The Acyclicity And Negative Acyclicity Of Fuzzy Relations

In this paper,we define and study the acyclicity and negative acyclicity of a fuzzy relation in detail.The study of acyclicity of a fuzzy relation consists of two parts: the acyclicity of a fuzzy relation and its strict part.In the study of T-acyclicity,we first investigate the relationships between T-acyclicity and T-transitive closure,irreflexivity of T-transitive closure under the condition that the involved t-norm is left continuous.Then we deal with the relationships between minT-acyclicity and minT-acyclicity of its cuts with the t-norm min.In studying the acyclicity of the strict part of a fuzzy relation,we first investigate the relationships between T-acyclicity and T-acyclicity of the strict part of a fuzzy relation,-asymmetry under the condition that the involved t-norm is min;Then we obtain an equivalent statement of TT-acyclicity under the condition that the involved t-norm is left continuous and n_T is a strong negation;Afterwards we investigate the relationships between T-acyclicity and S-transitivity;Finally,we investigate the relationships between T-acyclicity of the strict part of a fuzzy relation and negative S-transitivity.The study of the negative acyclicity of a fuzzy relation includes two parts: the negative acyclicity of a fuzzy relation and the relationships between T-acyclicity and negative T-acyclicity.In the study of negative acyclicity,we investigate the relationships between negative T-acyclicity and negatively T-transitive interior,reflexivity of negativelyT-transitive interior under the condition that the involved t-conorm is right continuous.In studying the relationships between the negative T-acyclicity and T-acyclicity of a fuzzy relation,we first investigate the relationships between negative T-acyclicity andT-acyclicity of the strict part of a fuzzy relation with the t-norm min and t-conorm max;Then we investigate the relationships of negative T-acyclicity,T-acyclicity of the strict part of a fuzzy relation and T-acyclicity of its complementary relation under a De Morgan triplet(T,S,n).