IJAMA: 4(2), December 2012, pp. 175-182 © Global Research Publications TRANSLATES OF ANTI FUZZY SUBGROUPS P. K. SHARMA Abstract: As an abstraction of the geometric notion of translation, we introduce two operators T � + and T � – called the fuzzy translation operator on the fuzzy set A. We investigate their properties and also study their action on Anti fuzzy subgroups (Anti fuzzy normal subgroups) of a group and show that they are invariant under these translation. However we show that anti fuzzy abelianness is not translation invariant. We also study the interaction of these operators with anti fuzzy coset formation and prove that operators commute. Mathematics Subject Classification: 03F72, 08A72, 20K27. Keywords: Fuzzy set (FS), Anti fuzzy subgroup (AFSG), Anti fuzzy Normal subgroup (AFNSG), Translation, Operator, Coset. 1. INTRODUCTION The concept of fuzzy set was initiated by Zadeh [11]. Then it become a vigorous area of research in engineering, medical science, social science, graph theory etc. Rosenfeld [5] gave the idea of fuzzy subgroups and Biswas [1] gave the idea of anti fuzzy subgroups. Palaniappan and Muthuraj [3] defined the homomorphism and anti-homomorphism of fuzzy and anti fuzzy subgroups. Palaniappan and Arjun [4] defined the homomorphism and anti-homomorphism of fuzzy and anti fuzzy ideals. Sharma [7] and [8] introduced the notion of anti fuzzy submodule of a module. In this paper, we introduce two operators T � + and T � – called the fuzzy translation operator on the fuzzy set A and investigate their properties. We also study the action of these operators on Anti fuzzy subgroups (Anti fuzzy normal subgroups) of a group and shown that they are invariant under these translation. However it is shown that anti fuzzy abelianness is not translation invariant. It is also shown that the interaction of these operators with anti fuzzy coset formation commute. 2. PRELIMINARIES Definition 2.1: A fuzzy set A of a set X is a function A : X � [0, 1]. Fuzzy sets taking the values 0 and 1 are called Crisp set. Let A and B be two fuzzy subsets of a set X. Then the following expressions are defined in [1], [11].