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The purpose of this paper is to define the notion of an interval valued Intuitionistic Fuzzy R-ideal (briefly, an i-v IF R-ideal) of a BCI - algebra. Necessary and sufficient conditions for an i-v Intuitionistic Fuzzy R-ideal are stated. Cartesian product of i-v Fuzzy ideals are discussed. I. Introduction: The notion of BCK-algebras was proposed by Imai and Iseki in 1996. In the same year, Iseki (6) introduced the notion of a BCI-algebra which is a generalization of a BCK-algebra. Since then numerous mathematical papers have been written investigating the algebraic properties of the BCK/BCI-algebras and their relationship with other universal structures including lattices and Boolean algebras. Fuzzy sets were initiated by Zadeh (10). In (9), 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. InZadeh also constructed a method of approximate inference using his i-v fuzzy sets. In Birwa's defined interval valued fuzzy subgroups of Rosenfelds nature, and investigated some elementary properties. The idea of "intuitionistic fuzzy set" was first published by Atanassov as a generalization of notion of fuzzy sets. After that many researchers considers the Fuzzifications of ideal and sub algebras in BCK/BCI-algebras. In this paper, using the notion of interval valued fuzzy set, we introduce the concept of an interval-valued intuitionistic fuzzy BCI-algebra of a BCI-algebra, and study some of their properties. Using an i-v level set of i-v intuitionistic fuzzy set, we state a characterization of an intuitionistic fuzzy R-ideal of BCI-algebra. We prove that every intuitionistic fuzzy R-ideal of a BCI-algebra X can be realized as an i-v level R-ideal of an i-v intuitionistic fuzzyR-ideal of X. in connection with the notion of homomorphism, we study how the images and inverse images of i-v intuitionistic fuzzy R-ideal become i-v intuitionistic fuzzy R-ideal. II. Preliminaries:

Intuitionistic Fuzzy R-Ideals of BCI-Algebras with Interval Valued Membership& Non Membership Functions