Research On The Properties And Construction Of Two Types Of Special Boolean Matrices

Boolean matrix is widely used in many disciplines,such as logic,computer science,communication engineering,electronic mapping,instrument control,management and decision-making.But up to now,there are still a lot of unsolved problems about Boolean matrices,including the fast judgment method of idempotency of Boolean matrices and the fast solution algorithm of transitive closure.Fuzzy matrix is a generalization of Boolean matrix,and idempotent fuzzy matrix and transitive closure of fuzzy matrix have important applications in fuzzy multi-attribute decision making and fuzzy clustering analysis.In this thesis,the properties and construction of idempotent Boolean matrices are studied.The method of identifying the idempotency of Boolean matrices is given.The method of constructing self-reflexive idempotent Boolean matrices is obtained.A fast algorithm for solving the transitive closure of Boolean matrices is proposed and numerical experiments are carried out.The relevant conclusions are extended to fuzzy matrices by using decomposition theorem.The specific work done in this thesis is as follows:1.The properties of idempotent Boolean matrices are deeply studied,and the necessary and sufficient conditions of idempotent Boolean matrices are given.This method can be implemented quickly and efficiently only by using set merging operations.For non-idempotent transitive Boolean matrices,an effective algorithm is proposed to modify them into idempotent matrices;2.Based on a class of special transfer matrices(maximal transfer relations),the construction method of self-reflexive idempotent Boolean matrices is obtained.This method can construct all self-reflexive idempotent Boolean matrices on the same order;3.A fast algorithm for calculating transitive closure of Boolean matrix is proposed,and numerical experiments are carried out with Python programming.The results show that the algorithm is fast and effective,especially for high-order sparse Boolean matrix;4.By using the decomposition theorem of fuzzy sets and the form of truncated matrices,the related conclusions of idempotency and transitive closure of Boolean matrices are extended to the fuzzy matrices and verified by examples.