︀)
(2) An almost bi-ideal of Γ-semigroup E is a non-empty set J such that (JΓeΓJ) ∩ J ≠ ∅︀.
(3) An almost quasi-ideal of Γ-semigroup E is a non-empty set J such that (eΓJ ∩ JΓe) ∩ J ≠ ∅︀.
(4) A left α-ideal of Γ-semigroup J is a non-empty set J such that EαJ ⊆ J. A right α-ideal of a Γ-semigroup E is a non-empty set J such that JβE ⊆ J.
(5) An (α, β)-ideal of Γ-semigroup E is a non-empty set J such that it is both a left α-ideal and right β-ideal of E.

For any m_i ∈ [0, 1] and , we define:

\underset{i \in A}{\lor} m_{i} : sup_{i \in A} {m_{i}} and \underset{i \in A}{\land} m_{i} : = inf_{i \in A} {m_{i}} .

For any m, n ∈ [0, 1], we have

m \lor n = max {m, n} and m \land n = min {m, n} .

Definition 2.2 [1]

A fuzzy set ξ of a non-empty set J is a function ξ : J → [0, 1].

For any two fuzzy sets ξ and ς of a non-empty set J, we define ≥, =, ∧, and ∨ as follows:

(1) ξ ≥ ς ⇔ ξ(j) ≥ ς(j) for all j ∈ J.
(2) ξ = ς ⇔ ξ ≥ ς and ς ≥ ξ.
(3) (ξ ∧ ς)(j) = ξ(j) ∧ ς(j) for all j ∈ J.
(4) (ξ ∨ ς)(j) = ξ(k) ∨ ς(j) for all j ∈ J.

For any two fuzzy sets of ξ and ς of a non-empty of J, we define

ξ \subseteq ς if ξ (u) \leq ς (u), (ξ \cup ς) (u) = ξ (u) \land ς (u), and (ξ \cap ς) (u) = ξ (u) \lor ς (u) for all u \in J .

For any element j in a semigroup E, we define the set F_j as F_j := {(m, n) ∈ E × E | j = mn}.

For two fuzzy sets ξ and ς on a semigroup E, we define the product ξ ○ ς for all j ∈ E as follows:

(ξ \circ ς) (j) = {\begin{array}{l} \underset{(m, n) \in F_{j}}{⋁} {ξ (m) \land ς (n) ∣ (m, n) \in F_{j}}, & if F_{j} \neq \emptyset, \\ 0 & if F_{j} = \emptyset . \end{array}

The following conditions define the types of fuzzy almost ideal on semigroups.

Definition 2.3 [8]

A fuzzy set ξ of a semigroup J is considered to be:

(1) a fuzzy almost left (right) ideal of S if ℰ ○ ξ ∩ ξ ≠ 0 (ξ ○ ℰ ∩ξ ≠ 0).
(2) a fuzzy almost ideal of E if it is both a fuzzy almost left and fuzzy almost right ideal of E,
(3) a fuzzy almost bi-ideal of E if ξ ○ ℰ ○ ξ ∩ ξ ≠ 0.
(4) a fuzzy almost quasi-ideal of E if (ℰ ○ ξ ∩ ξ ○ ℰ) ∩ ξ ≠ 0.

The following conditions define the types of fuzzy sub-semigroups on Γ-semigroups.

Definition 2.4 [9]

A fuzzy set ξ of a Γ-semigroup E is said to be

(1) A fuzzy sub-semigroup of E if ξ(uγv) ≥ ξ(u) ∧ ξ(v) for all u, v ∈ E and γ ∈ Γ.
(2) A fuzzy left (right) ideal of E if ξ(uγv) ≥ ξ(v) (ξ(uγv) ≥ ξ(u)) for all u, v ∈ E and γ ∈ Γ.
(3) A fuzzy ideal of E if it is both a fuzzy left and fuzzy right ideal of E.
(4) A fuzzy bi-ideal of E if ξ is a fuzzy sub-semigroup of E and ξ(uγvβw) ≥ ξ(u) ∧ ξ(w) for all u, v,w ∈ E and γ, β ∈ Γ.

Now, we review the definition of the intuitionistic fuzzy sets and their basic properties, which are discussed in the following section.

Definition 2.5 [2]

Let μ and γ be fuzzy sets. An intuitionistic fuzzy (IF ) set on a non-empty set E is a function

G : = {(j, μ (j), γ (j)) | j \in E}

which satisfies 0 ≤ μ(j) + γ(j) ≤ 1.

For IF sets J₁ = (μ_J_₁, γ_J_₁) and J₂ = (μ_J_₂, γ_J_₂), we define the following relations:

(1) J₁ ⊆ J₂ if and only if μ_J_₁(j) ≤ μ_J_₂(j) and γ_J_₁(j) ≥ γ_J_₂(j).
(2) J₁ = J₂ if and only if J₁ ⊆ J₂ and J₂ ⊆ J₁.
(3) J₁ ∩ J₂ = {(μ_J_₁∩ μ_J_₂)(j), (γ_J_₁∪ γ_J_₂)(j) | j ∈ E}.
(4) J₁ ∪ J₂ = {(μ_J_₁∪ μ_J_₂)(j), (γ_J_₁∩ γ_J_₂)(j) | j ∈ E}.

For k ∈ E, we define F_k = {(y, z) ∈ E × E | k = yz}.

For IF sets J₁ = (μ_J_₁, γ_J_₁) and J₂ = (μ_J_₂, γ_J_₂), we define the product J₁ ○ J₂ = (μ_J_₁○_J_₂, γ_J_₁○_J_₂) as follows:

(μ_{J_{1}} \circ μ_{J_{2}}) (j) = {\begin{array}{l} \underset{(m, n) \in F_{j}}{\lor} {μ_{J_{1}} (m) \land μ_{J_{2}} (n)}, & if F_{k} \neq \emptyset, \\ 0, & if otherwise. \end{array}

and

(γ_{J_{1}} \circ γ_{J_{2}}) (j) = {\begin{array}{l} \underset{(m, n) \in F_{k}}{\land} {γ_{J_{1}} (m) \lor γ_{J_{2}} (n)}, & if F_{k} \neq \emptyset, \\ 0, & if otherwise. \end{array}

For any element j in a Γ-semigroup E, we define the set F_j_{_α}as F_j_{_α}:= {(m, α, n) ∈ E × Γ × E | j = mαn}.

For the IF sets J₁ = (μ_J_₁, γ_J_₁) and J₂ = (μ_J_₂, γ_J_₂), we define the product ξ ○_α ς for all k ∈ S as follows:

(μ_{J_{1}} \circ_{α} μ_{J_{2}}) (j) = {\begin{array}{l} \underset{(m, α, n) \in F_{j_{α}}}{⋁} {μ_{J_{1}} (m) \land μ_{J_{2}} (n)}, & if F_{j_{α}} \neq \emptyset, \\ 0, & if otherwise. \end{array}

and

(γ_{J_{1}} \circ_{α} γ_{J_{2}}) (j) = {\begin{array}{l} \underset{(m, α, n) \in F_{j_{α}}}{⋀} {γ_{J_{1}} (m) \lor γ_{J_{2}} (n)}, & if F_{j_{α}} \neq \emptyset, \\ 0, & if otherwise. \end{array}

Definition 2.6

Let J be a non-empty set of a semigroup E. An intuitionistic characteristic function is defined as χ_J = (μ_χ_{_J}, γ_χ_{_J}), where

μ_{χ_{J}} (j) : = {\begin{array}{l} 1, & j \in J, \\ 0, & j \notin j, \end{array}

and

γ_{χ_{J}} (j) : = {\begin{array}{l} 0, & j \in J, \\ 1, & j \notin J . \end{array}

Remark 2.7

For the sake of simplicity, we use the symbol χ_E = (μ_χ_{_E}, γ_χ_{_E}) for the IF set χ_E := {(u, μ_χ_{_E}(u), γ_χ_{_E}(u)) | u ∈ E}.

Definition 2.8 [2]

An IF set J = (μ_J, γ_J ) on a Γ-semigroup E is defined as an:

(1) IF subsemigroup on E if it satisfies μ_J (uαv) ≥ μ_J (u) ∧ μ_J (v) and γ_J (uαv) ≤ γ_J (u) ∨ γ_J (v)
(2) IF left ideal on E if it satisfies μ_J (uαv) ≥ μ_J (v) and γ_J (uαv) ≤ γ_J (v)
(3) IF right ideal on E if it satisfies μ_J (uαv) ≥ μ_J (u) and γ_J (uαv) ≤ γ_J (u)
(4) IF ideal on E if it is both intuitionsitic fuzzy left ideal and right ideal of E

for all u, v ∈ E and α ∈ Γ.

3. New Types of Intuitionistic Ideals

In this section, we define the bipolar fuzzy (α, β)-ideal and study its basic properties.

Definition 3.1

Let J = (μ_J, γ_J ) be an IF set on a Γ-semigroup E and α, β ∈ Γ. Then J = (μ_J, γ_J ) is defined as:

(1) An IF left α-ideal of E if μ_J (uαv) ≥ μ_J (v) and γ_J (uαv) ≤ γ_J (v) for all u, v ∈ E.
(2) An IF right β-ideal of E if μ_J (uβv) ≥ μ_J (v) and γ_J (uβv) ≤ γ_J (v) for all u, v ∈ E.
(3) An IF (α, β)-ideal of E if it is both an IF left α-ideal and an IF right β-ideal of E.
(4) An IF α-ideal of E if it is an IF (α, α)-ideal of E.

Theorem 3.2

Proof

Let us suppose that J is a left α-ideal of E and u, v ∈ E.

If v ∈ J, then uαv ∈ J. Thus, μ_χ_{_J}(v) = μ_χ_{_J}(uαv) = 1 and

γ_χ_{_J}(v) = γ_χ_{_J}(uαv) = 0. Hence, μ_χ_{_J}(uαv) ≥ μ_χ_{_J}(v) and γ_χ_{_J}(uαv) ≤ γ_χ_{_J}(v).

Theorem 3.3

The intersection and union of any two IF left α-ideals (right β-ideals, (α, β)-ideals) of a Γ-semigroup E is an IF left α-ideal (right β-ideal, (α, β)-ideal) of E.

Proof

Let J₁ = (μ_J_₁, γ_J_₁) and J₂ = (μ_J_₂, γ_J_₂) be IF left α-ideals of E and let u, v ∈ E. Then,

(μ_{J_{1}} \cup μ_{J_{2}}) (u α v) = μ_{J_{1}} (u α v) \land μ_{J_{2}} (u α v) \geq μ_{J_{1}} (v) \land μ_{J_{2}} (v),

and

(γ_{J_{1}} \cap γ_{J_{2}}) (u α v) = γ_{J_{1}} (u α v) \lor γ_{J_{2}} (u α v) \leq γ_{J_{1}} (v) \lor γ_{J_{2}} (v) .

Thus, J₁ ∩ J₂ is an IF left α-ideal of E. Similarly,

(μ_{J_{1}} \cup μ_{J_{2}}) (u α v) = μ_{J_{1}} (u α v) \lor μ_{J_{2}} (u α v) \geq μ_{J_{1}} (v) \lor μ_{J_{2}} (v),

and

(γ_{J_{1}} \cup γ_{J_{2}}) (u α v) = γ_{J_{1}} (u α v) \land γ_{J_{2}} (u α v) \leq γ_{J_{1}} (v) \land γ_{J_{2}} (v) .

Thus, J₁ ∪ J₂ is an IF left α-ideal of E.

Next, we provide the definitions for IF (α, β)-bi-ideals and study their basic properties.

Definition 3.4

Theorem 3.5

Let J be a non-empty subset of Γ-semigroup E. Then, J is an (α, β)-bi-ideal of E if and only if the characteristic function χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-bi-ideal of E.

Proof

Let us suppose that J is an (α, β)-bi-ideal of E and JαEβJ ⊆ J.

Therefore, χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-bi-ideal of E.

Theorem 3.6

The intersection and union of any two IF (α, β)-bi-ideals of a Γ-semigroup E is an (α, β)-bi-ideal of E.

Proof

Let J₁ = (μ_J_₁, γ_J_₁) and J₂ = (μ_J_₂, γ_J_₂) be (α, β)-bi-ideals of E and u ∈ E. Then,

\begin{array}{l} ((μ_{J_{1}} \cap μ_{J_{2}}) \circ_{α} μ_{χ_{E}} \circ_{β} (μ_{J_{1}} \cap μ_{J_{2}})) (u) \\ \geq (μ_{J_{1}} \circ_{α} μ_{χ_{E}} \circ_{β} μ_{J_{1}}) (u) \land (μ_{J_{2}} \circ_{α} μ_{χ_{J}} \circ_{β} μ_{J_{2}}) (u) \\ \geq (μ_{J_{1}} \cap μ_{J_{2}}) (u), \end{array}

and

\begin{array}{l} ((γ_{J_{1}} \cap γ_{J_{2}}) \circ_{α} γ_{χ_{E}} \circ_{β} (γ_{J_{1}} \cap γ_{J_{2}})) (u) \\ \leq γ_{J_{1}} \circ_{α} (γ_{χ_{E}} \circ_{β} γ_{J_{1}}) (u) \lor (γ_{J_{2}} γ_{J_{2}} γ_{J_{2}}) (u) \\ \leq (γ_{J_{1}} \cap γ_{J_{2}}) (u) . \end{array}

Thus, J₁ ∩ J₂ is an IF α-bi-ideal of S.

Next, we define conditions for IF (α, β)-quasi-ideal and study its basic properties.

Definition 3.7

Theorem 3.8

If L = (μ_L, γ_L) and R = (μ_R, γ_R) are IF left and right α-ideals of a Γ-semigroup E, respectively, then L ∩ R is an IF α-quasi-ideal of E.

Proof

\begin{array}{l} μ_{χ_{E}} \circ_{α} (μ_{L} \cap μ_{R}) \cap (μ_{L} \cap μ_{R}) \circ_{α} μ_{χ_{E}} \\ \subseteq μ_{χ_{E}} \circ_{α} (μ_{L} \cap μ_{R}) \circ_{α} μ_{χ_{E}} \subseteq μ_{L} \cap μ_{R}, \end{array}

and

\begin{array}{l} γ_{χ_{E}} \circ_{α} (γ_{L} \cap γ_{R}) \cap (γ_{L} \cap γ_{R}) \circ_{α} γ_{χ_{E}} \\ \supseteq γ_{χ_{E}} \circ_{α} (γ_{L} \cup γ_{R}) \circ_{α} γ_{χ_{E}} \supseteq γ_{L} \cup γ_{R} . \end{array}

Thus, L ∩ R is an IF α-quasi-ideal of S.

Theorem 3.9

Every IF (α, β)-quasi-ideal of a Γ-semigroup E is an intersection of an IF left α-ideal and IF right β-ideal of S

Proof

\begin{array}{l} μ_{χ_{E}} \circ_{α} μ_{L} = μ_{χ_{E}} \circ_{α} (μ_{Q} \cup (μ_{χ_{E}} \circ_{α} μ_{Q})) \\ = (μ_{χ_{E}} \circ_{α} μ_{Q}) \cup (μ_{χ_{E}} \circ_{α} (μ_{χ_{E}} \circ_{α} μ_{Q})) \\ = (μ_{χ_{E}} \circ_{α} μ_{Q}) \cup ((μ_{χ_{E}} \circ_{α} μ_{χ_{E}}) \circ_{α} μ_{Q}) \\ = (μ_{χ_{E}} \circ_{α} μ_{Q}) \cup (μ_{χ_{E}} \circ_{α} μ_{Q}) \\ \subseteq μ_{Q} \cup (μ_{χ_{E}} \circ_{α} μ_{Q}) = μ_{L} . \end{array}

Additionally, we obtain:

\begin{array}{l} μ_{R} \circ_{β} μ_{χ_{E}} = (μ_{Q} \cup (μ_{Q} \circ_{β} μ_{χ_{E}})) \circ_{β} μ_{χ_{E}} \\ = (μ_{Q} \circ_{α} μ_{χ_{E}}) \cup ((μ_{Q} \circ_{β} μ_{χ_{E}}) \circ_{α} μ_{χ_{E}}) \\ = (μ_{Q} \circ_{α} μ_{χ_{E}}) \cup (μ_{Q} \circ_{β} (μ_{χ_{E}} \circ_{α} μ_{χ_{E}})) \\ = (μ_{Q} \circ_{α} μ_{χ_{E}}) \cup (μ_{Q} \circ_{β} μ_{χ_{E}}) \\ \subseteq μ_{Q} \cup (μ_{Q} \circ_{β} μ_{χ_{E}}) = μ_{R} . \end{array}

\begin{array}{l} μ_{Q} \subseteq (μ_{Q} \cup (μ_{χ_{E}} \circ_{α} μ_{Q})) \cap (μ_{Q} \cup (μ_{Q} \circ_{β} μ_{χ_{E}})) \\ = μ_{L} \cap μ_{R}, \end{array}

and

\begin{array}{l} μ_{L} \cap μ_{R} = (μ_{Q} \cup (μ_{χ_{E}} \circ_{α} μ_{Q})) \cap (μ_{Q} \cup (μ_{Q} \circ_{β} μ_{χ_{E}})) \\ = μ_{Q} \cap ((μ_{χ_{E}} \circ_{α} μ_{Q}) \cup (μ_{Q} \circ_{β} μ_{χ_{E}})) \\ \subseteq μ_{Q} \cap μ_{Q} = μ_{Q} . \end{array}

Hence, μ_Q = μ_L ∩ μ_R. Similarly, we can show that γ_Q = γ_L ∩ γ_R.

Theorem 3.10

Let J be a non-empty subset of a Γ-semigroup E. Then, J is an (α, β)-quasi-ideal of E if and only if the characteristic function χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-quasi-ideal of S.

Proof

Let us suppose that J is an (α, β)-quasi-ideal of S and u ∈ E.

Therefore, χ_J = (μ_χ_{_J}, γ_χ_{_J}) is an IF (α, β)-quasi-ideal of E.

4. New Types of Intuitionistic Almost Ideals

Definition 4.1

Let J = (μ_J, γ_J ) be an IF set of a Γ-semigroup E and α, β ∈ Γ, then is said to be

(1) An IF almost left α-ideal of E if (μ_χ_{_e}○_α μ_J ) ∧ μ_J ≠ 0 and (γ_χ_{_e}○_α γ_J ) ∨ γ_J ≠ 1 for all e ∈ E.
(2) An IF almost right β-ideal of E if (μ_χ_{_e}○_β μ_J ) ∧ μ_J ≠ 0 and (γ_χ_{_e}○_β γ_J ) ∨ γ_J ≠ 1 for all e ∈ E.
(3) An IF almost (α, β)-ideal of E if it is both an IF almost left α-ideal and an IF almost right β-ideal of E.

Here, χ_e = (μ_χ_{_e}, γ_χ_{_e}) is an IF set of E mapping every element of E to e.

Theorem 4.2

Proof

Theorem 4.3

Proof

Next, we review the definition of supp(J) and study the relation between supp(J) and IF almost left α-ideal (right β-ideal, (α, β)-ideal) of Γ-semigroups.

Let J = (μ_J, γ_J ) be an IF set of a non-empty of E. Then the support of J is determined rather than supp(J) = {u ∈ E | J(u) ≠ 0}, where μ_J (u) ≠ 0 and γ_J (u) ≠ 1 for all u ∈ E.

Theorem 4.4

Proof

Definition 4.5

An ideal I of a Γ-semigroup S is called a minimal if for every ideal of J of S, for which J ⊆ I, we obtain J = I.

Definition 4.6

Theorem 4.7

Proof

Next, we provide the definitions for IF almost (α, β)-quasi-ideals and study their properties.

Definition 4.8

Theorem 4.9

If J = (μ_J, γ_J ) is an IF almost (α, β)-quasi-ideal of a Γ-semigroup E and K = (μ_K, γ_K) is an IF set of E such that J ⊆ K, then K = (μ_K, γ_K) is an IF (α, β)-quasi-ideal of E.

Proof

Theorem 4.10

Proof

In the following section, we study the properties and relationship between supp(J) and IF almost (α, β)-quasi-ideal of Γ-semigroups.

Theorem 4.11

Let J = (μ_J, γ_J ) be an IF set of a non-empty Γ-semigroup E. Then, J = (μ_J, γ_J ) is an IF almost (α, β)-quasi-ideal of E if and only if supp(J) is an almost (α, β)-quasi-ideal of E.

Proof

Definition 4.12

An almost ideal I of a Γ-semigroup S is called a minimal if for every almost ideal of J of S, where J ⊆ I, we have J = I.

Definition 4.13

An IF almost (α, β)-quasi-ideal J = (μ_J, γ_J ) of a Γ-semigroup E is minimal if for all IF almost (α, β)-quasi-ideal K = (μ_K, γ_K) of E, where K ⊆ J, supp(K) = supp(J).

Theorem 4.14

Let J be a non-empty subset of a Γ-semigroup E. Then J is a minimal almost (α, β)-quasi-ideal if and only if χ_J = (μ_χ_{_J}, γ_χ_{_J}) is a minimal IF almost (α, β)-quasi-ideal of E.

Proof

Next, we present the definitions of IF almost (α, β)-bi-ideals and study their properties.

Definition 4.15

Theorem 4.16

If J = (μ_J, γ_J ) is an IF almost (α, β)-bi-ideal of a Γ-semigroup E and K = (μ_K, γ_K) is an IF set of E such that J ⊆ K, then K = (μ_K, γ_K) is an IF (α, β)-bi-ideal of E.

Proof

Theorem 4.17

Proof

In the next section, we study properties and relationship between supp(ξ) and IF almost (α, β)-bi-ideal of Γ-semigroups.

Theorem 4.18

Let J = (μ_J, γ_J ) be an IF of a non-empty oΓ-semigroup E. Then, J = (μ_J, γ_J ) is an IF almost (α, β)-bi-ideal of S if and only if supp(J) is an almost (α, β)-bi-ideal of E.

Proof

Definition 4.19

An IF almost (α, β)-bi-ideal J = (μ_J, γ_J ) of a Γ-semigroup E is minimal if for all IF almost (α, β)-bi-ideal K = (μ_K, γ_K) of E, where K ⊆ J, supp(K) = supp(J).

Theorem 4.20

Let J be a non-empty subset of a Γ-semigroup E. Then, J is a minimal almost (α, β)-bi-ideal if and only if χ_J = (μ_χ_{_J}, γ_χ_{_J}) is a minimal IF almost (α, β)-quasi-ideal of E.

Proof

5. Conclusion

Abstract
1. Introduction
2. Preliminaries
3. New Types of Intuitionistic Ideals
4. New Types of Intuitionistic Almost Ideals
5. Conclusion

References

Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Sardar, SK, Majumder, SK, and Mandal, M. (2011) . Atanassov’s intuitionistic fuzzy ideals of gamma semigroups. [Online]. Available: https://arxiv.org/abs/1101.3080
Chinram, R (2016). On quasi-gamma-ideals in gamma-semigroups. Science Asia. 32, 351-353. https://doi.org/10.2306/scienceasia1513-1874.2006.32.351
Jana, C, Pal, M, and Saied, B (2017). (∊, ∊ ∨ q)-bipolar fuzzy BCK/BCI-algebras. Missouri Journal of Mathematical Sciences. 29, 139-160. https://doi.org/10.35834/mjms/1513306827
Jana, C, and Pal, M (2017). On (∊, ∊ ∨ qβ)-fuzzy soft BCI-algebras. Missouri Journal of Mathematical Sciences. 29, 197-215. https://doi.org/10.35834/mjms/1513306831
Jana, C, and Pal, M (2017). Generalized intuitionistic fuzzy ideals of BCK/BCI-algebras based on 3-valued logic and its computational study. Fuzzy Information and Engineering. 9, 455-478. https://doi.org/10.1016/j.fiae.2017.05.002
Jana, C, Senapati, T, and Pal, M (2016). (∊, ∊ ∨ q)-intuitionistic fuzzy BCI-subalgebras of a BCI-algebra. Journal of Intelligent & Fuzzy Systems. 31, 613-621. https://doi.org/10.3233/IFS-162175
Simuen, A, Iampan, A, and Chinram, R (2021). A novel of ideals and fuzzy ideals of gamma-semigroups. Journal of Mathematics. 2021. article no. 6638299
Mordeson, JN, Malik, DS, and Kuroki, N (2003). Fuzzy Semigroups. Heidelberg, Germany: Springer

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International Journalof Fuzzy Logic andIntelligent Systems

Original Article

New Types of Intuitionistic Fuzzy Ideals in Γ-Semigroups

Abstract

1. Introduction

2. Preliminaries

Definition 2.1 [8]

Definition 2.2 [1]

Definition 2.3 [8]

Definition 2.4 [9]

Definition 2.5 [2]

Definition 2.6

Remark 2.7

Definition 2.8 [2]

3. New Types of Intuitionistic Ideals

Definition 3.1

Theorem 3.2

Theorem 3.3

Definition 3.4

Theorem 3.5

Theorem 3.6

Definition 3.7

Theorem 3.8

Theorem 3.9

Theorem 3.10

4. New Types of Intuitionistic Almost Ideals

Definition 4.1

Theorem 4.2

Theorem 4.3

Theorem 4.4

Definition 4.5

Definition 4.6

Theorem 4.7

Definition 4.8

Theorem 4.9

Theorem 4.10

Theorem 4.11

Definition 4.12

Definition 4.13

Theorem 4.14

Definition 4.15

Theorem 4.16

Theorem 4.17

Theorem 4.18

Definition 4.19

Theorem 4.20

5. Conclusion

Conflict of Interest

References

Biographies

Article

Original Article

New Types of Intuitionistic Fuzzy Ideals in Γ-Semigroups

Abstract

1. Introduction

2. Preliminaries

Definition 2.1 [8]

Definition 2.2 [1]

Definition 2.3 [8]

Definition 2.4 [9]

Definition 2.5 [2]

Definition 2.6

Remark 2.7

Definition 2.8 [2]

3. New Types of Intuitionistic Ideals

Definition 3.1

Theorem 3.2

Theorem 3.3

Definition 3.4

Theorem 3.5

Theorem 3.6

Definition 3.7

Theorem 3.8

Theorem 3.9

Theorem 3.10

4. New Types of Intuitionistic Almost Ideals

Definition 4.1

Theorem 4.2

Theorem 4.3

Theorem 4.4

International Journal
of Fuzzy Logic and
Intelligent Systems