Research On The Related Properties Of The Quantale System And Its Subsystems

Steven Vickers has proposed the topological system in his work《Topology via Logic》by synthesizing the methods of topology and the theory of logics,which also is applied to study of computer’s theory.Quantales were introduced by Mulvey in 1986 with the purpose of studying C*-algebra,it can be considered as a generalization of frame in the non-commutative situation.The theory of Quantale system has been investigated again,the categorical properties of Q-Localification and spatialization have been given.The subsystem is defined by congruence relation,the equalizer and coequalizer of QSys have been got.The subsystem of Quantale system is defined by nucleus,the embedding theorem of Quantale system has been got.In the end,|=-filters of a topological system have been preliminarily investigated,some properties have been got.The structure of this thesis is organized as follows:Chapter one:Preliminaries.In this chapter,we recapitulate the basic concepts and relevant conclusions of Quantale,topological systems,Quantale systems and the category theory which will be used throughout this thesis.Chapter two:The category properties of Quantale system.The category prop-erties of Q-Localification and spatialization have been got.Chapter three:Subsystem of Quantale system.The definition of subsystem of Quantale system is given,with the structure of Quantale,the properties of sub-system have been investigated.The subsystem is defined by congruence relation,the equalizer and coequalizer of QSys have been got.The subsystem of Quantale system is defined by nucleus.The embedding between Quantale systems is defined,with the structure of Quantale,the embedding theorem of Quantale system has been got.The embedding of spatilization of Quantale system has been investigated.It is proved that the set of all subsystems of a Quantale system with the defined order is a complete lattice.Chapter four:|=filters of a topological system.|=filters of a topological system have been preliminarily investigated in this chapter.Firstly,the concept of |=-filters is introduced.It is proved that |=-filters are usual filters,but usual filters are not|=-filters.Secondly,it is got that |=-filters have the reversing order.The infimum of|=-filters has been got with the method of logic,|=-Fil(QD)is a frame.Thirdly,the concrete forms of meet and join in |=-Fil(ΩD)have been given.