On Normal Sub-Intuitionistic Fuzzy Multigroups

Vol.19, No.2, 2019, 127-139

ISSN: 2279-087X (P), 2279-0888(online) Published on 5 April 2019

www.researchmathsci.org

DOI: http://dx.doi.org/10.22457/apam.581v19n2a1

127

Annals of

On Normal Sub-Intuitionistic Fuzzy Multigroups

1

Department of Mathematics, Federal University Dutse P.M.B. 7156 Dutse, Jigawa State, Nigeria Email: [email protected]

2

Department of Mathematical Sciences, Kaduna State University-Kaduna P.M.B. 2339 Kaduna State, Nigeria

Email: [email protected]

3

Department of Mathematics, Ahmadu Bello University Zaria P.M.B. 1094 Samaru-Zaria, Kaduna State, Nigeria

Email: [email protected]

Received 16 February 2019; accepted 1 April 2019

Abstract. This paper developed the concept of normal sub-intuitionistic fuzzy

Multigroups. It then established that intersection, union, of any two normal intuitionistic fuzzy multigroups is also normal. The inverse of any normal sub-intuitionistic fuzzy multigroup is normal and for any normal sub-sub-intuitionistic fuzzy multigroup, the root (support) set of normal sub-intuitionistic fuzzy multigroup is a normal sub-intuitionistic fuzzy group. It also showed that under the isomorphism function between any two groups, the image of a normal sub-intuitionistic fuzzy multigroup under the isomorphism is a normal sub-intuitionistic fuzzy multigroup and the inverse image of a normal intuitionistic fuzzy multigroup under the isomorphism is a normal sub-intuitionistic fuzzy multigroup.

Keywords: Multiset, fuzzy multiset, intuitionistic fuzzy multiset, multi groups, fuzzy

multigroups, intuitionistic fuzzy multi group, isomorphism function, normal sub- intuitionistic fuzzy multigroup

AMS Mathematics Subject Classification (2010): 08A72

1. Introduction

I.M.Adamu, Y.Tella and A.J.Alkali

128

factor, Zadeh [5] introduced Fuzzy sets in 1965, in which a membership function assigns to each element of the universe of discourse, a number from the unit interval [0,1] to indicate the degree of belongingness to the set under consideration. Fuzzy sets were introduced with a view to reconcile mathematical modeling and human knowledge in the engineering sciences. Since then, a considerable body of literature has blossomed around the concept of fuzzy sets in an incredibly wide range of areas, from mathematics and logics to traditional and advanced engineering methodologies. In 1983, Atanassov [6, 7] introduced the concept of Intuitionistic Fuzzy sets. The same time a theory called ‘Intuitionistic Fuzzy set theory’ was independently introduced by Takeuti and Titani [8] as a theory developed in (a kind of) Intuitionistic logic. An Intuitionistic Fuzzy set is characterized by two functions expressing the degree of membership and the degree of nonmembership of elements of the universe to the Intuitionistic Fuzzy set. Among the various notions of higher-order Fuzzy sets, Intuitionistic Fuzzy sets proposed by Atanassov provide a flexible framework to explain uncertainty and vagueness. It is well-known that in the beginning of the last century L.Brouwer introduced the concept of Intuitionism. The name Intuitionistic Fuzzy set is due to George Gargove, with themotivation that their fuzzification denies the law of excluded middle-one of the main ideas of Intuitionism. As a generalization of multiset, Yager [9] introduced fuzzy multisets and suggested possible applications to relational databases. An element of a Fuzzy Multiset can occur more than once with possibly the same or different membership values. The concept of Intuitionistic Fuzzy Multiset is introduced in [10] which have applications in medical diagnosis and robotics. In mathematics, abstract algebra is the study of algebraic structures and more specifically the term algebraic structure generally refers to a set (called carrier set or underlying set) with one or more finitely operations defined on it. Examples of algebraic structures include groups, rings, fields, and lattices. The algebraic structures of Fuzzy multisets are introduced in [11]. Adamu et al. [12] developed the concepts of normal and soft normal groups under multisets and soft multisets context. They defined the concepts of normal submultigroups and soft normal Multigroups and proved some of their related algebraic structures. In this paper we are extending these algebraic structures on intuitionistic fuzzy multisets and intuitionistic fuzzy multigroups by introducing a new concept named normal sub intuitionistic fuzzy multigroups.

2. Preliminaries

Definition 2.1. [13] Let ܺ be a set. A multiset (mset) ܯ drawn from ܺ is represented by a function count ܯ or ܥ_ெ defined as ܥ_ெ ∶ ܺ ⟶ ሼ0,1, 2, 3, … ሽ. For eachݔ ∈ ܺ, ܥ_ெሺݔሻ is the characteristic value of ݔ in ܯ. Here ܥ_ெሺݔሻ denotes the number of occurrences of ݔ in ܯ.

Definition 2.2. [14] LetX be a group. A multi set ܩ overX is a multi-group over X if

the count of ܩsatisfies the following two conditions. i. ܥ_ீሺݔݕሻ ≥ ܥ_ீሺݔሻ ∧ ܥ_ீሺݕሻ∀ݔ, ݕ ∈ ܺ; ii. ܥ_ீሺݔିଵሻ ≥ ܥ_ீሺݔሻ∀ݔ ∈ ܺ.

129

Definition 2.4. [16] Let ܩ be a group and ߤ ∈ ܨܲሺܩሻ (fuzzy power set of ܩ), then μ is called fuzzy subgroup of ܩ if

i. ߤሺݔݕሻ ≥ ߤሺݔሻ ∧ ߤሺݕሻ∀ݔ, ݕ ∈ ܺ and ii. ߤሺݔିଵሻ ≥ ߤሺݔሻ∀ݔ ∈ ܺ.

Definition 2.5. [10] Let ܺ be a nonempty set. An Intuitionistic Fuzzy Multiset ܣ denoted by IFMS drawn from ܺ is characterized by two functions : ‘count membership’ of

ܣሺܥܯ஺ሻ and ‘count non membership’ of ܣሺܥܰ_஺ሻ given respectively by ܥܯ_஺∶ ܺ ⟶ ܳ and ܥܰ_஺∶ ܺ ⟶ ܳ where ܳ is the set of all crisp multisets drawn from the unit interval

ሾ0,1ሿ such that for eachݔ ∈ ܺ, the membership sequence is defined as a decreasingly ordered sequence of elements in ܣሺܥܯ_஺ሻ which is denoted by (ߤ_஺ଵሺݔሻ, ߤ_஺ଶሺݔሻ, … , ߤ_஺௣ሺݔሻ) where (ߤ_஺ଵሺݔሻ ≥ ߤ_஺ଶሺݔሻ ≥, … , ߤ_஺௣ሺݔሻ) and the corresponding non membership sequence will be denoted by

( ߭_஺ଵሺݔሻ, ߭_஺ଶሺݔሻ, … , ߭_஺௣ሺݔሻ) such that 0 < ߤ_஺௜ሺݔሻ + ߭_஺௜ሺݔሻ < 1 for every ݔ ∈ ܺ and

݅ = 1,2, … , ݌.

An IFMS ܣ is denoted by

ܣ = ൛< ݔ ∶ ሺߤ஺ଵሺݔሻ, ߤ஺ଶሺݔሻ, … , ߤ஺௣ሺݔሻሻ , ሺ ߭஺ଵሺݔሻ, ߭஺ଶሺݔሻ, … , ߭஺௣ሺݔሻሻ >∶ ݔ ∈ ܺൟ

Definition 2.6. [10] Length of an element ݔ in an IFMS ܣ is defined as the Cardinality of

ܥܯ஺ሺݔሻ or ܥܰ_஺ሺݔሻ for which 0 < ߤ_஺௜ሺݔሻ + ߭_஺௜ሺݔሻ < 1 and it is denoted by ܮሺݔ ∶ ܣሻ. That is

ܮሺݔ ∶ ܣሻ = ⎸ܥܯ஺ሺݔሻ⎹ = ⎸ܥܰ஺ሺݔሻ⎹

ܮሺݔ ∶ ܣሻ = |ܥܯܣሺݔሻ| = |ܥܰܣሺݔሻ|

Definition 2.7. [10] If ܣ and ܤ are IFMSs drawn from ܺ then

ܮሺݔ ∶ ܣ, ܤሻ = ܯܽݔሼܮሺݔ ∶ ܣሻ, ܮሺݔ ∶ ܤሻሽ. Alternatively we use ܮሺݔሻ for ܮሺݔ ∶ ܣ, ܤሻ.

Definition 2.8. [11] For any two IFMSs ܣ and ܤ drawn from a set ܺ, the following operations and relations will hold. Let

ܣ = ൛< ݔ ∶ ሺߤ஺ଵሺݔሻ, ߤ஺ଶሺݔሻ, … , ߤ஺௣ሺݔሻሻ , ሺ ߭஺ଵሺݔሻ, ߭஺ଶሺݔሻ, … , ߭஺௣ሺݔሻሻ >∶ ݔ ∈ ܺൟ

And

ܤ = ൛< ݔ ∶ ሺߤ஺ଵሺݔሻ, ߤ஺ଶሺݔሻ, … , ߤ஺௣ሺݔሻሻ , ሺ ߭஺ଵሺݔሻ, ߭஺ଶሺݔሻ, … , ߭஺௣ሺݔሻሻ >∶ ݔ ∈ ܺൟ

Then

i. Inclusion

ܣ ⊂ ܤ ⇔ ߤ_஺௝ሺݔሻ ≤ ߤ_஺௝ሺݔሻ and ߭_஺௝ሺݔሻ + ߭_஺௝ሺݔሻ; ݆ = 1,2, … , ܮሺݔሻ, ݔ ∈ ܺ

ܣ = ܤ ⇔ ܣ ⊂ ܤand ܤ ⊂ ܣ ii. Complement

¬ܣ = ൛< ݔ ∶ ሺ ߭஺ଵሺݔሻ, ߭஺ଶሺݔሻ, … , ߭஺௣ሺݔሻሻ , ሺߤ஺ଵሺݔሻ, ߤ஺ଶሺݔሻ, … , ߤ஺௣ሺݔሻሻ >∶ ݔ ∈ ܺൟ

iii. Union ሺܣ ∪ ܤሻ

In ܣ ∪ ܤthe membership and non-membership values are obtained as follows.

I.M.Adamu, Y.Tella and A.J.Alkali

130

߭_஺∪஻௝ ሺݔሻ = ߭஺௝ሺݔሻ ∧ ߭஺௝ሺݔሻ, ݆ = 1,2, … , ܮሺݔሻ, ݔ ∈ ܺ.

iv. Intersection ሺܣ ∩ ܤሻ

In ܣ ∩ ܤ the membership and non-membership values are obtained as follows.

ߤ_஺∩஻௝ ሺݔሻ = ߤ_஺௝ሺݔሻ ∧ ߤ_஻௝ሺݔሻ ߭_஺∩஻௝ ሺݔሻ = ߭_஺௝ሺݔሻ ∨ ߭_஺௝ሺݔሻ, ݆ = 1,2, … , ܮሺݔሻ, ݔ ∈ ܺ.

Definition 2.9. [11] If ܺ and ܻ are two nonempty sets and ݂ ∶ ܺ → ܻ be a mapping. Then

i. The image of the FMS ܣ ∈ ܨܯሺܺሻ under the mapping ݂ is denoted by ݂ሺܣሻ, or

ܥܯ௙ሾ஺ሿሺݕሻ = ൜⋁௙ሺ௫ሻୀ௬ܥܯ஺ሺݔሻ ݂݅ ݂

ିଵ_{ሺݕሻ ≠ ߶}

0 ݋ݐℎ݁ݎݓ݅ݏ݁

ii. The inverse image of the FMS ܤ ∈ ܨܯሺܻሻ under the mapping ݂ is denoted by

݂ିଵ_{ሺܤሻ or݂}ିଵ_{ሾܤሿ, where ܥܯ}

௙షభ_ሾ஻ሿሺݔሻ = ܥܯ_஻݂ሾݔሿ.

Definition 2.10. [11] Let ܺ be a group. A fuzzy multiset ܩ over ܺ is a fuzzy multi group (FMG) over ܺ if the count (count membership) of ܩ satisfies the following two conditions.

i. ܥܯ_ீሺݔݕሻ ≥ ܥܯ_ீሺݔሻ ∧ ܥܯ_ீሺݕሻ ∀ݔ, ݕ ∈ ܺ. ii. ܥܯ_ீሺݔିଵሻ = ܥܯ_ீሺݔሻ ∀ݔ ∈ ܺ

Definition 2.11. [12] A submgroup ܪ of a mgroup ܯ ∈ ܯܩሺܺሻ is said to be a normal submgroup iff for any ℎ ∈ ܪ∗,

ܥுሺݔିଵℎݔሻ ≥ ܥுሺℎሻ, ∀ ݔ ∈ ܯ∗.

Theorem 2.12. [12] Let ܯ_ଵand ܯ_ଶ be submgroups of a mgroup

ܯ ∈ ܯܩሺܺሻsuch that ܯ_ଵ, ܯ_ଶ∈ ⊿℘ሺܯሻ, then i. ܯ_ଵ∩ ܯ_ଶ∈ ⊿℘ሺܯሻ;

ii. ܯ_ଵ∪ ܯ_ଶ∈ ⊿℘ሺܯሻ.

Theorem 2.13 [12] If ܪ ∈ ⊿℘ሺܯሻ then ܪ∗ is a normal subgroup of ܯ∗

Theorem 2.14. [12] Let ܺ be a group and ܯ ∈ ܯܩሺܺሻ. If ܯ ∈ ⊿℘ሺܯሻ, then ܯିଵ∈ ⊿℘ሺܯሻ.

Theorem 2.15. [12] Let ܺ, ܻbe two groups and ݂ ∶ ܺ ⟶ ܻ be an isomorphism of groups. Suppose ܯ_ଵ∈ ܯܩሺܺሻ, ܯ_ଶ∈ ܯܩሺܻሻ and ݂ሺܯ_ଵሻ ⊆ ܯ_ଶ

i. if ܪ ∈ ⊿℘ሺܯ_ଵሻ, then ݂ሺܪሻ ∈ ⊿℘ሺܯ_ଶሻ; ii. if ܪ ∈ ⊿℘ሺܯ_ଶሻ then ݂ିଵሺܪሻ ∈ ⊿℘ሺܯ_ଵሻ.

Definition 2.16. [17] Let ܺ be a group. An intuitionistic fuzzy multiset ܩ over ܺ is an intuitionistic fuzzy multi group (IFMG) over ܺ if the counts (count membership and non-membership) of ܩ satisfies the following two conditions.

131 ii. ܥܯ_ீሺݔିଵሻ ≥ ܥܯ_ீሺݔሻ ∀ݔ ∈ ܺ.

iii. ܥܰ_ீሺݔݕሻ ≤ ܥܯ_ீሺݔሻ ∧ ܥܯ_ீሺݕሻ ∀ݔ, ݕ ∈ ܺ. iv. ܥܰ_ீሺݔିଵሻ ≤ ܥܯ_ீሺݔሻ ∀ݔ ∈ ܺ.

Definition 2.17. [17] Let ܩ ∈ ܨܯܵሺܺሻ. Then define

ܩ∗_{= ሼݔ ∈ ܺ ∶ ܥܯ}

ீሺݔሻ = ܥܯீሺ݁ሻ ܽ݊݀ ܥܰீሺݔሻ = ܥܰீሺ݁ሻሽ.

Proposition 2.18. [17] Let ܩ ∈ ܨܯܩሺܺሻ. Then ܩ∗ is a subgroup of ܺ.

Definition 2.19. [17] Let ܩ ∈ ܨܯܵሺܺሻ. Let݆ ∈ ℕ. Then define

ܩ௝_{= ൛ݔ ∈ ܺ ∶ ߤ}

ீ௝ሺݔሻ ≥ 0, ߤீ௝ାଵሺݔሻ = 0 ܽ݊݀ ߭ீ௝ሺݔሻ = 0 ൟ .

Theorem 2.20. [17] Let ܩ ∈ ܨܯܩሺܺሻ. Then ܩ௝ is a subgroup of ܺ iff

ߤ_ீ௝ାଵሺݔݕିଵ_{ሻ = 0 ܽ݊݀ ߭}

ீ௝ାଵሺݔݕିଵሻ = 0 ∀ݔ, ݕ ∈ ܩ௝

3. Normal sub-intuitionistic fuzzy multigroups

Throughout this section, let ܺ be a group with binary operation and the identity element is ݁. Also we assume that the intuitionistic fuzzy multisets and intuitionistic fuzzy Multigroups are taken from ܫܨܯܵሺܺሻ and ܫܨܯܩሺܺሻrespectively.

Definition 3.1. A sub-intuitionistic fuzzy multigroup ܪ of an intuitionistic fuzzy multigroup ܩ ∈ ܫܨܯܩሺܺሻis said to be a normal sub-intuitionistic fuzzy Multigroups iff for any ℎ ∈ ܪ∗,

i. ܥܯ_ுሺݔିଵℎݔሻ ≥ ܥܯ_ுሺℎሻ, ∀ ݔ ∈ ܩ∗. ii. ܥܰ_ுሺݔିଵℎݔሻ ≤ ܥܰ_ுሺℎሻ∀ ݔ ∈ ܩ∗.

We denote the set of all normal sub-intuitionistic fuzzy Multigroups of an intuitionistic fuzzymgroup ܩ ∈ ܫܨܯܩሺܺሻ by ⊿℘ܵܫܨܯܩሺܩሻ and the normality of ܪ over ܩ ∈ ܫܨܯܩሺܺሻ byܪ ⊴ ܩ

Example 3.2. ሺΖ_ସ, +_ସሻ is a group. Then

ܩ = ൞

< 2: ሺ0.6,0.4,0.3,0.1ሻ, ሺ0.4,0.6,0.7,0.9ሻ >, < 1: ሺ0.8,0.7,0.7,0.5,0.1,0.1ሻ, ሺ0.2,0.3,0.3,0.5,0.9,0.9ሻ >, < 3: ሺ0.8,0.7,0.7,0.5,0.1,0.1ሻ, ሺ0.2,0.3,0.3,0.5,0.9,0.9ሻ >, < 0 ∶ ሺ0.9,0.8,0.7,0.5,0.1,0.1ሻ, ሺ0.1,0.8,0.3,0.5,0.9,0.9ሻ >, ൢ

is an intuitionistic fuzzy multi group. And

ܪ = ൞

< 2: ሺ0.6,0.4,0.3,0.1ሻ, ሺ0.4,0.6,0.7,0.9ሻ >, < 1: ሺ0.7,0.6,0.5,0.5,0.1,0.1ሻ, ሺ0.3,0.4,0.5,0.5,0.9,0.9ሻ >, < 3: ሺ0.7,0.6,0.5,0.5,0.1,0.1ሻ, ሺ0.3,0.4,0.5,0.5,0.9,0.9ሻ >, < 0 ∶ ሺ0.9,0.8,0.7,0.5,0.1,0.1ሻ, ሺ0.1,0.2,0.3,0.5,0.9,0.9ሻ > ൢ

is a normal sub-intuitionistic fuzzy multigroup of the intuitionistic fuzzy multigroup ܩ.

I.M.Adamu, Y.Tella and A.J.Alkali

132 i. ܩ_ଵ∩ ܩ_ଶ∈ ⊿℘ܵܫܨܯܩሺܩሻ; ii. ܩ_ଵ∪ ܩ_ଶ∈ ⊿℘ܵܫܨܯܩሺܩሻ.

Proof:

i. ܩ_ଵand ܩ_ଶ are sub-intuitionistic fuzzy mgroups, we have;

ܥܯ ீభሺݔݕሻ ≥ ܥܯீభሺݔሻ ∧ ܥܯ ீభሺݕሻ, ܥܯ ீభሺݔିଵሻ = ܥܯீభሺݔሻ ∀ ݔ, ݕ ∈ ܺ , ܥܯ ீమሺݔݕሻ ≥ ܥܯீమሺݔሻ ∧ ܥܯ ீమሺݕሻ, ܥܯ ீమሺݔିଵሻ = ܥܯீమሺݔሻ ∀ ݔ, ݕ ∈ ܺ .

∴ ܥܯ ீభ∩ீమሺݔݕሻ = ⋀൛ܥܯ ீభሺݔݕሻ, ܥܯீమሺݔݕሻൟ ≥ ⋀൛ൣܥீభሺݔሻ ∧ ܥீభሺݕሻ൧, ൣܥீమሺݔሻ ∧ ܥீమሺݕሻ൧ൟ = ܥܯீభሺݔሻ ∧ ܥܯ ீభሺݕሻ ∧ ܥܯீమሺݔሻ ∧ ܥܯ ீమሺݕሻ = ൣܥܯீభሺݔሻ ∧ ܥܯ ீమሺݔሻ൧ ∧ ൣܥܯீభሺݕሻ ∧ ܥܯ ீమሺݕሻ൧

= ܥܯ ீభ∩ீమሺݔሻ ∧ ܥܯ ீభ∩ீమሺݕሻ (1) ܥܯ ீభ∩ீమሺݔିଵሻ = ܥܯ ீభሺݔିଵሻ ∧ ܥܯ ீమሺݔିଵሻ

= ܥܯ ீభሺݔሻ ∧ ܥܯ ீమሺݔሻ

= ܥܯ ீభ∩ீమሺݔሻ (2)

Next we also have;

ܥܰீభሺݔݕሻ ≤ ܥܰீభሺݔሻ ∧ ܥܰீభሺݕሻ, ܥܰீభሺݔିଵሻ = ܥܰீభሺݔሻ ∀ ݔ, ݕ ∈ ܺ , ܥܰ ீమሺݔݕሻ ≤ ܥܰீమሺݔሻ ∧ ܥܰ ீమሺݕሻ, ܥܰ ீమሺݔିଵሻ = ܥܰீమሺݔሻ ∀ ݔ, ݕ ∈ ܺ .

∴ ܥܰ ீభ∩ீమሺݔݕሻ = ⋀൛ܥܰ ீభሺݔݕሻ, ܥܰீమሺݔݕሻൟ ≤ ⋁൛ൣܥܰீభሺݔሻ ∧ ܥܰ ீభሺݕሻ൧, ൣܥܰீమሺݔሻ ∧ ܥܰ ீమሺݕሻ൧ൟ

= ܥܰீభሺݔሻ ∧ ܥܰ ீభሺݕሻ ∧ ܥܰீమሺݔሻ ∧ ܥܰ ீమሺݕሻ = ൣܥܰீభሺݔሻ ∧ ܥܰ ீమሺݔሻ൧ ∧ ൣܥܰீభሺݕሻ ∧ ܥܰ ீమሺݕሻ൧

= ܥܰ ீభ∩ீమሺݔሻ ∧ ܥܰ ீభ∩ீమሺݕሻ (3) ܥܰ ீభ∩ீమሺݔିଵሻ = ܥܰ ீభሺݔିଵሻ ∧ ܥܰ ீమሺݔିଵሻ

= ܥܰ ீభሺݔሻ ∧ ܥܰ ீమሺݔሻ

= ܥܰ ீభ∩ீమሺݔሻ (4)

Thus, from (1), (2), (3) and (4), we have

ܩଵ∩ ܩଶ∈ ܫܨܯܩሺܺሻ (5)

Next

Since ܩ_ଵ⊆ ܩ and ܩ_ଶ⊆ ܩ, then ܩ_ଵ∩ ܩ_ଶ⊆ ܩ Since ܩ_ଵ, ܩ_ଶ∈ ⊿℘ܵܫܨܯܩሺܯሻ, we have

ܥܯ ீభሺݔିଵݕଵݔሻ ≥ ܥܯீభሺݕଵሻ, ∀ ݔ ∈ ܩ∗ ,ݕଵ∈ ܩଵ∗; ܥܯீమሺݔିଵݕଶݔሻ ≥ ܥܯீమሺݕଶሻ, ∀ ݔ ∈ ܩ∗, ݕଶ∈ ܩଶ∗;

Now

ܥܯ ீభ∩ீమሺݔିଵݕݔሻ = ⋀൛ܥܯ ீభሺݔିଵݕݔሻ, ܥܯீమሺݔିଵݕݔሻൟ ≥ ⋀൛ܥܯ ீభሺݕሻ, ܥܯீమሺݕሻൟ∀ ݔ ∈ ܩ∗, ݕ ∈ ሺܩଵ∩ ܩଶሻ∗= ܩଵ∗∩ ܩଶ∗

Since ܥܯ _ீభሺݔିଵݕ_ଵݔሻ ≥ ܥܯ_ீభሺݕ_ଵሻ and ܥܯ_ீమሺݔିଵݕ_ଶݔሻ ≥ ܥܯ_ீమሺݕ_ଶሻ

∴ ܥܯ ீభ∩ீమሺݔିଵݕݔሻ ≥ ܥܯ ீభሺݕሻ ∧ ܥܯீమሺݕሻ = ܥܯ ீభ∩ீమሺݕሻ

Thus ܥܯ _{ீభ∩ீమ}ሺݔିଵݕݔሻ ≥ ܥܯ _{ீభ∩ீమ}ሺݕሻ, ∀ ݔ ∈ ܩ∗, ݕ ∈ ܩ_ଵ∩ ܩ_ଶ (6) Similarly,

133 Therefore,

ܥܰ ீభ∩ீమሺݔିଵݕݔሻ = ⋁൛ܥܰ ீభሺݔିଵݕݔሻ, ܥܰீమሺݔିଵݕݔሻൟ ≤ ⋁൛ܥܰ ீభሺݕሻ, ܥܰீమሺݕሻൟ∀ ݔ ∈ ܩ∗, ݕ ∈ ሺܩଵ∩ ܩଶሻ∗= ܩଵ∗∩ ܩଶ∗

Since ܥܰ _ீభሺݔିଵݕ_ଵݔሻ ≤ ܥܰ_ீభሺݕ_ଵሻ and ܥܰ_ீమሺݔିଵݕ_ଶݔሻ ≤ ܥܰ_ீమሺݕ_ଶሻ

∴ ܥܰ ீభ∩ீమሺݔିଵݕݔሻ ≤ ܥܰ ீభሺݕሻ ∧ ܥܰீమሺݕሻ = ܰሺݕሻ

Thus ܥܰ _{ீభ∩ீమ}ሺݔିଵݕݔሻ ≤ ܥܰ _{ீభ∩ீమ}ሺݕሻ, ∀ ݔ ∈ ܩ∗, ݕ ∈ ܩ_ଵ∩ ܩ_ଶ (7) From equations (5), (6) and (7) we have, ܩ_ଵ∩ ܩ_ଶ∈ ⊿℘ܵܫܨܯܩሺܯሻ.

ii. Since ܩ_ଵand ܩ_ଶ are sub-intuitionistic fuzzy mgroups, we have;

ܥܯ ீభሺݔݕሻ ≥ ܥܯீభሺݔሻ ∧ ܥܯ ீభሺݕሻ, ܥܯ ீభሺݔିଵሻ = ܥܯீభሺݔሻ ∀ ݔ, ݕ ∈ ܺ , ܥܯ ீమሺݔݕሻ ≥ ܥܯீమሺݔሻ ∧ ܥܯ ீమሺݕሻ, ܥܯ ீమሺݔିଵሻ = ܥܯீమሺݔሻ ∀ ݔ, ݕ ∈ ܺ .

∴ ܥܯ ீభ∪ீమሺݔݕሻ = ⋁൛ܥܯ ீభሺݔݕሻ, ܥܯீమሺݔݕሻൟ ≥ ⋁൛ൣܥܯீభሺݔሻ ∧ ܥܯ ீభሺݕሻ൧, ൣܥܯீమሺݔሻ ∧ ܥܯ ீమሺݕሻ൧ൟ

= ൣܥܯீభሺݔሻ ∧ ܥܯ ீమሺݔሻ൧⋁ൣܥܯீభሺݕሻ ∧ ܥܯ ீమሺݕሻ൧ = ൣܥܯீభሺݔሻ⋁ܥܯ ீమሺݔሻ൧⋀ൣܥܯீభሺݕሻ⋁ܥܯ ீమሺݕሻ൧

= ܥܯ ீభ∪ீమሺݔሻ ∧ ܥܯ ீభ∪ீమሺݕሻ (1) ܥܯ ீభ∪ீమሺݔିଵሻ = ܥܯ ீభሺݔିଵሻ⋁ܥܯ ீమሺݔିଵሻ

= ܥ ெభሺݔሻ⋁ܥ ெమሺݔሻ = ܥ ெభ∪ெమሺݔሻ

∴ ܥܯ ீభ∪ீమሺݔିଵሻ = ܥܯ ீభ∪ீమሺݔሻ (2)

We also have;

ܥܰ ீభሺݔݕሻ ≥ ܥܰீభሺݔሻ ∧ ܥܰ ீభሺݕሻ, ܥܰ ீభሺݔିଵሻ = ܥܰீభሺݔሻ ∀ ݔ, ݕ ∈ ܺ , ܥܰ ீమሺݔݕሻ ≥ ܥܰீమሺݔሻ ∧ ܥܰ ீమሺݕሻ, ܥܰ ீమሺݔିଵሻ = ܥܰீమሺݔሻ ∀ ݔ, ݕ ∈ ܺ .

∴ ܥܰ ீభ∪ீమሺݔݕሻ = ⋁൛ܥܰ ீభሺݔݕሻ, ܥܰீమሺݔݕሻൟ ≤ ⋁൛ൣܥܰீభሺݔሻ ∧ ܥܰ ீభሺݕሻ൧, ൣܥܰீమሺݔሻ ∧ ܥܰ ீమሺݕሻ൧ൟ

= ൣܥܰீభሺݔሻ ∧ ܥܰ ீమሺݔሻ൧⋁ൣܥܰீభሺݕሻ ∧ ܥܰ ீమሺݕሻ൧ = ൣܥܰீభሺݔሻ⋁ܥܰ ீమሺݔሻ൧⋀ൣܥܰீభሺݕሻ⋁ܥܰ ீమሺݕሻ൧

= ܥܰ ீభ∪ீమሺݔሻ ∧ ܥܰ ீభ∪ீమሺݕሻ (3) ܥܰ ீభ∪ீమሺݔିଵሻ = ܥܰ ீభሺݔିଵሻ⋁ܥܰ ீమሺݔିଵሻ

= ܥܰ ீభሺݔሻ⋁ܥܰ ீమሺݔሻ = ܥܰ ீభ∪ீమሺݔሻ

∴ ܥܰ ீభ∪ீమሺݔିଵሻ = ܥܰ ீభ∪ீమሺݔሻ (4)

Thus, from (1), (2), (3) and (4), we have

ܩଵ∪ ܩଶ∈ ܫܨܯܩܯܩሺܺሻ (5)

Next

Since ܩ_ଵ⊆ ܩ and ܩ_ଶ⊆ ܩ, then ܩ_ଵ∪ ܩ_ଶ⊆ ܩ Since ܩ_ଵ, ܩ_ଶ∈ ⊿℘ܵܫܨܯܩሺܯሻ, we have

ܥܯ ீభሺݔିଵݕଵݔሻ ≥ ܥܯீభሺݕଵሻ, ∀ ݔ ∈ ܩ∗ ,ݕଵ∈ ܩଵ∗; ܥܯீమሺݔିଵݕଶݔሻ ≥ ܥܯீమሺݕଶሻ, ∀ ݔ ∈ ܩ∗, ݕଶ∈ ܩଶ∗;

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Since ܥܯ _ீభሺݔିଵݕ_ଵݔሻ ≥ ܥܯ_ீభሺݕ_ଵሻ and ܥܯ_ீమሺݔିଵݕ_ଶݔሻ ≥ ܥܯ_ீమሺݕ_ଶሻ

∴ ܥܯ ீభ∪ீమሺݔିଵݕݔሻ ≥ ܥܯ ீభሺݕሻ⋁ܥܯீమሺݕሻ = ܥܯ ீభ∪ீమሺݕሻ

Thus ܥܯ _{ீభ∪ீమ}ሺݔିଵݕݔሻ ≥ ܥܯ _{ீభ∪ீమ}ሺݕሻ, ∀ ݔ ∈ ܩ∗, ݕ ∈ ܩ_ଵ∪ ܩ_ଶ (6) Similarly,

ܥܰ ீభሺݔିଵݕଵݔሻ ≤ ܥܰீభሺݕଵሻ, ∀ ݔ ∈ ܩ∗ ,ݕଵ∈ ܩଵ∗; ܥܰீమሺݔିଵݕଶݔሻ ≤ ܥܰீమሺݕଶሻ, ∀ ݔ ∈ ܩ∗, ݕଶ∈ ܩଶ∗;

∴ ܥܰ ீభ∪ீమሺݔିଵݕݔሻ = ⋁൛ܥܰ ீభሺݔିଵݕݔሻ, ܥܰீమሺݔିଵݕݔሻൟ ≤ ⋁൛ܥܰ ீభሺݕሻ, ܥܰீమሺݕሻൟ∀ ݔ ∈ ܩ∗, ݕ ∈ ሺܩଵ∪ ܩଶሻ∗= ܩଵ∗∪ ܩଶ∗

Since ܥܰ _ீభሺݔିଵݕ_ଵݔሻ ≤ ܥܰ_ீభሺݕ_ଵሻ and ܥܰ_ீమሺݔିଵݕ_ଶݔሻ ≤ ܥܰ_ீమሺݕ_ଶሻ

∴ ܥܰ ீభ∪ீమሺݔିଵݕݔሻ ≤ ܥܰ ீభሺݕሻ ∧ ܥܰீమሺݕሻ = ܥܰ ீభ∪ீమሺݕሻ

Thus ܥܰ _{ீభ∪ீమ}ሺݔିଵݕݔሻ ≤ ܥܰ _{ீభ∪ீమ}ሺݕሻ, ∀ ݔ ∈ ܩ∗, ∈ ܩ_ଵ∪ ܩ_ଶ (7) From equations (5), (6) and (7) we have, ܩ_ଵ∪ ܩ_ଶ∈ ⊿℘ܵܫܨܯܩሺܯሻ.

Remark 3.4. Let ሼܩ_௜ ∶ ܩ_௜ ∈ ⊿℘ܵܫܨܯܩሺܯሻ, ݅ ∈ ∆ሽ then ⋂_௜∈∆ܩ_௜ ∈ ⊿℘ܵܫܨܯܩሺܯሻ and ⋃_௜∈∆ܯ_௜ ∈ ⊿℘ሺܯሻ.

Proposition 3.5. If ܪ ∈ ⊿℘ܵܫܨܯܩሺܩሻ then ܪ∗ is a normal sub-intuitionistic group of ܩ∗.

Proof:

Since ܪ ⊆ ܩ ⟹ ܪ∗⊆ ܩ∗

Thus, ܪ∗ is a sub-intuitionistic group of ܩ∗

Since ܪ ∈ ⊿℘ܵܫܨܯܩሺܩሻ, then for any ݔ ∈ ܩ∗and ℎ ∈ ܪ∗

ܥܯுሺݔିଵℎݔሻ ≥ ܥܯுሺℎሻ > 0

∴ ܥܯுሺݔିଵℎݔሻ > 0

Similarly,

ܥܰுሺݔିଵℎݔሻ ≤ ܥܰுሺℎሻ = 0

∴ ܥܯுሺݔିଵℎݔሻ = 0

⟹ ሺݔିଵ_{ℎݔሻ ∈ ܪ}∗

∴ ܪ∗_{is a normal sub-intuitionistic group of ܩ}∗_.

Proposition 3.6. Let ܩ ∈ ܨܯܩሺܺሻand ܪ ∈ ⊿℘ܵܫܨܯܩሺܩሻ. Then ܪ௝∗ is a normal subgroup of ܩ∗ iff

ߤ_ு௝ାଵሺݔିଵ_{ݕݔሻ = 0 ܽ݊݀ ߭}

ு௝ାଵሺݔିଵݕݔሻ = 0 ∀ݔ ∈ ܩ∗ ܽ݊݀ ݕ ∈ ܪ௝∗.

Proof: Let ݔ ∈ ܩ∗ ܽ݊݀ ݕ ∈ ܪ௝∗. It implies that

ߤ_ு௝ሺݔିଵ_{ݕݔሻ ≥ 0 And ߤ}

ு௝ାଵሺݔିଵݕݔሻ = 0 (by definition 2.19)

߭_ு௝ሺݔିଵ_{ݕݔሻ = 0 and ߭}

ு௝ାଵሺݔିଵݕݔሻ = 0

Assume ߤ_ு௝ାଵሺݔିଵݕݔሻ = 0 ܽ݊݀ ߭_ு௝ାଵሺݔିଵݕݔሻ = 0 ∀ݔ ∈ ܩ∗ ܽ݊݀ ݕ ∈ ܪ௝∗ Then by the above proposition,

ߤ_ு௝ሺݔିଵ_{ݕݔሻ > 0 and ߭}

ு௝ሺݔିଵݕݔሻ = 0

⟹ ݔିଵ_{ݕݔ ∈ ܪ}௝∗_{. Then ܪ}௝∗ _{is a normal subgroup of ܩ}∗_{. Hence the proof}

Conversely,

ܪ௝∗_{is a normal subgroup ofܩ}∗_{. Thenݔ ∈ ܩ}∗ _{, ݕ ∈ ܪ}௝∗ _{⟹ ݔ}ିଵ_{ݕݔ ∈ ܪ}௝∗

⟹ ߤ_ு௝ାଵሺݔିଵ_{ݕݔሻ = 0 ܽ݊݀ ߭}

135

Proposition 3.7. Let ܺ be a group and ܩ ∈ ܫܨܯܩሺܺሻ. If ܩ ∈ ⊿℘ܵܫܨܯܩሺܩሻ, then

ܩିଵ _{∈ ⊿℘ܵܫܨܯܩሺܩሻ.}

Proof: Since ܩ ∈ ⊿℘ܵܫܨܯܩሺܩሻ, then we have

ܥܯெሺݔିଵݕݔሻ ≥ ܥܯெሺݕሻ, ∀ ݔ ∈ ܺ and ݕ ∈ ܩ∗ Also

ܥܯ_ீషభሺݔݕሻ = ܥܯ_ீሾሺݔݕሻିଵሿ = ܥܯ_ீሺݔݕሻ ≥ ܥܯ_ீሺݔሻ ∧ ܥܯ_ீሺݕሻ

= ܥܯீሺݔିଵሻ ∧ ܥܯீሺݕିଵሻ

= ܥܯ_ீషభሺݔሻ ∧ ܥܯ_ீషభሺݕሻ ∴ ܥܯீషభሺݔݕሻ ≥ ܥܯ_ீሺݔିଵሻ ∧ ܥܯ_ீሺݕିଵሻ ܥܯ_ீషభሺݔିଵሻ = ܥܯ_ீሾሺݔିଵሻିଵሿ = ܥܯ_ீሺݔሻ

= ܥܯீሺݔିଵሻ = ܥܯீషభሺݔሻ ∴ ܥܯீషభሺݔିଵሻ = ܥܯ_ீషభሺݔሻ

Thus, ܩିଵ∈ ܫܨܯܩሺܺሻ

ܥܯீషభሺݔିଵݕݔሻ = ܥܯ_ீሾሺݔିଵݕݔሻିଵሿ = ܥܯ_ீሺݔିଵݕݔሻ ≥ ܥܯ_ீሺݕሻ, ∀ ݔ ∈ ܺ

And ݕ ∈ ܩ∗

∴ ܥܯ_ீషభሺݔିଵݕݔሻ = ܥܯ_ீሾሺݔିଵݕݔሻିଵሿ = ܥܯீሺݔିଵݕݔሻ ≥ ܥܯீሺݕሻ = ܥܯீషభሺݕሻ ⟹ ܥܯீషభሺݔିଵݕݔሻ ≥ ܥܯ_ீషభሺݕሻ(1)

We also have;

ܥܰீሺݔିଵݕݔሻ ≥ ܥܰீሺݕሻ, ∀ ݔ ∈ ܺand ݕ ∈ ܩ∗ And

ܥܰ_ீషభሺݔݕሻ = ܥܰ_ீሾሺݔݕሻିଵሿ = ܥܰ_ீሺݔݕሻ ≥ ܥܰ_ீሺݔሻ ∧ ܥܰ_ீሺݕሻ = ܥܰீሺݔିଵሻ ∧ ܥܰீሺݕିଵሻ

= ܥܰ_ீషభሺݔሻ ∧ ܥܰ_ீషభሺݕሻ

∴ ܥܰீషభሺݔݕሻ ≥ ܥܰ_ீሺݔିଵሻ ∧ ܥܰ_ீሺݕିଵሻ ܥܰ_ீషభሺݔିଵሻ = ܥܰ_ீሾሺݔିଵሻିଵሿ = ܥܰ_ீሺݔሻ = ܥܰீሺݔିଵሻ = ܥܰீషభሺݔሻ

∴ ܥܰ_ீషభሺݔିଵሻ = ܥܰ_ீషభሺݔሻ

Thus, ܩିଵ∈ ܫܨܯܩሺܺሻ

ܥܰீషభሺݔିଵݕݔሻ = ܥܰ_ீሾሺݔିଵݕݔሻିଵሿ = ܥܰ_ீሺݔିଵݕݔሻ ≥ ܥܰ_ீሺݕሻ, ∀ ݔ ∈ ܺ

and ݕ ∈ ܩ∗

∴ ܥܰ_ீషభሺݔିଵݕݔሻ = ܰሾሺݔିଵݕݔሻିଵሿ = ܥܰீሺݔିଵݕݔሻ ≥ ܥܰீሺݕሻ = ܥܰீషభሺݕሻ ⟹ ܥܰீషభሺݔିଵݕݔሻ ≥ ܥܰ_ீషభሺݕሻ(2)

Hence from (1) and (2) we have ܩିଵ∈ ⊿℘ܵܫܨܯܩሺܩሻ

Proposition 3.8. Let ܺ, ܻ be two groups and ݂ ∶ ܺ ⟶ ܻ be an isomorphism of groups. Suppose ܩ_ଵ∈ ܫܨܯܩሺܺሻ, ܩ_ଶ∈ ܫܨܯܩሺܻሻ and ݂ሺܩ_ଵሻ ⊆ ܩ_ଶ

i. if ܪ ∈ ⊿℘ܵܫܨܯܩሺܩ_ଵሻ, then ݂ሺܪሻ ∈ ⊿℘ܵܫܨܯܩሺܩ_ଶሻ. ii. ifܪ ∈ ⊿℘ܵܫܨܯܩሺܩ_ଶሻ then ݂ିଵሺܪሻ ∈ ⊿℘ܵܫܨܯܩሺܩ_ଵሻ.

Proof: i. Let ܪ ∈ ⊿℘ܵܫܨܯܩሺܩ_ଵሻ. Clearly ܪ is a sub-intuitionistic fuzzy mgroup of

ܩଵ(by definition).

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ܥܯ௙ሺுሻሺݔݕሻ =⋁I(})MLN](|) Since G() ≠ ∅ (by definition 2.9. (i)) But G is 1-1 and onto (by hypothesis), thusG() = |.

In particular,

I(])() = ](|) = ]!G()$ = ](G()G()) (1) (G is an isomorphism). But

](G()G()) ≥ ]!G()$⋀]!G()$ (2) ([is submgroup of ). But

]!G()$⋀]!G()$ = _!IYZ_$YZ_(])()⋀_!IYZ_$YZ_(])() (3) (by definition 2.9.(ii)) and

_!IYZ_$YZ_(])()⋀_!IYZ_$YZ_(])() = I(])()⋀I(])() (4) Thus using (1), (2), (3) and (4), we have:

I(])() ≥ I(])()⋀I(])() (5)

I(])() = ⋁I(N)MLYZ]() (Since G() ≠ ∅ (by definition 2.9.(i))

But G is 1-1 and onto (by hypothesis), thusG() = . In particular,

I(])() = ]() = ]!G()$ = ]!G()$€ (6) But

]!G()$€ ≥ ]!G()$ = _!IYZ_$YZ_(])() = I(])() (7) Thus from (6) and (7) we have

I(])() ≥ I(])() (8)

Also

+I(])() = +](|) = +]!G()$ = +](G()G()) (9) ( Gis an isomorphism). But

+](G()G()) ≤ +]!G()$⋀+]!G()$ (10) ([is submgroup of ). But

+]!G()$⋀+]!G()$ = +_!IYZ_$YZ_(])()⋀+_!IYZ_$YZ_(])() (11) (by definition 2.9.(ii)) and

+_!IYZ_$YZ_(])()⋀+_!IYZ_$YZ_(])() = +I(])()⋀+I(])() (12) Thus using (9), (10), (11) and (12), we have:

+I(])() ≤ +I(])()⋀+I(])() (13)

+I(])() = +]() = +]!G()$ = +]!G()$€ (14) But +_]!G()$€ ≤ _]!G()$ = +_!IYZ$YZ_(])() = +_I(])() (15)

Thus from (14) and (15) we have

+I(])() ≤ +I(])() (16)

From (5), (8),(13) and (16) we have G([) ∈ f((F) (17) But G() ⊆ _-, since [ ⊆ , then G([) ⊆

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i.e. _I(])() ≥ _I(])() for all ∈ _-∗ and ∈ !G([)$∗ and +_I(])() ≤ _I(])() for all ∈ _-∗ and ∈ !G([)$∗ Now

I(])() = ⋁I(})MLYZ_NL](|)= _]!G()$ (19) (SinceG is 1-1 and onto by definition)

But _]!G()$ = _]!G()G()G()$ and

]!G()G()G()$ = ]!G()$G()G()‚ (20) (since G is an isomorphism by hypothesis)

But

]!G()$G()G()‚ ≥ ]!G()$ = I(])() (21) From (19), (20) and (21), we have

I(])() ≥ I(])() (22) Similarly,

+I(])() = ⋁I(})MLYZ_NL+](|)= +_]!G()$ (23) (SinceG is 1-1 and onto by definition)

But +_]!G()$ = +_]!G()G()G()$ and

+]!G()G()G()$ = +]!G()$G()G()‚ (24) (since G is an isomorphism by hypothesis)

But

+]!G()$G()G()‚ ≤ +]!G()$ = +I(])() (25) From (23), (24) and (25), we have

+I(])() ≤ +I(])() (26) Hence, using (18), (22) and (26) we deduce that G([) ∈ ⊿℘af((_-)

ii. Let [ be a sub-intuitionistic fuzzy mgroup of _-.

We show that G([) is a sub-intuitionisticfuzzy mgroup of . But _IYZ_(])() = _]%G()& (by definition)

= ]%G()G()& (Gis an isomorphism)

≥ ]%G()& ∧ ]%G()& (by definition)

= _IYZ_(])() ∧ _IYZ_(])()

∴ _IYZ_(])() ≥ _IYZ_(])() ∧ _IYZ_(])() (27)

_IYZ_(])() = _]%G()& (by definition)

= ]ƒ!G()$„ (Gis an isomorphism)

= ]%G()& ([is a submgroup)

= _IYZ_(])() (by definition)

Thus _IYZ_(])() = _IYZ_(])() (28)

I.M.Adamu, Y.Tella and A.J.Alkali

138

≤ +]%G()& ∧ +]%G()& (by definition)

= +IYZ_(])() ∧ +_IYZ_(])()

∴ +_IYZ_(])() ≤ +_IYZ_(])() ∧ +_IYZ_(])() (29)

+_IYZ_(])() = +_]%G()& (by definition)

= +]ƒ!G()$„ (Gis an isomorphism)

= +]%G()& ([is a submgroup)

= +_IYZ_(])() (by definition)

Thus +_IYZ_(])() = +_IYZ_(])() (30) Now for all ∈ ∗ and ∈ %G([)&∗ we have

_IYZ_(])() = _]%G()& (by definition)

= ]%(G())G()G()& (G is an isomorphism)

≥ ]%G()& ([ ∈ ⊿℘af((_-)by hypothesis)

= IYZ_(])() (by definition)

∴ IYZ_(])() ≥ _IYZ_(])() (31)

_IYZ_(])() = _]%G()& (by definition)

= +]%(G())G()G()& (G is an isomorphism)

≤ ]%G()& ([ ∈ ⊿℘af((_-)by hypothesis)

= +_IYZ_(])() (by definition)

∴ +IYZ_(])() ≤ +_IYZ_(])()(6)

Hence from (27) to (31) we have G([) is a normal subintuitionistic fuzzy mgroup of .

4. Conclusion

The paper developed the concept of normal sub-intuitionistic fuzzy Multigroup with some of its related algebraic structures such as intersection, union, of any two normal sub-intuitionistic fuzzy multigroups are also normal. It also showed that the inverse of any normal intuitionistic fuzzy multigroup is normal and for any normal sub-intuitionistic multigroup, the root (support) set of normal sub-sub-intuitionistic fuzzy multigroup is a normal sub-intuitionistic fuzzy group. Finally, it showed that under the isomorphism function between any two groups, the image of a normal sub-intuitionistic fuzzy multigroup under the isomorphism is a normal sub-intuitionistic fuzzy multigroup and the inverse image of a normal sub-intuitionistic fuzzy multigroup under the isomorphism is a normal sub-intuitionistic fuzzy multigroup.

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