On Some Properties of Anti-Q-Fuzzy Normal Subgroups

arXiv: General Mathematics, 2019

the notation of intuitionistic Q-fuzzy semi-prime ideal in a semigroup. Solairaju and Nagarajan, in 2009, defined the notation of Q-fuzzy subgroup. Further, in 2013, Sharma [4] introduced new algebraic structure of α-fuzzy subgroup. Palaniappan and Muthuraj, in 2004, proved many results in homomorphism and anti-homomorphism in fuzzy subgroups. The purpose of this work is to define notations of α-Qfuzzy subset and α-Q-fuzzy subgroup, and we prove some elementary algebraic properties related to these two notations. Furthermore, the concept of anti-homomorphism in α-Q-fuzzy subgroup between two groups is defined and some properties related to an α-Q-fuzzy abelian and cyclic α-Q-fuzzy subgroups are studied. 2. PRELIMINARY Definition 2.1. Zadeh (1965)-Let X be a non-empty set. A fuzzy subset θ of the set X is a mapping θ: X→[0, 1]. Definition 2.2. Rosenfeld (1971)-Let (G, *) be a group. A fuzzy subset θ of G said to be a fuzzy subgroup of G, if the following conditions hold: i. θ(xy) ≥ min {θ(x), θ(y)} and ii. θ(x −1) ≥ θ(x), for all x and y in G. Definition 2.3. Solairaju and Nagarajan (2006)-Let X and Q be non-empty sets. A Q-fuzzy subset θ of the set X is a mapping. i. θ: X×Q→[0, 1]. Definition 2.4. Solairaju and Nagarajan (2006)-Let (G, *) be a group and Q be a non-empty set. A Q-fuzzy subset θ of G is said to be a Q-fuzzy subgroup of G if the following conditions are satisfied: i. θ(xy, q) ≥ min {θ(x, q), θ(y, q)} and ii. θ(x −1 , q) ≥ θ(x, q), for all x and y in G and q in Q. Proposition 2.5. Solairaju and Nagarajan (2006)-Let θ and σ be any two Q-fuzzy subsets of a non-empty set X. Then, for all x∈X and q∈Q, the followings are true: i. θ⊆σ ⇔ θ(x, q) ≤ σ(x, q), ii. θ=σ ⇔ θ(x, q) = σ(x, q), iii. (θ∪σ)(x, q) = max {θ(x, q), σ(x, q)}, iv. (θ∩σ)(x, q) = min {θ(x, q), σ(x, q)}. Definition 2.6. Solairaju and Nagarajan (2006)-If θ is a Q-fuzzy subgroup of a group G, then C[θ] is the complement of Q-fuzzy subgroup θ, and is defined by C[θ(x, q)] = 1-θ (x, q), for all x in G and q in Q. Definition 2.7. Palaniappan and Muthuraj (2004)-If (G, *) and (G′,ο) are any two groups, then the function f: G→G′ is called a homomorphism if f (x * y) = f(x) ο f(y), for all x and y in G. Definition 2.8. Palaniappan and Muthuraj (2004) If (G, *) and (G′,ο) are any two groups, then the function f: G→G′ is called an anti-homomorphism if f(x*y) = f(y) ο f(x), for all x and y in G. Definition 2.9. Palaniappan and Muthuraj (2004) Let f: X→X′ be any mapping from a non-empty set X into a non

2014

This paper is to further investigate some properties of an anti fuzzy subgroup of a group in relation to pseudo coset. It also uses isomorphism theorems to establish some results in relation to level subgroups of a fuzzy subgroup μ of a group G.

International Journal of Computer Applications, 2010

In this paper, we have given Independent Proof of Several theorems on pseudo fuzzy cossets of fuzzy normal subgroups We investigate the notion of pseudo fuzzy double cossets, pseudo fuzzy middle cossets of a group and consider its fundamental properties. Cartesian product fuzzy relations are also discussed.

2012

On the basis of fuzzy sets introduced by L.A. Zadeh, we first gave the definition of -anti fuzzy set and then defined -anti fuzzy subgroups and -anti fuzzy normal subgroups and finally, defined quotient group of the -anti fuzzy cosets of an -anti fuzzy normal subgroup. This paper proves a necessary and sufficient condition of -anti fuzzy subgroup (normal subgroup) to be anti fuzzy subgroup (normal subgroup). Some properties of quotient group of -anti fuzzy normal subgroups are also discussed.

International Journal of Algebra and Statistics, 2016

More general form of "quasi-coincident with" relation (q) is introduced, and related properties are investigated. The notions of (∈, ∈ ∨ q δ 0)-fuzzy subgroups, q δ 0-level sets and ∈ ∨ q δ 0-level sets are introduced, and related results are investigated. Relations between an (∈, ∈)-fuzzy subgroup and an (∈, ∈ ∨ q δ 0)-fuzzy subgroup are discussed, and characterizations of (∈, ∈ ∨ q δ 0)-fuzzy subgroups are displayed by using level sets and ∈ ∨ q δ 0-level sets. The concepts of ∈ ∨ q δ 0-admissible fuzzy sets, admissible (∈, ∈ ∨ q δ 0)-fuzzy subgroups and δ-characteristic fuzzy sets are introduced. Using these notions, characterizations of admissible (∈, ∈ ∨ q δ 0)-fuzzy subgroups and admissible subgroups are considered. 2010 Mathematics Subject Classification. 20N25, 03E72. Keywords. (∈, ∈ ∨ q δ 0)-fuzzy subgroup, q δ 0-level set, ∈ ∨ q δ 0-admissible fuzzy set.

Thai Journal of Mathematics, 2019

In this paper, proof of some results relating to fuzzy subgroup have been provided. Some new theorems on fuzzy cosets of fuzzy subgroup of a group are also stated and proved.

International Journal of Computer Applications, 2013

In this paper, proofs of some theorems relating to fuzzy subgroups, pseudo fuzzy cosets and pseudo fuzzy double cosets have been provided. Some new theorems are also stated and proved.

Using the notion of anti fuzzy points and its besideness to and non-quasi-coincidence with a fuzzy set, new concepts of an anti fuzzy subgroup are introduced and their inter-relations are investigated.

In this manuscript, we present notions of anti bipolar Q-fuzzy normal semi groups and we present characterization of anti bipolar Q – fuzzy normal semi groups. Further, we have characterized a semi group that a semi lattice of groups in terms of anti bipolar Q – fuzzy ideals, generalized anti bipolar Q – fuzzy bi – ideals and anti bipolar Q – fuzzy quasi – ideals. Keywords: semi lattice of groups, semi lattice of archimedean semi groups, bipolar Q – fuzzy sets, anti bipolar Q – fuzzy ideals, anti bipolar Q – fuzzy co – normal, anti bipolar Q – fuzzy bi – normal

2019

In this research, we introduce a hyper Q-fuzzy HX subgroup of a HX group ν notion also we discuss a few of its properties. Also we define a hyper Qfuzzy normal HX subgroup, normal level subgroup HX of hyper Q-fuzzy normal HX subgroup of a HX group also self conjugate and we discussed some of their important properties.