Acyclicity And Negative Acyclicity Measures Of Fuzzy Relations

In this paper,we deal with the measures of acyclicity and negative acyclicity of fuzzy relations.we define and study the acyclicity indicator and negative acyclicity indicators of fuzzy relations.Thus,some results in the literature regarding acyclicity and negative acyclicity are extended to their degree description.The study of acyclicity indicator of fuzzy relations consists of two parts:the a-cyclicity indicators of a fuzzy relation and its strict part.In the study of acyclici-ty indicator,we first investigate the relationships between T-acyclicity indicator and T-asymrmetry indicator,irreflexivity indicator of T-transitive closure under the condi-tion that the involved t-norm is left continuous.Then we deal with the relationships be-tween T-acyclicity indicator and T-transitivity indicator,T-irreflexivity indicator with the t-norm min.In studying the negative acyclicity indicator of the strict parts of the fuzzy relations,we first investigate the relationships between the T-acyclicity indicator of a fuzzy relation and the T-acyclicity indicator of its strict part,its T-asymmetry indicator with a general t-norm T.Then we focus on the study of the min-acyclicity indicators of the strict part of a fuzzy relation and its transitive indicator with the t-norm min.Finally,we deal with the relationships between the min-acyclicity indicator of strict part and T-negative transitivity indicator.Next we turn to the study of the negative acyclicity indicator of fuzzy relations.We first give the relationship between S-negative acyclicity indicator and S-strong com-pleteness indicator under a general t-conorm S.Then we discuss the connection of the S-negative acyclicity indicator and the reflexivity indicator of S-negatively transitive interior indicator under the condition that the involved t-conorm is right continuous.Finally,we investigate the relationships between the indicators of S-negative acyclicity and T-acyclicity of its complementary relation,and between the T-acycticity indicator of a fuzzy relation and the S-negative acyclicity indicator of its complementary relation under a De Morgan triplet(T,S,n).