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INTERVAL VALUED (α, β)-INTUITIONISTIC FUZZY BI-IDEALS OF SEMIGROUPS
INTERVAL VALUED (α, β)-INTUITIONISTIC FUZZY BI-IDEALS OF SEMIGROUPS
Journal of Applied Mathematics & Informatics. 2016. Jan, 34(1_2): 115-143
Copyright © 2016, Korean Society of Computational and Applied Mathematics
  • Received : February 12, 2015
  • Accepted : May 26, 2015
  • Published : January 30, 2016
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SALEEM ABDULLAH
MUHAMMAD ASLAM
SHAH HUSSAIN

Abstract
The concept of quasi-coincidence of interval valued intuitionistic fuzzy point with an interval valued intuitionistic fuzzy set is considered. By using this idea, the notion of interval valued ( α, β )-intuitionistic fuzzy bi-ideals, (1,2)ideals in a semigroup introduced and consequently, a generalization of interval valued intuitionistic fuzzy bi-ideals and intuitionistic fuzzy bi-ideals is defined. In this paper, we study the related properties of the interval valued ( α, β )-intuitionistic fuzzy bi-ideals, (1,2) ideals and in particular, an interval valued (Є, Є ∨ q )-fuzzy bi-ideals and (1,2) ideals in semigroups will be investigated. AMS Mathematics Subject Classification : 65H05, 65F10.
Keywords
1. Introduction
The concept of a fuzzy set was first initiated by Zadeh [1] . Fuzzy set theory has been shown to be a useful tool to describe situations in which the data are imprecise or vague. Fuzzy sets handle such situations by attributing a degree to which a certain object belongs to a set. The fuzzy algebraic structures play a prominent role in mathematics with wide applications in many other branches such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological spaces, logic, set theory, group theory, real analysis, measure theory etc. After the introduction of the concept of fuzzy sets by Zadeh, several researches conducted the researches on the generalizations of the notion of fuzzy set with huge applications in computer, logics, automata and many branches of pure and applied mathematics. The notion of i-v fuzzy sets was first introduced by Zadeh [1] as an extension of fuzzy sets in which the values of the membership degrees are intervals of numbers instead of the numbers. Thus, i-v fuzzy sets provide a more adequate description of uncertainty than the traditional fuzzy sets. I-v fuzzy set theory has been shown to be a useful tool to describe situations in which the data are imprecise or vague. Rosenfeld studied fuzzy subgroups of a group [2] . The study of fuzzy semigroups was studied by Kuroki in his classical paper [3] and Kuroki initiated fuzzy ideals, bi-ideals, semi-prime ideals, quasi-ideals of semigroups [4 , 5 , 6 , 7 , 8 , 9 , 10] . A systematic exposition of fuzzy semigroup was given by Mordeson et.al. [11] , and they have find theoretical results on fuzzy semigroups and their use in fuzzy finite state machines, fuzzy languages and fuzzy coding. Mordeson and Malik studied monograph in [12] deals with the application of fuzzy approch to the concepts of formal languages and automata. In 2008, Shabir and Khan introduced the concept of i-v fuzzy ideals generated by i-v fuzzy subset in ordered semigroup [43] . Using the notions ” belong to ” relation (∈) introduced by Pu and Lia [19] . In [20] , Morali proposed the concept of a fuzzy point belonging to a fuzzy subset under natural equivalence on fuzzy subset. Bhakat and Das introduced the concepts of ( α, β )-fuzzy subgroups by using the ” belong to ” relation (∈) and ” quasi-coincident with ” relation ( q ) between a fuzzy point and a fuzzy subgroup, and defined an (∈, ∈ ∨ q )-fuzzy subgroup of a group [21] . Kazanci and Yamak [22] studied generalized types fuzzy bi-ideals of semigroups and defined
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bi-ideals of semigroups. Jun and Song [23] studied generalized fuzzy interior ideals of semigroups. In [24] , Shabir et. al. characterized regular semigroups by the properties of ( α, β )-fuzzy ideals, bi-ideals and quasi-ideals. In [25] , Shabir and Yasir characterized regular semigroups by the properties of
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ideals, generalized bi-ideals and quasi-ideals of a semigroup. M. Shabir et. al. defined some types of (∈, ∈ ∨ qk )-fuzzy ideals of semigroups and characterized regular semigroups by these ideals [26] . In [27] , M. Shabir and T. Mehmood studied (∈, ∈ ∨ qk )-fuzzy h-ideals of hemirings and characterized different classes of hemirings by the properties of (∈, ∈ ∨ qk )-fuzzy h-ideals. Recently, M. Aslam et al. [28] initiated the concept of ( α,β )-fuzzy Γ-ideals of Γ-LA-semigroups and given some characterization of Γ-LA-semigroups by ( α,β )-fuzzy Γ-ideals.
In fuzzy sets theory, there is no means to incorporate the hesitation or uncertainty in the membership degrees. In 1986, Atanassov [29] premised the concept of an intuitionistic fuzzy set (IFS) and more operations defined in [30] . An Atanassov intuitionistic fuzzy set is deliberated as a generalization of fuzzy set [1] and has been found to be useful to deal with vagueness. In the sense of an IFS is characterized by a pair of functions valued in [0, 1]: the membership function and the non-membership function. The evaluation degrees of membership and non-membership are independent. Thus, an Atanassov intuitionistic fuzzy set is more material and concise to describe the essence of fuzziness, and Atanassov intuitionistic fuzzy set theory may be more suitable than fuzzy set theory for dealing with imperfect knowledge in many problems. The concept has been applied to various algebraic structures. Atanassov and Gargov [31] initiated the notion of i-v intuitionistic fuzzy sets which is a generalization of both intuitionistic fuzzy sets and interval valued fuzzy sets. In [?], Akram and Dudek have defined interval valued intuitionistic fuzzy Lie ideals of Lie algebras and some interesting results are abtained. Biswas [32] introduced the notion of intuitionistic fuzzy subgroup of a group by using the notion of intuitionistic fuzzy sets. In [33] , Kim and Jun defined intuitionistic fuzzy ideals of semigroups. Kim and Lee [34] studied intuitionistic fuzzy bi-ideals of semigroups. Kim and Jun initiated the concept intuitionistic fuzzy interior ideals of semigroups [35] . Coker and Demirci introduced the notion of intuitionistic fuzzy point [36] of 1995. Jun [37] introduced the notion of (Φ, Ψ)-intuitionistic fuzzy subgroups, where Φ,Ψ are any two of {∈, q , ∈ ∨ q , ∈ ∧ q } with Φ ≠∈ ∧ q , and related properties are investigated. Aslam and Abdullah [38] introduced the concept of (Φ, Ψ)-intuitionistic fuzzy ideals of semigroups and obtained some properties of (Φ, Ψ)-intuitionistic fuzzy ideals. Recently, Abdullah et.al., initiated the concept of ( α, β )-intuitionistic fuzzy ideals of hemirings by using the ” belong to ” relation (∈) and ” quasi-coincident with ” relation ( q ) between an intuitionistic fuzzy point and an intuitionistics fuzzy set, and they defined prime (semi-prime) ( α, β )-intuitionistic fuzzy ideals of hemirings [39] .
In this paper, we introduce the concept of interval valued (∈, ∈ ∨ q )-intuitionistic fuzzy bi-ideal of semigroup and ( α, β )-intuitionistic fuzzy bi-ideal of semigroup where Φ,Ψ are any two of {∈, q , ∈ ∨ q , ∈ ∧ q } with α ≠ ∈ ∧ q , by using belong to relation (∈) and quasi-coincidence with relation ( q ) between intuitionistic fuzzy point and intuitionistic fuzzy set, and investigated related properties. We also prove that in regular semigroup, every (∈, ∈ ∨ q )-intuitionistic fuzzy (1, 2) ideal of semigroup S is an (∈, ∈ ∨ q )-intuitionistic fuzzy bi-ideal of semigroup S .
2. Preliminaries
Definition 1 ( [1 , 2] ). Let X be a non-empty fixed set. An interval valued intuitionistic fuzzy set (briefly, IVIFS) A is an object having the form
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where the functions
PPT Slide
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denote the degree of membership
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and the degree of non-membership
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of each element x X to the set A , respectively, and
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for all x S for the sake of simplicity, we use the symbol
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for the IVIFS
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Definition 2 ( [4] ). Let c be a point in a non-empty set X . If
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are two interval numbers such that
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, and at same time both values
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does not less than
PPT Slide
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. Then, the IFS
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is called an interval valued intuitionistic fuzzy point (IVIFP for short) in X , where
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is the degree of membership (resp, non-membership) of
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and c X is the support of
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be an IVIFP in X. and let
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be an interval valued IFS in X . Then,
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is said to belong to A , written
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. We say that
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is quasi-coincident with A , written
PPT Slide
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. To say that
PPT Slide
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means that
PPT Slide
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means that
PPT Slide
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does not hold and
PPT Slide
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Definition 3. An IVIFS
PPT Slide
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in S is called an intuitionistic fuzzy subsemigroup of S if the following conditions hold:
PPT Slide
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Definition 4. An IVIFS
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in S is called an intuitionistic fuzzy right ideal of S if it satisfy
PPT Slide
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and λA ( xy ) ≤ λA ( x ) for all x, y S .
Definition 5. An IVIFS
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in S is called an intuitionistic fuzzy left ideal of S if it satisfy
PPT Slide
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and λA ( xy ) ≤ λA ( y ) for all x, y S .
Theorem 1 ( [13] ). An IVIFS
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in a semigroup S is an interval valued (∈, ∈ ∨ q ) -intuitionistic fuzzy left (resp. right) ideal of a semigroup S if and only if the following conditions hold .
PPT Slide
Lager Image
3. Interval valued (α, β)-Intuitionistic Fuzzy bi-ideals
Definition 6. An IVIFS
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in a semigroup S is said to be an interval valued ( α, β )-intuitionistic fuzzy subsemigroup of a semigroup S if the following condition holds:
PPT Slide
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Definition 7. An IVIFS
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in a semigroup S is said to be an interval valued ( α, β )-intuitionistic fuzzy left (resp, right) ideal of semigroup S if ∀ x, y S and
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or
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, the following hold.
PPT Slide
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An IVIFS
PPT Slide
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in a semigroup S is said to be an interval valued ( α, β )-intuitionistic fuzzy ideal of a semigroup S , if
PPT Slide
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is an interval valued ( α, β )-intuitionistic fuzzy left ideal and interval valued ( α, β )-intuitionistic fuzzy right ideal of a semigroup S .
Definition 8. An IVIFS
PPT Slide
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in a semigroup S is said to be an interval valued ( α, β )-intuitionistic fuzzy bi-ideal of a semigroup S , where α, β are any two of {∈, q , ∈ ∨ q , ∈ ∧ q } with α ≠ ∈ ∧ q , if for all x, y, z S ,
PPT Slide
Lager Image
or
PPT Slide
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, the following conditions hold:
PPT Slide
Lager Image
Definition 9. An IVIFS
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in semigroup S is said to be an interval valued ( α, β )-intuitionistic fuzzy (1, 2) ideal of semigroup S if for all
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or
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, the following conditions hold.
PPT Slide
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Theorem 2. Let
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be a non-zero interval valued ( α, β ) -intuitionistic fuzzy subsemigroup of S. Then, the set
PPT Slide
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is a subsemigroup of S .
Proof . Let
PPT Slide
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. Suppose that
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, then
PPT Slide
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but
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So,
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, which is a contradiction. Now, let
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for β ∈ {∈, q , ∈ ∨ q , ∈ ∧ q }, which is a contaradiction. Hence,
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is a subsemigroup of S . □
Theorem 3. Let
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be a non-zero interval valued ( α, β ) -intuitionistic fuzzy subsemigroup of S. Then, the set
PPT Slide
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is a subsemigroup of S .
Proof . The proof follows from Theorem 2. □
Theorem 4 . Let
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be a non-zero interval valued ( α, β ) -intuitionistic fuzzy bi-ideal of S. Then, the set Let
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is a bi-ideal of S .
Proof . Let Let
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be a non-zero interval valued ( α, β )-intuitionistic fuzzy bi-ideal of S . Then, by Theorem 2,
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is a subsemigroup of S . Now, let x, z
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and y S . Then,
PPT Slide
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. Suppose that
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, then
PPT Slide
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but
PPT Slide
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which implies that
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for β ∈ {∈, q , ∈ ∨ q , ∈ ∧ q }, which is a contradiction. Now, let
PPT Slide
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for β ∈ {∈, q , ∈ ∨ q , ∈ ∧ q }, which is a contaradiction. Hence
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is a bi-ideal of S . □
Theorem 5. Let
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be a non-zero interval valued ( α, β ) -intuitionistic fuzzy (1, 2) ideal of S. Then, the set
PPT Slide
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is a (1, 2) ideal of S .
Proof . Straightforward. □
Theorem 6. Let L be a left (resp. right) ideal of S and let
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be an IVIFS such that
PPT Slide
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Then,
PPT Slide
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is an interval valued ( q , ∈ ∨ q ) -intuitionistic fuzzy left (resp. right) ideal of S .
Proof . (For α = q ), let x, y S and
PPT Slide
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be such that
PPT Slide
Lager Image
. So, y L . Therefore, xy L . Thus, if
PPT Slide
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and so
PPT Slide
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and
PPT Slide
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. Therefore,
PPT Slide
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does not occur. From the fact that
PPT Slide
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, it follows that the case
PPT Slide
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does not occur. Hence,
PPT Slide
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is an interval valued ( q , ∈ ∨ q )-intuitionistic fuzzy left ideal of S . □
Theorem 7. Let B be a subsemigroup of S and let
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be an IVIFS such that
PPT Slide
Lager Image
Then,
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is an interval valued ( q , ∈ ∨ q ) -intuitionistic fuzzy sub-semigroup of S .
Proof . Straightforward. □
Theorem 8. Let B be a bi-ideal of a semigroup S and let
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be an IVIFS of S such that
PPT Slide
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Then,
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is an interval valued ( q , ∈ ∨ q ) -intuitionistic fuzzy bi-ideal of S .
Proof . ( i ) (For α = q ), let x, y S and
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be such that
PPT Slide
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. Then,
PPT Slide
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, and
PPT Slide
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. Thus, x, y B . Since B is subsemi-group. So, xy B . Thus,
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and
PPT Slide
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and
PPT Slide
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. So,
PPT Slide
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and
PPT Slide
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. Since
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does not occur. From the fact that
PPT Slide
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and
PPT Slide
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it follows that
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does not occur. Hence,
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is an interval valued ( q , ∈ ∨ q )-intuitionistic fuzzy subsemigroup of S . Let x, y, z S and
PPT Slide
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be such that
PPT Slide
Lager Image
. Then,
PPT Slide
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, and
PPT Slide
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. Thus, x, z B . Since B is a bi-ideal. So, xyz B . Thus,
PPT Slide
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and
PPT Slide
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.
PPT Slide
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and
PPT Slide
Lager Image
. Since
PPT Slide
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does not occur. From the fact that
PPT Slide
Lager Image
, it follows that
PPT Slide
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does not occur. Hence,
PPT Slide
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is an interval valued ( q , ∈ ∨ q )-intuitionistic fuzzy bi-ideal of S . □
Theorem 9. Let B be a (1, 2) ideal of a semigroup S and let
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be an IVIFS of S such that
PPT Slide
Lager Image
Then,
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is an interval valued ( q , ∈ ∨ q ) -intuitionistic fuzzy (1, 2) ideal of S .
Proof . Proof follow from Theorem 8. □
4. Interval Valued Intutionistic Fuzzy Bi-Ideals of type (∈, ∈ ∨q)
Definition 10. An IVIFS
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in semigroup S is said to be an interval valued (∈, ∈ ∨ q )-intuitionistic fuzzy bi-ideal of semigroup S if ∀ x, y, a S ,
PPT Slide
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or
PPT Slide
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, the following conditions hold.
PPT Slide
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PPT Slide
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Definition 11. An IVIFS
PPT Slide
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in a semigroup S is said to be an interval valued (∈, ∈ ∨ q )-intuitionistic fuzzy (1, 2) ideal of a semigroup S if for all x, y, z, a
PPT Slide
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