The Congruence And Topology Of Quantum B-algebras

Since Mulvey put forward the concept of Quantale,the research and application of Quantale theory has been greatly developed.Its ideas and methods have a profound impact on several branches of mathematics,logic and theoretical computer science.Based on the implication operators in quantale,Rump and Yang proposed the concept of quantum B-algebras in 2013,and carried out a series of researches on quantum B-algebras.Quantum B-algebras are a kind of non-commutative logical algebras,which provide a unified semantics for a wide range of non-commutative algebraic logics.The study on congruence and topology is helpful to understand the internal structures and properties of quantum B-algebras.Based on this,this thesis focuses on the congruence and topology of quantum B-algebras.The main contents and innovations are listed as follows:The first part mainly studies congruence relations,homomorphisms and uniform structures induced by a congruence relation on quantum B-algebras.Congruence rela-tions on logical algebras are usually induced by filters.Since the quantum B-algebra is a algebra structure with a partial order,the quotient structure under the congruence relation induced by a filter is not necessarily a quantum B-algebra.In this paper,a new congruence relation on quantum B-algebras is introduced by combining the definition of congruence relations on posets and algebras.It is proved that the quotient structure under the congruence relation is a quantum B-algebra.The corresponding definition of F-morphisms and its equivalent characterization are given,and the basic theorem of homomorphism on quantum B-algebras is obtained.A uniform structure on a quan-tum B-algebra is induced by a congruence,and it is proved that the topological space induced by the uniform structure is first-countable,zero-dimensional,locally compact and completely regular.Two binary operations on quantum B-algebras are proved to be continuous with respect to this topology,which implies the quantum B-algebra is a topological quantum B-algebra.Finally,the concept of uniform quantum B-algebras is given,and the uniform space induced by the congruence relation is proved to be a uniform quantum B-algebra.In the second part,we focus on the topology and its completion induced by the filterable family of congruence relations on quantum B-algebras.First,a topology on a quantum B-algebra is induced by a filterable family of congruence relations.The properties of the topological space and its quotient spaces are investigated.Then,the two binary operations on a quantum B-algebra are showed to be continuous with respect to this topology.A method for constructing topological quantum B-algebras satisfying₂separability is given.A class of Cauchy nets with the directed setas the index set on quantum B-algebras and its convergence are introduced.By defining an appropriate congruence relation on the set of Cauchy nets,an equivalent form of the inverse limit of inverse systems on the category of bounded quantum B-algebras is given.Finally,the definition of the completeness of a class of topological quantum B-algebras is introduced.In particular,a characterization of the completeness of topo-logical lattice-ordered quantum B-algebras induced by a filterable family of congruences is obtained.