| Since the concept of quantales was introduced by Mulvey in 1984,many math-ematicians and logicists have paid close attention to the theory of quantales.Based on the logical implication operators → and(?),Rump introduced a new concept-quantum B-algebras in 2013.Quantum B-algebras are a.class of non-commutative logic algebras,which can be seen as a generalization of quantales.Quantum B-algebras contain the majority of implication algebras like BCK-algebras,residu-ated lattices,partially ordered groups,BL-and MV-algebras,effect algebras and their non-commutative extensions.Due to the relationships between quantum B-algebras and other implication algebras,in this paper we invatigate dual quantum B-algebras,unital quantum B-algebras and enveloping quantales over quantum B-algebras.Specifically speaking,this dissertation is divided into the following several parts:Chapter One:Preliminaries.In this chapter some basic concepts and relative conclusions used in this paper are given.Chapter Two:Quantum B-algebra.In first part of this chapter,we study the dual quantum B-algebras.Firstly,we deeply discuss the properties of quantum B-algebras by a new symbol.Secondly,the concept and properties of dual quantum B-algebras are given and investigated,and the equivalen characterizations of dual quantum B-algebras are given.At last,we obtain the equivalent conditions that a integral and dual quantum B-algebra is a distributive lattice,and prove that a complete dual quantum B-algebra is equivalent to a dual Quantale.In second part of this chapter,the invertible elements of quantum B-algebras are stuided.The invertible elements of quantum B-algebras are investigated,and it is shown that the set of all invertible elements of a quantum B-algebras is a partially ordered group.The third part,we discuss the implication operators → and(?)over integral quantum B-algebras.Chapter Three:Enveloping Quantale.Firstly,relative conclusions of Power-set Quantales are given.Secondly,the characterization between morphism of quan-tum B-algebras and morphism of enveloping quantales are stuied.At last,we study the morphism of enveloping quantales by establishing cartesian product. |
