Some Studies On Countable Continuous Lattices And Heyting Systems

This thesis mainly discuss homomorphisms and its correspondingtwo extended theorems on countable continuous lattices!countable directed minimalsets and its dually countable filtered maximal sets, and factorization of morphismsbetween Heyting systems.Firstly, in this thesis, we introduce the related theory’s progress and current situ-ation of countable continuous lattices and Heyting systems. In addition, organizationstructure of this paper is also given.Secondly, in this thesis, we give the definition of countable directed minimalsets and obtain a characterization and two extended theorems of homomorphisms ofthe countable continuous lattices. We make further discussions on countable direct-ed minimal sets under countable continuous lattices. We also obtain some intrinsiccharacterizations of countable continuous lattices. We not only give a sufcient con-dition of complete distributive lattice, but also get some intrinsic characterizationsof completely distributive lattice. We study on countable filtered maximal sets un-der dually countable continuous lattices. Some intrinsic characterizations of duallycountable continuous lattices are given. In addition, we obtain a sufcient conditionwhich turning dually countable continuous lattices to be completely distributive andstudy several intrinsic characterizations of completely distributive lattices.Finally, the definitions of injective-subjective function and subjective-injectivefunction between Heyting system are introduced and examined. In addition, someexamples are given to illustrate the possibility and rationality of these definitions andsome good results are also obtained.