A Study Of Congruences And Filters On Quantum B Algebras

Quantum B-algebras are the logical algebras corresponding to quantum B logics,which provide a unified algebraic semantic for a wider range of substructure logics.Quantum B-algebras cover the vast majority of common logical algebras,such as partially ordered groups,generalized BL algebras,generalized MV algebras,cone algebras and their non-commutative extensions.Quantum B-algebras can be regarded as the implicational subreducts of quantales,and the study of quantum B algebras will further enrich the research contents of quantale theories.With the efforts of many scholars,the basic frameworks of quantum B-algebra theories have been initially formed,but there is still a lot of work to be done.This thesis mainly studies the internal structures of quantum B-algebras and conducts a series of studies on the congruences and filters of quantum B-algebras.The main contents of this thesis are divided into the following parts:Chapter One:Preliminaries.Some relative concepts and conclusions about lattice theory,quantales and quantum B-algebras uesed are given.Chapter Two:Congruences on quantum B-algebras.Firstly,we put forward the notions of regular congruences and strong regular congruences on quantum B-algebras,and their related properties are discussed.Secondly,by introducing the definition of quasi-chain modulo p on quantum B-algebras,it is given the equivalent characterizations of regular congruences and strong regular congruences.Finally,by proposing the concept of congruence product of the quantum B-algebra X,the specific form of join of congruences sets C（X）on a quantum B-algebra is given,and the join operation is used to give the specific form of the union of the regular congruence sets RC（X）.Chapter Three:Filters on quantum B-algebras.Firstly,the basic properties of filters in integral quantum B-algebras are introduced,and the mutual transformations between normal filters and congruence relations in integral quantum B-algebras are given.Secondly,we introduce the concepts of two-sided quantum B-algebras,implication-commutative quantum B-algcbras and lattice quantum Balgebras and their related properties are studied.For a two-sided quantum B-algebra X,it is proved that all filters and empty set are the quantale quotient of the logical quantale U（X）.Finally,the relevant properties of filters of locally unital quantum B-algebras are discussed.