| Domain theory originates at the end of 60's of the last century,it mainly researches the order relations and topology structures on a poset,and becomes the denotational se-mantics of a functional programming language.However,with the rapid development of computer and network,there are more and more demand for the nonsequential program-ming languages,and how to make the Domain theory as the computer programming lan-guage semantics provides more sophisticated quantitative models representation turns into a study focus.Therefore,the research of quantitative Domain is particularly impor-tant.Using fuzzy sets to study quantitative Domain theory is customarily called fuzzy poset.Fuzzy poset comes from two aspects:one is originally proposed by Belohlavek;another is defined by Fan and Zhang.Later on,Yao proves that these two kinds are equivalent to each other.In this thesis,first of all,standing on the the existing results,we further study the quantitative Domain theory,in the second chapter,based on a complete residuated lat-tice,algebraic fuzzy closure operators and algebraic fuzzy closure L-systems on a fuzzy complete lattice are defined and investigated.We establish a "one-to-one" correspon-dence between algebraic fuzzy closure operators and algebraic fuzzy closure L-systems under a condition on the fuzzy order.Moreover,it is shown that the category of(alge-braic)fuzzy closure operator spaces is isomorphic to the category of(algebraic)fuzzy closure L-system spaces.In the third and fourth chapter,we do some theoretical researches on two algebraic structures-the residual lattice and the Boolean ring,respectively.The notion of fuzzy extended filters is introduced on residuated lattices.By defining an operator between two arbitrary fuzzy filters in terms of fuzzy extended filters,two results are immediate-ly obtained:(1)The class of all fuzzy filters on a residuated lattice forms a complete Heyting algebra;(2)The connection between the fuzzy extended filters and the fuzzy generated filters is built,with which three other classes generating complete Heyting algebras,respectively,are presented.Finally,by the aid of fuzzy t-filters,we also devel-op the characterization theorems of the special algebras and quotient algebras via fuzzy extended filters.The L-fuzzy extended ideals are studied in a Boolean ring,besides,we build the relation between an L-fuzzy ideal and the class of its L-fuzzy extended ideals,the result that the family of all L-fuzzy ideals in a Boolean ring is a complete Heyting algebra is immediately obtained.Furthermore,the lattice structures of L-fuzzy extended ideals of an L-fuzzy ideal,L-fuzzy extended ideals relative to an L-fuzzy subset,L-fuzzy stable ideals relative to an L-fuzzy subset and their connections are discussed.In the fifth and sixth chapter,we further investigate the rough set theory on a lat-tice.Han et al.introduced a new pair of rough approximation operators via ideal on a CCD lattice in 2016,which is more general and accurate than Zhou and Hu's.In this paper,we further study its properties,and then the axiomatic approaches for the rough approximation operators are proposed.Through some of our axioms,the rough approximations via ideal on a complete atomic Boolean lattice can be viewed as special cases of rough approximation operators via ideal on a CCD lattice if the ideal is well given.The promotion of rough set model is an important part of the study for rough set theory,we further generalize rough set theory to a wider algebraic structure-a complete lattice,with the Galois ideal,and the properties of rough approximation operators on a complete lattice are discussed. |
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