Anti Q-fuzzy Subgroups under t-conorms

In this paper we introduce the notion of anti Q-fuzzy subgroups of G with respect to t-conorm C and study their important properties. Next we define the union, normal and direct product of two anti Q-fuzzy subgroups of G with respect to t-conorm C and we show that the union, normal and direct product of them is again an anti Q-fuzzy subgroup of G with respect to t-conorm C. It is also shown that the homomorphic image and pre image of anti Q-fuzzy subgroup of G with respect to t-conorm C is again an anti Q-fuzzy subgroup of G with respect to t-conorm C.


Introduction
Undoubtedly the notion of fuzzy set theory initiated by Zadeh [31] in 1965 in a seminal paper, plays the central role for further development. This notion tries to show that an object corresponds more or less to the particular category we want to assimilate it to; that was how the idea of defining the membership of an element to a set not on the Aristotelian pair { } 1 , 0 any more but on the continuous interval [ ] 1 , 0 was born. The notion of a fuzzy set is completely non-statistical in nature and the concept of fuzzy set provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership rather than the presence of random variables. Since the concept of fuzzy group was introduced by Rosenfeld in [27] 14 in 1971, the theories and approaches on different fuzzy algebraic structures developed rapidly. Yuan and Lee [30] defined the fuzzy subgroup and fuzzy subring based on the theory of falling shadows. Also Solairaju and Nagarajan [29] introduced the notion of Q-fuzzy groups. The triangular conorm (t-conorm) originated from the studies of probabilistic metric spaces [5,28] in which triangular inequalities were extended using the theory of t-conorm. The author by using norms, investigated some properties of fuzzy algebraic structures [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. In Section 2, some preliminary definitions and results regarding multisets have been introduced. In Section 3, we have studied anti Q-fuzzy subgroups of G with respect to t-conorm C. Also the union, normal and direct product of them has also been discussed here. In Section 4, the homomorphic image and pre image of them have been investigated under group homomorphisms and anti group homomorphisms.

Preliminaries
The following definitions and preliminaries are required in the sequel of our work and hence presented in brief. [3]). A group is a non-empty set G on which there is a binary

Definition 2.1 (See
such that (1) if a and b belong to G, then ab is also in G (closure), in which case the identity is denoted by 0, or the multiplicative notation, that is for which the identity is denoted by e.

15
(1) H is a subgroup of G.
be any two groups. The function ), G y x ∈ Definition 2.5 (See [4]). Let G be an arbitrary group with a multiplicative binary operation and identity e. A fuzzy subset of G, we mean a function from G into [ ].
and fuzzy pre-image (or fuzzy inverse image) of ν under φ is having the following four properties: dual to the drastic t-norm.
for any t-conorm C and all [ ].

Anti Q-fuzzy Subgroups and t-conorms
be a group and Q be a non empty set. Then is said to be an anti Q-fuzzy subgroup of G with respect to t-conorm C if the following conditions are satisfied: , and . Q q ∈ Throughout this paper the set of all anti Q-fuzzy subgroups of G with respect to t-conorm C will be denoted by ( ).

Remark 3.2. The condition (2) of Definition 3.1 implies that
We denote by ( )

G NAQFSC
the set of all normal anti Q-fuzzy subgroups of G with respect to t-conorm C. ). and ).
and C is idempotent t-conorm. Then we obtain the following statements:  q  e  e  xy   H  H  H  H   ,  ,  , , ,