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On intuitionistic fuzzy subhypernear-rings of hypernear-rings

In this paper, we introduce the concept of an intuitionistic fuzzy subhypernearring of a hypernear-ring and obtain some results in this connection. | Turk J Math 27 (2003) , 447 – 459. ¨ ITAK ˙ c TUB On Intuitionistic Fuzzy Subhypernear-rings of Hypernear-Rings Kyung Ho Kim Abstract In this paper, we introduce the concept of an intuitionistic fuzzy subhypernearring of a hypernear-ring and obtain some results in this connection. Key Words: Fuzzy subhypernear-ring,intuitionistic fuzzy subhypernear-ring, upper (resp. lower) t-level cut, homomorphism. 1. Introduction After the introduction of the concept of fuzzy sets by Zadeh [3], several researchers were conducted on the generalizations of the notion of fuzzy set. The idea of “intuitionistic fuzzy set” was first published by Atanassov [1], as a generalization of the notion of fuzzy set. In this paper, using Atanassov’s idea, we establish the intuitionistic fuzzification of the concept of subhypernear-rings in hypernear-rings and investigate some of their properties. Also, for any intuitionistic fuzzy set A = (µA , γA ) and a homomorphism f ) in R by f from hypernear-ring R to hypernear-ring R0 , we define IFS Af = (µfA , γA f (x) := γA (f(x)) for all x ∈ R. Then we show that If an IFS µfA (x) := µA (f(x)), γA 0 A = (µA , γA ) in R is an intuitionistic fuzzy subhypernear-ring of R0 , then an IFS f ) in R is an intuitionistic fuzzy subhypernear-ring of R. We consider the Af = (µfA , γA notion of equivalence relations on the family of all intuitionistic fuzzy subhypernear-rings of a hypernear-ring and investigate some related properties. 2000 Mathematics Subject Classification: 03F55, 06F05, 20M12, 03B52 447 KIM 2. Preliminaries First we shall present the fundamental definitions. A hyperstructure is a set H together with a map + : H × H −→ P ∗ (H) called hyperoperation, where P ∗ (H) denotes the set of all the nonempty subsets of H. A hypernear-ring is an algebraic structure (R, +, ·) which satisfies the following axioms: (H1) x + (y + z) = (x + y) + z, (H2) There is 0 ∈ R such that x + 0 = 0 + x = x. (H3) For every x ∈ R there exists one and only one x0