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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

© The Korean Institute of Intelligent Systems

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.

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

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 [47]. 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.

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 JΓJJ. A left (right) ideal of a Γ-semigroup E is a non-empty set J of E such that EΓJJ(JΓEJ). 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 JΓEEΓJJ. A sub-Γ-semigroup K of a Γ-semigroup E is called a bi-ideal of E if JΓEΓJJ.

Definition 2.1 [8]

Let E be a Γ-semigroup and J be a nonempty subset of E, for all eE 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 (eΓJJΓe) ∩ J ≠ ∅︀.

• (4) A left α-ideal of Γ-semigroup J is a non-empty set J such that EαJJ. A right α-ideal of a Γ-semigroup E is a non-empty set J such that JβEJ.

• (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 mi ∈ [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) ξςξ(j) ≥ ς(j) for all jJ.

• (2) ξ = ςξς and ςξ.

• (3) (ξς)(j) = ξ(j) ∧ ς(j) for all jJ.

• (4) (ξς)(j) = ξ(k) ∨ ς(j) for all jJ.

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 Fj as Fj := {(m, n) ∈ E × E | j = mn}.

For two fuzzy sets ξ and ς on a semigroup E, we define the product ξς for all jE 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 ○ ξξ ≠ 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 ○ ξξ ≠ 0.

• (4) a fuzzy almost quasi-ideal of E if (ℰ ○ ξξ ○ 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, vE and γ ∈ Γ.

• (2) A fuzzy left (right) ideal of E if ξ(uγv) ≥ ξ(v) (ξ(uγv) ≥ ξ(u)) for all u, vE 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,wE 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 = (μJ1, γJ1) and J2 = (μJ2, γJ2), we define the following relations:

• (1) J1J2 if and only if μJ1(j) ≤ μJ2(j) and γJ1(j) ≥ γJ2(j).

• (2) J1 = J2 if and only if J1J2 and J2J1.

• (3) J1J2 = {(μJ1μJ2)(j), (γJ1γJ2)(j) | jE}.

• (4) J1J2 = {(μJ1μJ2)(j), (γJ1γJ2)(j) | jE}.

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

For IF sets J1 = (μJ1, γJ1) and J2 = (μJ2, γJ2), we define the product J1J2 = (μJ1J2, γJ1J2) 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 Fjαas Fjα:= {(m, α, n) ∈ E × Γ × E | j = mαn}.

For the IF sets J1 = (μJ1, γJ1) and J2 = (μJ2, γJ2), we define the product ξα ς for all kS 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 χJ = (μχJ, γχJ), 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 χE = (μχE, γχE) for the IF set χE := {(u, μχE(u), γχE(u)) | uE}.

Definition 2.8 [2]

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

• (1) IF subsemigroup on E if it satisfies μJ (uαv) ≥ μJ (u) ∧ μJ (v) and γJ (uαv) ≤ γJ (u) ∨ γJ (v)

• (2) IF left ideal on E if it satisfies μJ (uαv) ≥ μJ (v) and γJ (uαv) ≤ γJ (v)

• (3) IF right ideal on E if it satisfies μJ (uαv) ≥ μJ (u) and γJ (uαv) ≤ γJ (u)

• (4) IF ideal on E if it is both intuitionsitic fuzzy left ideal and right ideal of E

for all u, vE 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 = (μJ, γJ ) be an IF set on a Γ-semigroup E and α, β ∈ Γ. Then J = (μJ, γJ ) is defined as:

• (1) An IF left α-ideal of E if μJ (uαv) ≥ μJ (v) and γJ (uαv) ≤ γJ (v) for all u, vE.

• (2) An IF right β-ideal of E if μJ (uβv) ≥ μJ (v) and γJ (uβv) ≤ γJ (v) for all u, vE.

• (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 χJ = (μχJ, γχJ) 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, vE.

If vJ, then uαvJ. Thus, μχJ(v) = μχJ(uαv) = 1 and

γχJ(v) = γχJ(uαv) = 0. Hence, μχJ(uαv) ≥ μχJ(v) and γχJ(uαv) ≤ γχJ(v).

If vJ, then uαvJ. Thus, μχJ(v) = 0, γχJ(v) = 1 and μχJ(uαv) = 1, γχJ(uαv) = 0. Hence, μχJ(uαv) ≥ μχJ(v) and γχJ(uαv) ≤ γχJ(v). Therefore, χJ = (μχJ, γχJ) is an IF left α-ideal of S.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF left α-ideal of E and u, vE with vJ. Then, μχJ(v) = μχJ(uαv) = 1 and γχJ(v) = γχJ(uαv) = 0. By assumption, μχJ(uαv) ≥ μχJ(v) and γχJ(uαv) ≤ γχJ(v). Thus, uαvJ. 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 = (μJ1, γJ1) and J2 = (μJ2, γJ2) be IF left α-ideals of E and let u, vE. 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, J1J2 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, J1J2 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 = (μJ, γJ ) be an IF set of a Γ-semigroup E and α, β ∈ Γ. Then, J = (μJ, γJ ) is called an IF (α, β)-bi-ideal of E if μJα μχEβ μJμJ and γJα γχEβ γJγJ, where χE = (μχE, γχE) 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 χJ = (μχJ, γχJ) is an IF (α, β)-bi-ideal of E.

Proof

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

If uJαEβJ, then μJ (u) = (μJα μχJβ μJ )(u) = 1 and γJ (u) = (γJα γχJβ γJ )(u) = 0. Hence, (μJα μχJβ μJ )(u) ≥ μJ (u) and (γJα γχJβ γJ )(u) ≤ γJ (u).

If uJαEβJ, then μJ (u) = 0, (μJα μχJβ μJ )(u) = 1 and γJ (u) = 1, (γJα γχJβ γJ )(u) = 0. Hence, (μJα μχJβ μJ )(u) ≥ μJ (u) and (γJα γχJβ γJ )(u) ≤ γJ (u).

Therefore, χJ = (μχJ, γχJ) is an IF (α, β)-bi-ideal of E.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF (α, β)-bi-ideal of E and uJαEβJ. Then, (μJαμχJβμJ )(u) = 1 and (γJα γχJβ γJ )(u) = 0. By assumption, (μJα μχJβ μJ )(u) ≥ μJ (u) and (γJα γχJβ γJ )(u) ≤ γJ (u). Thus, uJ. 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 = (μJ1, γJ1) and J2 = (μJ2, γJ2) be (α, β)-bi-ideals of E and uE. 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, J1J2 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 = (μJ, γJ ) be an IF set of a Γ-semigroup E and α, β ∈ Γ. Then, J = (μJ, γJ ) is called an IF (α, β)-quasi-ideal of E if μχEα μJμJβ μχEμJ and γχEα γJγJβ γχEγJ.

Theorem 3.8

If L = (μL, γL) and R = (μR, γR) are IF left and right α-ideals of a Γ-semigroup E, respectively, then LR is an IF α-quasi-ideal of E.

Proof

Let L = (μL, γL) and R = (μR, γR) be IF left and right α-ideals of E, respectively. Then, μLαμRμχEαμLμL and μLα μRμRα μχEμR. Similarly, γLα γRγχEα γLγL and γLα γRγRα γχEγR Thus, μLα μRμLμR and γLα γRγLγR. 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, LR 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 = (μQ, γQ) be an IF (α, β)-quasi-ideal of E. Considering μL = μQ ∪ (μχEα μQ) and γL = γQ ∪ (γχEα γQ), where L = (μL, γL) μR = μQ ∪ (μQβ μχE) and γR = γQ ∪ (γQβ γχE), where R = (μR, γR), we obtain:

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

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

Similarly, we can show that: γχEα γLγL and γRβ γχEγR. Thus, L = (μL, γL) and R = (μR, γR) 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, μQ = μLμR. Similarly, we can show that γQ = γLγR.

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 χJ = (μχJ, γχJ) is an IF (α, β)-quasi-ideal of S.

Proof

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

If u ∈ (EαJ) ∩ (JβE), then uJ. Thus, μχJ(u) = 1 and γχJ(u) = 0. Hence, ((μχJα μχE) ∩ (μχJβ μχE))(u) ≤ μχJ(u) and ((γχJα γχE) ∪ (γχEβ γχJ))(u) ≥ γχJ(u).

If u ∉ (EαJ) ∩ (JβE), then μχJ(u) = 0 and γχJ(u) = 1. Hence, ((μχJα μχE) ∩ (μχJβ μχE))(u) ≤ μχJ(u) and ((γχJα γχE) ∪ (γχEβ γχJ))(u) ≥ γχJ(u).

Therefore, χJ = (μχJ, γχJ) is an IF (α, β)-quasi-ideal of E.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF (α, β)-quasi-ideal of S and u ∈ (EαJ)∩(JβE), we obtain: ((μχJα μχE) ∩ (μχJβ μχE))(u) = 1, and ((γχJα γχE) ∪ (γχEβ γχJ))(u) = 0. By assumption, ((μχJα μχE) ∩ (μχJβ μχE))(u) ≤ μχJ(u), and ((γχJα γχE)∪(γχEβ γχJ))(u) ≥ γχJ(u). Thus, uJ. Hence, J is an (α, β)-quasi-ideal of E.

Definition 4.1

Let J = (μJ, γJ ) 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 (μχeα μJ ) ∧ μJ ≠ 0 and (γχeα γJ ) ∨ γJ ≠ 1 for all eE.

• (2) An IF almost right β-ideal of E if (μχeβ μJ ) ∧ μJ ≠ 0 and (γχeβ γJ ) ∨ γJ ≠ 1 for all eE.

• (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, χe = (μχe, γχe) is an IF set of E mapping every element of E to e.

Theorem 4.2

If J = (μJ, γJ ) is an IF almost left α-ideal (right β-ideal, (α, β)-ideal) of a Γ-semigroup E and K = (μK, γK) is an IF set of E such that JK, then K = (μK, γK) is an IF almost left α-ideal (right β-ideal, (α, β)-ideal) of E.

Proof

Let us suppose that J = (μJ, γJ ) is an IF almost left α-ideal of E and K = (μK, γK) is an IF set of E such that JK. Then, (μχeαμJ )∧μJ ≠ 1and (γχeα γJ )∨γJ ≠ 0. Thus (μχeα μJ ) ∧ μJ ⊆ (μχeα μK) ∧ μK ≠ 0 and (γχeα γJ ) ∨ γJ ⊆ (γχeα γK) ∨ γK ≠ 1. Hence, K = (μK, γK) 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 χJ = (μχJ, γχJ) 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, uαJJ ≠ ∅︀ for all uE. Thus, there exist vuαJ and vJ such thato (μχEα μχJ)(v) = μχJ(v) = 1 and (γχeαγχJ)(v) = γχJ(v) = 0. Hence, (μχeαμχJ)∧μχJ≠ 0 and (γχEα γχJ) ∨ γχJ≠ 1. Therefore, χJ = (μχJ, γχJ) is an IF almost left α-ideal of S.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF almost left α-ideal of S and uS, we obtain: (μχeαμχJ)∧μχJ≠ 0 and (γχeα γχJ) ∨ γχJ≠ 1. Thus, there exists an rS such that ((μχeαμχJ)∧μχJ)(r) ≠ 0 and ((γχeαγχJ)∨γχJ)(r) ≠ 1. Hence, ruαJJ Implies that uαJJ ≠ ∅︀. 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 = (μJ, γJ ) be an IF set of a non-empty of E. Then the support of J is determined rather than supp(J) = {uE | J(u) ≠ 0}, where μJ (u) ≠ 0 and γJ (u) ≠ 1 for all uE.

Theorem 4.4

Let J = (μJ, γJ ) be an IF set of a non-empty Γ-semigroup E. Then, J = (μJ, γJ ) 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 = (μJ, γJ ) be an IF almost left α-ideal of S and uE. Then, (μχeαμχJ)∧μχJ≠ 0 and (γχeαγχJ)∨γχJ≠ 1. Thus, there exists an rE such that ((μχeα μχJ) ∧ μχJ)(r) ≠ 0 and ((γχeα γχJ) ∨ γχJ)(r) ≠ 1. Hence, there exists a kE such that r = uαJ, μχJ(r) ≠ 0, γχJ(r) ≠ 1, and μχJ(k) ≠ 0, γχJ(k) ≠ 1. This implies that r, k ∈ supp(J). Thus, (μχeα μχsupp(J))(r) ≠ 0, (γχeα γχsupp(J))(r) ≠ 1, and μχsupp(J)≠ 0, γχsupp(J)≠ 1. Hence, (μχeα μχsupp(J)) ∧ μχJsupp(J)≠ 0 and (γχeα γχsupp(J))∨γχJsupp(J)≠ 1. Therefore, χsupp(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 χJsupp(J)is an IF almost left α-ideal of E. Thus, (μχeαμχsupp(J))∧μχsupp(J)≠ 0 and (γχeα γχsupp(J)) ∨ γχsupp(J)≠ 1. Hence, there exists an rE such that ((μχeα μχsupp(J)) ∧ μχsupp(J))(r) ≠ 0 and ((γχeα γχsupp(J)) ∨ γχsupp(J))(r) ≠ 1. This implies that (μχeα μχsupp(J))(r) ≠ 0, (γχeα γχsupp(J))(r) ≠ 0 and μχJ(r) ≠ 0,γχJ(r) ≠ 1. Thus, there exists a kE such that r = uαJ, μχsupp(J)(r) ≠ 0, γχsupp(J)(r) ≠ 1 and μχsupp(J)(k) ≠ 0, γχsupp(J)(k) ≠ 1. Hence, (μχeαμχsupp(J))∧ μχsupp(J)≠ 0 and (γχeα γχJsupp(J)) ∨ γχsupp(J)≠ 1. Therefore, J = (μJ, γJ ) 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 JI, we obtain J = I.

Definition 4.6

An IF almost left α-ideal (right β-ideal, (α, β)-ideal) J = (μJ, γJ ) 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 = (μK, γK) of E, where KJ, 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 χJ = (μχJ, γχJ) 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 χJ = (μχJ, γχJ) is an IF left α-ideal of E. Let K = (μK, γK) be an IF left α-ideal of E such that KJ. Then, using Theorem 4.4, it can be concluded that supp(Q) is an almost left α-ideal of E. Thus, supp(K) ⊆ supp(χJ) = J. By assumption, supp(K) = K =supp(χJ ). Thus, χJ = (μχJ, γχJ) is a minimal IF almost left α-ideal of E.

Conversely, let us suppose that χJ = (μχJ, γχJ) 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 KJ. Then, using Theorem 4.3, it can be concluded that χK = (μχK, γχK) is an IF left α-ideal of E such that χKχJ. Thus, K = supp(χK) = supp(χJ) = 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 = (μJ, γJ ) 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 (Jα χe) ∩ (χeβ J) ≠ 0 and (Jα χe) ∪ (χeβ J) ≠ 1.

Theorem 4.9

If J = (μJ, γJ ) is an IF almost (α, β)-quasi-ideal of a Γ-semigroup E and K = (μK, γK) is an IF set of E such that JK, then K = (μK, γK) is an IF (α, β)-quasi-ideal of E.

Proof

Let us suppose that J = (μJ, γJ ) is an IF almost (α, β)-quasi-ideal of E and K = (μK, γK) is an IF set of E such that JK. Then, (μχEα μJ ) ∧ (μJβ μχe) ≠ 0 and (γχeαγJ )∧(γJβγχe) ≠ 1. Thus, (μχeαμJ )∧(μJβμχe) ⊆ (μχeαμJ )∧(μKβμχe) ≠ 0 and (γχeαγJ )∧(γJβ γχE) ⊆ (γχeα γK) ∧ (γKβ γχe) ≠ 0 Hence, K = (μK, γK) 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 χJ = (μχJ, γχJ) 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 uE. Thus, there exist v ∈ (Jαu)∩(uβJ)∩J and vJ such that ((μχeαμJ )∧ (μJβ μχe))(v) ≠ 0 and ((γχeα γJ ) ∨ (γJβ γχe))(v) ≠ 1. Hence, (Jαχe)∩(χEβJ) ≠ 0 and (Jαχe)∪(χEβJ) ≠ 1. Therefore, χJ = (μχJ, γχJ) is an IF almost (α, β)-quasi-ideal of S.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF almost (α, β)-quasi-ideal of E and uE, we obtain (Jα χe) ∩ (χeβ J) ≠ 0 and (Jα χe) ∪ (χeβ J) ≠ 1. Thus there exists an rE such that ((μχeα μJ ) ∧ (μJβ μχe))(r) ≠ 0 and ((γχeα γJ ) ∧ (γJβ γχe))(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 = (μJ, γJ ) be an IF set of a non-empty Γ-semigroup E. Then, J = (μJ, γJ ) 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 = (μJ, γJ ) be an IF almost (α, β)-quasi-ideal of E and uE. Then, (Jα χE) ∩ (χEβ J) ≠ 0 and (Jα χe) ∪ (χeβ J) ≠ 1. Thus, there exists an rE such that ((μχeα μJ ) ∧ (μJβ μχE))(r) ≠ 0 and ((γχeα γJ ) ∧ (γJβ γχe))(r) ≠ 1. Therefore, there exist j1, j2E such that r = j1αu = uβj2, μJ (r) ≠ 0, γJ (r) ≠ 1 and μJ (j1) ≠ 0, γJ (j2) ≠ 1. This implies that r, j1, j2 ∈ supp(J). Thus, ((μχsupp(J)α μχe) ∩ (μχeβ μχsupp(J)))(r) ≠ 0 and μχJsupp(J)(r) ≠ 0. Similarly, ((γχsupp(J)α γχe) ∩ (γχeβ γχsupp(J)))(r) ≠ 1 and γχsupp(J)(r) ≠ 1. Hence, (μχsupp(J)α μχe)∩(μχeβ μχsupp(J)) ≠ 0 and (γχsupp(J)α γχe)∩(γχeβ γχsupp(J)) ≠ 1. Therefore, χsupp(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 χsupp(J) is an IF (α, β)-quasi-ideal of E. Thus, (μχsupp(J)α μχe) ∩ (μχeβ μχsupp(J)) ≠ 0 and (γχsupp(J)α γχe)∪(γχeβ γχsupp(J)) ≠ 1. Hence, there exists an rE such that ((μχsupp(J)α μχe) ∧ (μχeβ μχsupp(J)))(r) ≠ 0 and ((γχsupp(J)α γχe) ∨ (γχeβ γχsupp(J)))(r) ≠ 1. This implies that (μχsupp(J)α μχe)(r) ≠ 0, (μχeβ μχsupp(J))(r) ≠ 0 and μχsupp(J)(r) ≠ 0. Similarly, (γχsupp(J)α γχe)(r) ≠ 1, (γχeβ γχsupp(J))(r) ≠ 1 and γχsupp(J)(r) ≠ 1. Thus, there exist j1, j2E such that r = j1αu = uβj2, μJ (r) ≠ 0, γJ (r) ≠ 1 and μJ (j1) ≠ 0, γJ (j2) ≠ 1. Hence, (Jα χe) ∩ (χeβ J) ≠ 0 and (Jα χe) ∪ (χeβ J) ≠ 1. Therefore, J = (μJ, γJ ) 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 JI, we have J = I.

Definition 4.13

An IF almost (α, β)-quasi-ideal J = (μJ, γJ ) of a Γ-semigroup E is minimal if for all IF almost (α, β)-quasi-ideal K = (μK, γK) of E, where KJ, 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 χJ = (μχJ, γχJ) 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 χJ = (μχJ, γχJ) is an IF (α, β)-quasi-ideal of S. Let K = (μK, γK) be an IF (α, β)-quasi-ideal of E such that KJ. Then, using Theorem 4.9, we can conclude that supp(Q) is an almost (α, β)-quasi-ideal of E. Thus, supp(K) ⊆ supp(χJ) = J. By assumption, supp(K) = K = supp(χJ ). Thus, χJ = (μχJ, γχJ) is a minimal IF almost (α, β)-quasi-ideal of E.

Conversely, let us suppose that χJ = (μχJ, γχJ) 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 KJ. Then, using Theorem 4.11, we can conclude that χK = (μχK, γχK) is an IF (α, β)-quasi-ideal of E such that χKχJ. Thus, K = supp(χK) = supp(χJ) = 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 = (μJ, γJ ) be an IF set of a Γ-semigroup E and α, β ∈ Γ. is an IF almost (α, β)-bi-ideal of E if (μJα μχeβ μJ )∧μJ ≠ 0 and (γJα γχeβ γJ )∨γJ ≠ 1.

Theorem 4.16

If J = (μJ, γJ ) is an IF almost (α, β)-bi-ideal of a Γ-semigroup E and K = (μK, γK) is an IF set of E such that JK, then K = (μK, γK) is an IF (α, β)-bi-ideal of E.

Proof

Let us suppose that J = (μJ, γJ ) is an IF almost (α, β)-bi-ideal of a Γ-semigroup E and K = (μK, γK) is an IF set of E such that JK. Then, (μJα μχEβ μJ ) ∧ μJ ≠ 0 and (γJα γχeβ γJ ) ∨ γJ ≠ 1. Thus, (μJα μχeβ μJ ) ∧ μJ ⊆ ((μKα μχeβ μK) ∧ μJ ≠ 0 and (γJα γχEβ γJ ) ∨ γJ ⊆ (γKα γχeβ γK) ∨ γK ≠ 1. Hence, K = (μK, γK) 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 χJ = (μχJ, γχJ) 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, KαuβJJ ≠ ∅︀. for all uE. Thus, there exist vJαuβJ and vJ such that ((μχJαμχeχeβμχJ))(v) = μχJ(v) = 1 and ((γχJα γχeβ γχJ))(v) = γχJ(v) = 0. Hence, (μχJα μχeχeβ μχJ) ∧ μχJ≠ 0 and (γχJα γχeβ γχJ) ∨ γχJ≠ 1. Therefore, χJ = (μχJ, γχJ) is an IF almost (α, β)-bi-ideal of E.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF almost (α, β)-bi-ideal of E and uE, we obtain: (μχJα μχeχeβ μχJ)∧μχJ≠ 0 and (γχJαγχeβ γχJ)∨γχJ≠ 1. Thus, there exists an rE such that ((μχJαμχeχeβμχJ)∧μχJ)(r) ≠ 0 and ((γχJαγχeβγχJ)∨γχJ)(r) ≠ 1. Hence, rJαuβJJ implies that JαuβJJ ≠ ∅︀. 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 = (μJ, γJ ) be an IF of a non-empty oΓ-semigroup E. Then, J = (μJ, γJ ) 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 = (μJ, γJ ) be an IF almost (α, β)-bi-ideal of E and uE. Then, (μχJα μχeχeβ μχJ) ∧ μχJ≠ 0 and (γχJα γχeβ γχJ) ∨ γχJ≠ 1. Thus, there exists an rE such that ((μχJα μχeχeβ μχJ))(r) ≠ 0 and ((γχJα γχeβ γχJ))(r) ≠ 1. Therefore, there exist j1, j2E such that r = j1αβj2, μJ (r) ≠ 0, γJ (r) ≠ 1 and μJ (k) ≠ 0, γJ (k) ≠ 1. This implies that r, k1, k2 ∈ supp(J). Thus, (μχsupp(J)αμχEβ μχsupp(J))(r) ≠ 0 and μχsupp(J)≠ 0. Similarly, (γχsupp(J)αγχeβ γχsupp(J))(r) ≠ 1and γχsupp(J)≠ 1. Hence, ((μχsupp(J)α μχeβ μχsupp(J))) ∧ μχsupp(J)≠ 0 and ((γχsupp(J)αγχeβ γχsupp(J)))∨γχsupp(J)≠ 1. Thus, χsupp(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 χJsupp(J)is an IF almost (α, β)-bi-ideal of E. Thus, ((μχJsupp(J)α μχEβμχJsupp(J)))∧μχJsupp(J)≠ 0 and ((γχJsupp(J)αγχEβ γχJsupp(J))) ∨ γχJsupp(J)≠ 1. Hence, there exists an rE such that ((μχJsupp(J)αμχEβ μχJsupp(J)))∧μχJsupp(J)(r) ≠ 0 and ((γχJsupp(J)α γχEβ γχJsupp(J))) ∨ γχJsupp(J)(r) ≠ 1 and γχJsupp(J)≠ 1. This implies that (μχJsupp(J)α μχEβ μχJsupp(J))(r) ≠ 0 and μχJsupp(J)≠ 0. Similarly, (γχJsupp(J)αγχEβ γχJsupp(J))(r) ≠ 1 and γχJsupp(J)≠ 1. Therefore, there exists an rE such that ((μχJα μχEχEβ μχJ))(r) ≠ 0 and ((γχJα γχEβ γχJ))(r) ≠ 1. Thus, there exist j1, j2E such that r = j1αβj2, μJ (r) ≠ 0, γJ (r) ≠ 1 and μJ (k) ≠ 0, γJ (k) ≠ 1. Hence, (μχJα μχEχEβ μχJ) ∧ μχJ≠ 0 and (γχJα γχEβ γχJ) ∨ γχJ≠ 1. Therefore, J is a BF almost (α, β)-bi-ideal of S.

Definition 4.19

An IF almost (α, β)-bi-ideal J = (μJ, γJ ) of a Γ-semigroup E is minimal if for all IF almost (α, β)-bi-ideal K = (μK, γK) of E, where KJ, 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 χJ = (μχJ, γχJ) 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 χJ = (μχJ, γχJ) is an IF (α, β)-quasi-ideal of S. Let K = (μK, γK) be an IF (α, β)-bi-ideal of E such that KJ. Then, using Theorem 4.18, we can conclude that supp(Q) is an almost (α, β)-quasi-ideal of E. Thus, supp(K) ⊆ supp(χJ) = J. By assumption, supp(K) = K = supp(χJ ). Therefore, χJ = (μχJ, γχJ) is a minimal IF almost (α, β)-bi-ideal of E.

Conversely, let us suppose that χJ = (μχJ, γχJ) 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 KJ. Then, using Theorem 4.18, it can be deduced that χK = (μχK, γχK) is an IF (α, β)-bi-ideal of E such that χKχJ. Thus, K = supp(χK) = supp(χJ) = J. Therefore, J is a minimal almost (α, β)-bi-ideal of S.

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.

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

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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

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

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

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 [47]. 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

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 JΓJJ. A left (right) ideal of a Γ-semigroup E is a non-empty set J of E such that EΓJJ(JΓEJ). 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 JΓEEΓJJ. A sub-Γ-semigroup K of a Γ-semigroup E is called a bi-ideal of E if JΓEΓJJ.

Definition 2.1 [8]

Let E be a Γ-semigroup and J be a nonempty subset of E, for all eE 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 (eΓJJΓe) ∩ J ≠ ∅︀.

• (4) A left α-ideal of Γ-semigroup J is a non-empty set J such that EαJJ. A right α-ideal of a Γ-semigroup E is a non-empty set J such that JβEJ.

• (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 mi ∈ [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) ξςξ(j) ≥ ς(j) for all jJ.

• (2) ξ = ςξς and ςξ.

• (3) (ξς)(j) = ξ(j) ∧ ς(j) for all jJ.

• (4) (ξς)(j) = ξ(k) ∨ ς(j) for all jJ.

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 Fj as Fj := {(m, n) ∈ E × E | j = mn}.

For two fuzzy sets ξ and ς on a semigroup E, we define the product ξς for all jE 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 ○ ξξ ≠ 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 ○ ξξ ≠ 0.

• (4) a fuzzy almost quasi-ideal of E if (ℰ ○ ξξ ○ 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, vE and γ ∈ Γ.

• (2) A fuzzy left (right) ideal of E if ξ(uγv) ≥ ξ(v) (ξ(uγv) ≥ ξ(u)) for all u, vE 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,wE 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 = (μJ1, γJ1) and J2 = (μJ2, γJ2), we define the following relations:

• (1) J1J2 if and only if μJ1(j) ≤ μJ2(j) and γJ1(j) ≥ γJ2(j).

• (2) J1 = J2 if and only if J1J2 and J2J1.

• (3) J1J2 = {(μJ1μJ2)(j), (γJ1γJ2)(j) | jE}.

• (4) J1J2 = {(μJ1μJ2)(j), (γJ1γJ2)(j) | jE}.

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

For IF sets J1 = (μJ1, γJ1) and J2 = (μJ2, γJ2), we define the product J1J2 = (μJ1J2, γJ1J2) 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 Fjαas Fjα:= {(m, α, n) ∈ E × Γ × E | j = mαn}.

For the IF sets J1 = (μJ1, γJ1) and J2 = (μJ2, γJ2), we define the product ξα ς for all kS 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 χJ = (μχJ, γχJ), 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 χE = (μχE, γχE) for the IF set χE := {(u, μχE(u), γχE(u)) | uE}.

Definition 2.8 [2]

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

• (1) IF subsemigroup on E if it satisfies μJ (uαv) ≥ μJ (u) ∧ μJ (v) and γJ (uαv) ≤ γJ (u) ∨ γJ (v)

• (2) IF left ideal on E if it satisfies μJ (uαv) ≥ μJ (v) and γJ (uαv) ≤ γJ (v)

• (3) IF right ideal on E if it satisfies μJ (uαv) ≥ μJ (u) and γJ (uαv) ≤ γJ (u)

• (4) IF ideal on E if it is both intuitionsitic fuzzy left ideal and right ideal of E

for all u, vE 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 = (μJ, γJ ) be an IF set on a Γ-semigroup E and α, β ∈ Γ. Then J = (μJ, γJ ) is defined as:

• (1) An IF left α-ideal of E if μJ (uαv) ≥ μJ (v) and γJ (uαv) ≤ γJ (v) for all u, vE.

• (2) An IF right β-ideal of E if μJ (uβv) ≥ μJ (v) and γJ (uβv) ≤ γJ (v) for all u, vE.

• (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 χJ = (μχJ, γχJ) 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, vE.

If vJ, then uαvJ. Thus, μχJ(v) = μχJ(uαv) = 1 and

γχJ(v) = γχJ(uαv) = 0. Hence, μχJ(uαv) ≥ μχJ(v) and γχJ(uαv) ≤ γχJ(v).

If vJ, then uαvJ. Thus, μχJ(v) = 0, γχJ(v) = 1 and μχJ(uαv) = 1, γχJ(uαv) = 0. Hence, μχJ(uαv) ≥ μχJ(v) and γχJ(uαv) ≤ γχJ(v). Therefore, χJ = (μχJ, γχJ) is an IF left α-ideal of S.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF left α-ideal of E and u, vE with vJ. Then, μχJ(v) = μχJ(uαv) = 1 and γχJ(v) = γχJ(uαv) = 0. By assumption, μχJ(uαv) ≥ μχJ(v) and γχJ(uαv) ≤ γχJ(v). Thus, uαvJ. 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 = (μJ1, γJ1) and J2 = (μJ2, γJ2) be IF left α-ideals of E and let u, vE. 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, J1J2 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, J1J2 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 = (μJ, γJ ) be an IF set of a Γ-semigroup E and α, β ∈ Γ. Then, J = (μJ, γJ ) is called an IF (α, β)-bi-ideal of E if μJα μχEβ μJμJ and γJα γχEβ γJγJ, where χE = (μχE, γχE) 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 χJ = (μχJ, γχJ) is an IF (α, β)-bi-ideal of E.

Proof

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

If uJαEβJ, then μJ (u) = (μJα μχJβ μJ )(u) = 1 and γJ (u) = (γJα γχJβ γJ )(u) = 0. Hence, (μJα μχJβ μJ )(u) ≥ μJ (u) and (γJα γχJβ γJ )(u) ≤ γJ (u).

If uJαEβJ, then μJ (u) = 0, (μJα μχJβ μJ )(u) = 1 and γJ (u) = 1, (γJα γχJβ γJ )(u) = 0. Hence, (μJα μχJβ μJ )(u) ≥ μJ (u) and (γJα γχJβ γJ )(u) ≤ γJ (u).

Therefore, χJ = (μχJ, γχJ) is an IF (α, β)-bi-ideal of E.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF (α, β)-bi-ideal of E and uJαEβJ. Then, (μJαμχJβμJ )(u) = 1 and (γJα γχJβ γJ )(u) = 0. By assumption, (μJα μχJβ μJ )(u) ≥ μJ (u) and (γJα γχJβ γJ )(u) ≤ γJ (u). Thus, uJ. 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 = (μJ1, γJ1) and J2 = (μJ2, γJ2) be (α, β)-bi-ideals of E and uE. 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, J1J2 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 = (μJ, γJ ) be an IF set of a Γ-semigroup E and α, β ∈ Γ. Then, J = (μJ, γJ ) is called an IF (α, β)-quasi-ideal of E if μχEα μJμJβ μχEμJ and γχEα γJγJβ γχEγJ.

Theorem 3.8

If L = (μL, γL) and R = (μR, γR) are IF left and right α-ideals of a Γ-semigroup E, respectively, then LR is an IF α-quasi-ideal of E.

Proof

Let L = (μL, γL) and R = (μR, γR) be IF left and right α-ideals of E, respectively. Then, μLαμRμχEαμLμL and μLα μRμRα μχEμR. Similarly, γLα γRγχEα γLγL and γLα γRγRα γχEγR Thus, μLα μRμLμR and γLα γRγLγR. 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, LR 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 = (μQ, γQ) be an IF (α, β)-quasi-ideal of E. Considering μL = μQ ∪ (μχEα μQ) and γL = γQ ∪ (γχEα γQ), where L = (μL, γL) μR = μQ ∪ (μQβ μχE) and γR = γQ ∪ (γQβ γχE), where R = (μR, γR), we obtain:

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

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

Similarly, we can show that: γχEα γLγL and γRβ γχEγR. Thus, L = (μL, γL) and R = (μR, γR) 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, μQ = μLμR. Similarly, we can show that γQ = γLγR.

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 χJ = (μχJ, γχJ) is an IF (α, β)-quasi-ideal of S.

Proof

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

If u ∈ (EαJ) ∩ (JβE), then uJ. Thus, μχJ(u) = 1 and γχJ(u) = 0. Hence, ((μχJα μχE) ∩ (μχJβ μχE))(u) ≤ μχJ(u) and ((γχJα γχE) ∪ (γχEβ γχJ))(u) ≥ γχJ(u).

If u ∉ (EαJ) ∩ (JβE), then μχJ(u) = 0 and γχJ(u) = 1. Hence, ((μχJα μχE) ∩ (μχJβ μχE))(u) ≤ μχJ(u) and ((γχJα γχE) ∪ (γχEβ γχJ))(u) ≥ γχJ(u).

Therefore, χJ = (μχJ, γχJ) is an IF (α, β)-quasi-ideal of E.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF (α, β)-quasi-ideal of S and u ∈ (EαJ)∩(JβE), we obtain: ((μχJα μχE) ∩ (μχJβ μχE))(u) = 1, and ((γχJα γχE) ∪ (γχEβ γχJ))(u) = 0. By assumption, ((μχJα μχE) ∩ (μχJβ μχE))(u) ≤ μχJ(u), and ((γχJα γχE)∪(γχEβ γχJ))(u) ≥ γχJ(u). Thus, uJ. Hence, J is an (α, β)-quasi-ideal of E.

Definition 4.1

Let J = (μJ, γJ ) 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 (μχeα μJ ) ∧ μJ ≠ 0 and (γχeα γJ ) ∨ γJ ≠ 1 for all eE.

• (2) An IF almost right β-ideal of E if (μχeβ μJ ) ∧ μJ ≠ 0 and (γχeβ γJ ) ∨ γJ ≠ 1 for all eE.

• (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, χe = (μχe, γχe) is an IF set of E mapping every element of E to e.

Theorem 4.2

If J = (μJ, γJ ) is an IF almost left α-ideal (right β-ideal, (α, β)-ideal) of a Γ-semigroup E and K = (μK, γK) is an IF set of E such that JK, then K = (μK, γK) is an IF almost left α-ideal (right β-ideal, (α, β)-ideal) of E.

Proof

Let us suppose that J = (μJ, γJ ) is an IF almost left α-ideal of E and K = (μK, γK) is an IF set of E such that JK. Then, (μχeαμJ )∧μJ ≠ 1and (γχeα γJ )∨γJ ≠ 0. Thus (μχeα μJ ) ∧ μJ ⊆ (μχeα μK) ∧ μK ≠ 0 and (γχeα γJ ) ∨ γJ ⊆ (γχeα γK) ∨ γK ≠ 1. Hence, K = (μK, γK) 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 χJ = (μχJ, γχJ) 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, uαJJ ≠ ∅︀ for all uE. Thus, there exist vuαJ and vJ such thato (μχEα μχJ)(v) = μχJ(v) = 1 and (γχeαγχJ)(v) = γχJ(v) = 0. Hence, (μχeαμχJ)∧μχJ≠ 0 and (γχEα γχJ) ∨ γχJ≠ 1. Therefore, χJ = (μχJ, γχJ) is an IF almost left α-ideal of S.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF almost left α-ideal of S and uS, we obtain: (μχeαμχJ)∧μχJ≠ 0 and (γχeα γχJ) ∨ γχJ≠ 1. Thus, there exists an rS such that ((μχeαμχJ)∧μχJ)(r) ≠ 0 and ((γχeαγχJ)∨γχJ)(r) ≠ 1. Hence, ruαJJ Implies that uαJJ ≠ ∅︀. 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 = (μJ, γJ ) be an IF set of a non-empty of E. Then the support of J is determined rather than supp(J) = {uE | J(u) ≠ 0}, where μJ (u) ≠ 0 and γJ (u) ≠ 1 for all uE.

Theorem 4.4

Let J = (μJ, γJ ) be an IF set of a non-empty Γ-semigroup E. Then, J = (μJ, γJ ) 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 = (μJ, γJ ) be an IF almost left α-ideal of S and uE. Then, (μχeαμχJ)∧μχJ≠ 0 and (γχeαγχJ)∨γχJ≠ 1. Thus, there exists an rE such that ((μχeα μχJ) ∧ μχJ)(r) ≠ 0 and ((γχeα γχJ) ∨ γχJ)(r) ≠ 1. Hence, there exists a kE such that r = uαJ, μχJ(r) ≠ 0, γχJ(r) ≠ 1, and μχJ(k) ≠ 0, γχJ(k) ≠ 1. This implies that r, k ∈ supp(J). Thus, (μχeα μχsupp(J))(r) ≠ 0, (γχeα γχsupp(J))(r) ≠ 1, and μχsupp(J)≠ 0, γχsupp(J)≠ 1. Hence, (μχeα μχsupp(J)) ∧ μχJsupp(J)≠ 0 and (γχeα γχsupp(J))∨γχJsupp(J)≠ 1. Therefore, χsupp(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 χJsupp(J)is an IF almost left α-ideal of E. Thus, (μχeαμχsupp(J))∧μχsupp(J)≠ 0 and (γχeα γχsupp(J)) ∨ γχsupp(J)≠ 1. Hence, there exists an rE such that ((μχeα μχsupp(J)) ∧ μχsupp(J))(r) ≠ 0 and ((γχeα γχsupp(J)) ∨ γχsupp(J))(r) ≠ 1. This implies that (μχeα μχsupp(J))(r) ≠ 0, (γχeα γχsupp(J))(r) ≠ 0 and μχJ(r) ≠ 0,γχJ(r) ≠ 1. Thus, there exists a kE such that r = uαJ, μχsupp(J)(r) ≠ 0, γχsupp(J)(r) ≠ 1 and μχsupp(J)(k) ≠ 0, γχsupp(J)(k) ≠ 1. Hence, (μχeαμχsupp(J))∧ μχsupp(J)≠ 0 and (γχeα γχJsupp(J)) ∨ γχsupp(J)≠ 1. Therefore, J = (μJ, γJ ) 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 JI, we obtain J = I.

Definition 4.6

An IF almost left α-ideal (right β-ideal, (α, β)-ideal) J = (μJ, γJ ) 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 = (μK, γK) of E, where KJ, 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 χJ = (μχJ, γχJ) 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 χJ = (μχJ, γχJ) is an IF left α-ideal of E. Let K = (μK, γK) be an IF left α-ideal of E such that KJ. Then, using Theorem 4.4, it can be concluded that supp(Q) is an almost left α-ideal of E. Thus, supp(K) ⊆ supp(χJ) = J. By assumption, supp(K) = K =supp(χJ ). Thus, χJ = (μχJ, γχJ) is a minimal IF almost left α-ideal of E.

Conversely, let us suppose that χJ = (μχJ, γχJ) 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 KJ. Then, using Theorem 4.3, it can be concluded that χK = (μχK, γχK) is an IF left α-ideal of E such that χKχJ. Thus, K = supp(χK) = supp(χJ) = 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 = (μJ, γJ ) 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 (Jα χe) ∩ (χeβ J) ≠ 0 and (Jα χe) ∪ (χeβ J) ≠ 1.

Theorem 4.9

If J = (μJ, γJ ) is an IF almost (α, β)-quasi-ideal of a Γ-semigroup E and K = (μK, γK) is an IF set of E such that JK, then K = (μK, γK) is an IF (α, β)-quasi-ideal of E.

Proof

Let us suppose that J = (μJ, γJ ) is an IF almost (α, β)-quasi-ideal of E and K = (μK, γK) is an IF set of E such that JK. Then, (μχEα μJ ) ∧ (μJβ μχe) ≠ 0 and (γχeαγJ )∧(γJβγχe) ≠ 1. Thus, (μχeαμJ )∧(μJβμχe) ⊆ (μχeαμJ )∧(μKβμχe) ≠ 0 and (γχeαγJ )∧(γJβ γχE) ⊆ (γχeα γK) ∧ (γKβ γχe) ≠ 0 Hence, K = (μK, γK) 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 χJ = (μχJ, γχJ) 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 uE. Thus, there exist v ∈ (Jαu)∩(uβJ)∩J and vJ such that ((μχeαμJ )∧ (μJβ μχe))(v) ≠ 0 and ((γχeα γJ ) ∨ (γJβ γχe))(v) ≠ 1. Hence, (Jαχe)∩(χEβJ) ≠ 0 and (Jαχe)∪(χEβJ) ≠ 1. Therefore, χJ = (μχJ, γχJ) is an IF almost (α, β)-quasi-ideal of S.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF almost (α, β)-quasi-ideal of E and uE, we obtain (Jα χe) ∩ (χeβ J) ≠ 0 and (Jα χe) ∪ (χeβ J) ≠ 1. Thus there exists an rE such that ((μχeα μJ ) ∧ (μJβ μχe))(r) ≠ 0 and ((γχeα γJ ) ∧ (γJβ γχe))(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 = (μJ, γJ ) be an IF set of a non-empty Γ-semigroup E. Then, J = (μJ, γJ ) 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 = (μJ, γJ ) be an IF almost (α, β)-quasi-ideal of E and uE. Then, (Jα χE) ∩ (χEβ J) ≠ 0 and (Jα χe) ∪ (χeβ J) ≠ 1. Thus, there exists an rE such that ((μχeα μJ ) ∧ (μJβ μχE))(r) ≠ 0 and ((γχeα γJ ) ∧ (γJβ γχe))(r) ≠ 1. Therefore, there exist j1, j2E such that r = j1αu = uβj2, μJ (r) ≠ 0, γJ (r) ≠ 1 and μJ (j1) ≠ 0, γJ (j2) ≠ 1. This implies that r, j1, j2 ∈ supp(J). Thus, ((μχsupp(J)α μχe) ∩ (μχeβ μχsupp(J)))(r) ≠ 0 and μχJsupp(J)(r) ≠ 0. Similarly, ((γχsupp(J)α γχe) ∩ (γχeβ γχsupp(J)))(r) ≠ 1 and γχsupp(J)(r) ≠ 1. Hence, (μχsupp(J)α μχe)∩(μχeβ μχsupp(J)) ≠ 0 and (γχsupp(J)α γχe)∩(γχeβ γχsupp(J)) ≠ 1. Therefore, χsupp(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 χsupp(J) is an IF (α, β)-quasi-ideal of E. Thus, (μχsupp(J)α μχe) ∩ (μχeβ μχsupp(J)) ≠ 0 and (γχsupp(J)α γχe)∪(γχeβ γχsupp(J)) ≠ 1. Hence, there exists an rE such that ((μχsupp(J)α μχe) ∧ (μχeβ μχsupp(J)))(r) ≠ 0 and ((γχsupp(J)α γχe) ∨ (γχeβ γχsupp(J)))(r) ≠ 1. This implies that (μχsupp(J)α μχe)(r) ≠ 0, (μχeβ μχsupp(J))(r) ≠ 0 and μχsupp(J)(r) ≠ 0. Similarly, (γχsupp(J)α γχe)(r) ≠ 1, (γχeβ γχsupp(J))(r) ≠ 1 and γχsupp(J)(r) ≠ 1. Thus, there exist j1, j2E such that r = j1αu = uβj2, μJ (r) ≠ 0, γJ (r) ≠ 1 and μJ (j1) ≠ 0, γJ (j2) ≠ 1. Hence, (Jα χe) ∩ (χeβ J) ≠ 0 and (Jα χe) ∪ (χeβ J) ≠ 1. Therefore, J = (μJ, γJ ) 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 JI, we have J = I.

Definition 4.13

An IF almost (α, β)-quasi-ideal J = (μJ, γJ ) of a Γ-semigroup E is minimal if for all IF almost (α, β)-quasi-ideal K = (μK, γK) of E, where KJ, 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 χJ = (μχJ, γχJ) 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 χJ = (μχJ, γχJ) is an IF (α, β)-quasi-ideal of S. Let K = (μK, γK) be an IF (α, β)-quasi-ideal of E such that KJ. Then, using Theorem 4.9, we can conclude that supp(Q) is an almost (α, β)-quasi-ideal of E. Thus, supp(K) ⊆ supp(χJ) = J. By assumption, supp(K) = K = supp(χJ ). Thus, χJ = (μχJ, γχJ) is a minimal IF almost (α, β)-quasi-ideal of E.

Conversely, let us suppose that χJ = (μχJ, γχJ) 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 KJ. Then, using Theorem 4.11, we can conclude that χK = (μχK, γχK) is an IF (α, β)-quasi-ideal of E such that χKχJ. Thus, K = supp(χK) = supp(χJ) = 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 = (μJ, γJ ) be an IF set of a Γ-semigroup E and α, β ∈ Γ. is an IF almost (α, β)-bi-ideal of E if (μJα μχeβ μJ )∧μJ ≠ 0 and (γJα γχeβ γJ )∨γJ ≠ 1.

Theorem 4.16

If J = (μJ, γJ ) is an IF almost (α, β)-bi-ideal of a Γ-semigroup E and K = (μK, γK) is an IF set of E such that JK, then K = (μK, γK) is an IF (α, β)-bi-ideal of E.

Proof

Let us suppose that J = (μJ, γJ ) is an IF almost (α, β)-bi-ideal of a Γ-semigroup E and K = (μK, γK) is an IF set of E such that JK. Then, (μJα μχEβ μJ ) ∧ μJ ≠ 0 and (γJα γχeβ γJ ) ∨ γJ ≠ 1. Thus, (μJα μχeβ μJ ) ∧ μJ ⊆ ((μKα μχeβ μK) ∧ μJ ≠ 0 and (γJα γχEβ γJ ) ∨ γJ ⊆ (γKα γχeβ γK) ∨ γK ≠ 1. Hence, K = (μK, γK) 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 χJ = (μχJ, γχJ) 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, KαuβJJ ≠ ∅︀. for all uE. Thus, there exist vJαuβJ and vJ such that ((μχJαμχeχeβμχJ))(v) = μχJ(v) = 1 and ((γχJα γχeβ γχJ))(v) = γχJ(v) = 0. Hence, (μχJα μχeχeβ μχJ) ∧ μχJ≠ 0 and (γχJα γχeβ γχJ) ∨ γχJ≠ 1. Therefore, χJ = (μχJ, γχJ) is an IF almost (α, β)-bi-ideal of E.

Conversely, assuming that χJ = (μχJ, γχJ) is an IF almost (α, β)-bi-ideal of E and uE, we obtain: (μχJα μχeχeβ μχJ)∧μχJ≠ 0 and (γχJαγχeβ γχJ)∨γχJ≠ 1. Thus, there exists an rE such that ((μχJαμχeχeβμχJ)∧μχJ)(r) ≠ 0 and ((γχJαγχeβγχJ)∨γχJ)(r) ≠ 1. Hence, rJαuβJJ implies that JαuβJJ ≠ ∅︀. 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 = (μJ, γJ ) be an IF of a non-empty oΓ-semigroup E. Then, J = (μJ, γJ ) 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 = (μJ, γJ ) be an IF almost (α, β)-bi-ideal of E and uE. Then, (μχJα μχeχeβ μχJ) ∧ μχJ≠ 0 and (γχJα γχeβ γχJ) ∨ γχJ≠ 1. Thus, there exists an rE such that ((μχJα μχeχeβ μχJ))(r) ≠ 0 and ((γχJα γχeβ γχJ))(r) ≠ 1. Therefore, there exist j1, j2E such that r = j1αβj2, μJ (r) ≠ 0, γJ (r) ≠ 1 and μJ (k) ≠ 0, γJ (k) ≠ 1. This implies that r, k1, k2 ∈ supp(J). Thus, (μχsupp(J)αμχEβ μχsupp(J))(r) ≠ 0 and μχsupp(J)≠ 0. Similarly, (γχsupp(J)αγχeβ γχsupp(J))(r) ≠ 1and γχsupp(J)≠ 1. Hence, ((μχsupp(J)α μχeβ μχsupp(J))) ∧ μχsupp(J)≠ 0 and ((γχsupp(J)αγχeβ γχsupp(J)))∨γχsupp(J)≠ 1. Thus, χsupp(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 χJsupp(J)is an IF almost (α, β)-bi-ideal of E. Thus, ((μχJsupp(J)α μχEβμχJsupp(J)))∧μχJsupp(J)≠ 0 and ((γχJsupp(J)αγχEβ γχJsupp(J))) ∨ γχJsupp(J)≠ 1. Hence, there exists an rE such that ((μχJsupp(J)αμχEβ μχJsupp(J)))∧μχJsupp(J)(r) ≠ 0 and ((γχJsupp(J)α γχEβ γχJsupp(J))) ∨ γχJsupp(J)(r) ≠ 1 and γχJsupp(J)≠ 1. This implies that (μχJsupp(J)α μχEβ μχJsupp(J))(r) ≠ 0 and μχJsupp(J)≠ 0. Similarly, (γχJsupp(J)αγχEβ γχJsupp(J))(r) ≠ 1 and γχJsupp(J)≠ 1. Therefore, there exists an rE such that ((μχJα μχEχEβ μχJ))(r) ≠ 0 and ((γχJα γχEβ γχJ))(r) ≠ 1. Thus, there exist j1, j2E such that r = j1αβj2, μJ (r) ≠ 0, γJ (r) ≠ 1 and μJ (k) ≠ 0, γJ (k) ≠ 1. Hence, (μχJα μχEχEβ μχJ) ∧ μχJ≠ 0 and (γχJα γχEβ γχJ) ∨ γχJ≠ 1. Therefore, J is a BF almost (α, β)-bi-ideal of S.

Definition 4.19

An IF almost (α, β)-bi-ideal J = (μJ, γJ ) of a Γ-semigroup E is minimal if for all IF almost (α, β)-bi-ideal K = (μK, γK) of E, where KJ, 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 χJ = (μχJ, γχJ) 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 χJ = (μχJ, γχJ) is an IF (α, β)-quasi-ideal of S. Let K = (μK, γK) be an IF (α, β)-bi-ideal of E such that KJ. Then, using Theorem 4.18, we can conclude that supp(Q) is an almost (α, β)-quasi-ideal of E. Thus, supp(K) ⊆ supp(χJ) = J. By assumption, supp(K) = K = supp(χJ ). Therefore, χJ = (μχJ, γχJ) is a minimal IF almost (α, β)-bi-ideal of E.

Conversely, let us suppose that χJ = (μχJ, γχJ) 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 KJ. Then, using Theorem 4.18, it can be deduced that χK = (μχK, γχK) is an IF (α, β)-bi-ideal of E such that χKχJ. Thus, K = supp(χK) = supp(χJ) = J. Therefore, J is a minimal almost (α, β)-bi-ideal of S.

5. Conclusion

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.

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