Interval-valued Fuzzy Ideal of N(2,2,0) Algebra

Abstract

The purpose of this paper is to define the notion of an interval-valued fuzzy ideal of N (2, 2,0) algebra. Necessary and sufficient conditions for an i-v fuzzy set to be i-v fuzzy ideals are stated. It is proved that the intersection and direct product in N (2, 2,0) algebra of i-v fuzzy ideal are also i-v fuzzy ideal. The results enriched fuzzy theory in N (2, 2,0) algebra. Keywords-N(2,2,0) algebra; interval-valued fuzzy set; interval- valued fuzzy ideal; intersection; direct product I. INTRODUCTION The N (2, 2,0) algebra is an algebra system with two dual semigroups. The related properties of N (2, 2,0 ) algebra have been discussed in literatures(1-8). The concept of a fuzzy set, which was introduced by L.A.Zadeh (9), provides a natural framework for generalizing many of the concepts of general mathematics. Since then these fuzzy ideal and fuzzy subalgebra have been applied to other algebraic structures such as semigroups, groups, rings, BCI/BCH-algebra, etc. In (10), Zadeh made an extension of the concept of a fuzzy set by an interval-valued fuzzy set. This interval-valued fuzzy set is referred to as an i-v fuzzy set. In (10), Zadeh also constructed a method of approximate inference using i-v fuzzy sets. Biswas defined interval-valued fuzzy subgroups and investigated some elementary properties. In this paper, using the notion of interval-valued fuzzy set, the concept of i-v fuzzy ideal is introduced of N(2,2,0) algebra. Necessary and sufficient conditions for i-v fuzzy set to be i-v fuzzy ideals are stated. It is proved that the intersection and direct product in N(2,2,0) algebra of i-v fuzzy ideal are also i-v fuzzy ideal. II. PRELIMINARIES