The Research On Some Logical Algebra Systems Associated With Residuated Lattices

Since Boolean algebras were introduced as an algebraic correspondence with classical two-valued logic system,a lot of other algebraic structures,which are associated with some other logic systems,have been introduced and studied extensively, for example,MV-algebras associated with Lukasiewicz infinite valued logic system, BL-algebras associated with Basic Logic which were introduced by Hajek,Heyting algebras associated with intuitionistic propositional logic system,R₀ algebras associated with formal logic system L^* which were introduced by Professor Wang Guojun and also orthoalgebras and effect algebras which are mathematical models of quantum logic,et al.Many results have been obtained.All these algebras, which are associated with logic systems,are called logical algebra systems.They are closely interconnected.As an example,MV-algebras,BL-algebras and R₀ algebras are closely related to residuated lattices.In fact,all of them are subclasses of residuated lattices.Inspired by it,in this thesis we devote to investigating the structures,properties and relations of some logical algebra systems which are associated with residuated lattices.The results obtained in this thesis conduce to further researches on the logical algebra systems.On the other hand,since effect algebras are proposed as a new mathematical model for quantum logic,it has attracted a great deal of attention.In particular, with the development of effect algebras,some researchers begin to pay their attention to partial algebra structures and some new results are obtained.In the thesis, based on these results on partial algebra structures and residuated structures,we introduced a new partial algebra structure,called partial residuated structure. Some basic properties,structures and the relationship between the new structure and some quantum structures are studied.This thesis consists of the following four parts.The first part includes Chapter 1.Chapter 1 reviews main directions of research and their development of logical algebra systems.We focus on those logic algebras that are considered in this thesis.The second part consists of Chapter 2 and Chapter 3.Properties and structures of partial residuated lattices are considered in this part.In Chapter 2,the notions of partial residuated lattices are introduced by firstly defining partial adjoint pairs.Following this,properties of partial residuated lattices are discussed. In particular,we investigate the relationship between partial residuated lattices and lattice effect algebras,and prove that lattice effect algebras are partial residuated lattice and a sufficient and necessary condition for a partial residuated lattice to be a lattice effect algebra is proposed.Finally,dropping the commutativity, some similar results are obtained.In Chapter 3,we discuss some basic structures of partial residuated lattices.We focus on substructures,product structures and quotient.Morphisms of partial residuated lattices,subpartial residuated lattices, direct product and filter product of partial residuated lattices and congruence relation of partial residuated lattices,et al.are defined.In particular,we prove that all partial residuated lattices form a strong variety of partial algebras.The third part includes Chapter 4.In this chapter,we mainly study R₀ algebras.Firstly,we give a direct product decomposable form of R₀ algebras, which generalizes the result existed.Secondly,the concepts of R₀ algebras of fractions relative to a∧-closed system are introduced and Gabriel filters of R₀ algebras are studied.We show that a congruence relation can be obtained from a Gabriel filter on an R₀ algebra.All of these lay a good foundation for investigating localization of R₀ algebras.Thirdly,the prime and maximal spectra of R₀ algebras are considered by recalling some basic properties of filters of R₀ algebras.We show that the prime spectra of R₀ algebras is a compact T₀ topological space and the maximal spectra of R₀ algebras is a compact Hausdorff topological space. Finally,Based on these results above,weak Boolean product and Boolean product of R₀ algebras are defined and discussed,and some Boolean representations of R₀ algebras are obtained.The fourth part includes Chapter 5.The localization bounded commutative Re-monoid of a bounded commutative Re-monoid is discussed in this part.By defining F-multiplier,where F is a Gabriel filter,we construct the localization bounded commutative Re-monoid with respect to the Gabriel filter F.Finally, localization bounded commutative Re-monoids in some special instances are described.