Hits:

Indexed by:期刊论文

Date of Publication:2010-05-01

Journal:COMPUTERS & MATHEMATICS WITH APPLICATIONS

Volume:59

Issue:9

Page Number:3117-3129

ISSN No.:0898-1221

Key Words:Algebra; Intuitionistic fuzzy subsets; Fuzzy points; The cut sets of intuitionistic fuzzy sets; 3-valued logic; Intuitionistic fuzzy subgroups

Abstract:In this paper, a new kind of intuitionistic fuzzy subgroup theory, which is different from that of Ma, Zhan and Davvaz (2008) [22,23], is presented. First, based on the concept of cut sets on intuitionistic fuzzy sets, we establish the neighborhood relations between a fuzzy point x(a) and an intuitionistic fuzzy set A. Then we give the definitions of the grades of x(a), belonging to A, x(a) quasi-coincident with A, x(a) belonging to and quasi-coincident with A and x(a) belonging to or quasi-coincident with A, respectively. Second, by applying the 3-valued Lukasiewicz implication, we give the definition of (alpha, beta)-intuitionistic fuzzy subgroups of a group G for alpha, beta is an element of {is an element of, q, is an element of boolean AND q, E boolean OR q}, and we show that, in 16 kinds of (alpha, beta)-intuitionistic fuzzy subgroups, the significant ones are the (is an element of, is an element of)-intuitionistic fuzzy subgroup, the (is an element of, is an element of boolean OR q)-intuitionistic fuzzy subgroup and the (is an element of boolean AND q, is an element of)-intuitionistic fuzzy subgroup. We also show that A is a (is an element of, is an element of)-intuitionistic fuzzy subgroup of G if and only if, for any a is an element of (0, 1], the cut set A(a) of A is a 3-valued fuzzy subgroup of G. and A is a (is an element of, is an element of boolean OR q)-intuitionistic fuzzy subgroup (or (is an element of, is an element of boolean OR q)-intuitionistic fuzzy subgroup) of G if and only if, for any a is an element of (0, 0.5] (or for any a is an element of (0.5, 1]), the cut set A(a) of A is a 3-valued fuzzy subgroup of G. At last, we generalize the (is an element of, is an element of)-intuitionistic fuzzy subgroup, (is an element of, is an element of boolean OR q)-intuitionistic fuzzy subgroup and (is an element of boolean AND q, is an element of)-intuitionistic fuzzy subgroup to intuitionistic fuzzy subgroups with thresholds, i.e., (s, t]-intuitionistic fuzzy subgroups. We show that A is a (s, t]-intuitionistic fuzzy subgroup of G if and only if, for any a is an element of (s, t]. the cut set A(a) of A is a 3-valued fuzzy subgroup of G. We also characterize the (s, t]-intuitionistic fuzzy subgroup by the neighborhood relations between a fuzzy point x(a) and an intuitionistic fuzzy set A. (C) 2010 Elsevier Ltd. All rights reserved.