| As the important algebraic systems in studying quantum mechanics, orthomod-ular lattices and quantales have the same properties. However, they are different algebraic systems. In1991, L. Roman and B. Rumbos constructed a new category-quantic lattice category, which contains quantales as its objects. And they pointed out that for every orthomodular lattice L, if a&b (?)(a∧b(?))∧b for every a, b in L, then L is a quantic lattice. So the research on quantic lattices not only can help study the interior structures of orthomodular lattices but also can enrich, to some extent, the theory of quantales. The structure of this thesis is organized as follows:Chapter One:Preliminaries. In this chapter, we give some basic concepts and results of the quantic lattice theory and the fuzzy set theory which will be used throughout the thesis.Chapter Two:Prime elements and S-prime elements of quantic lattices. Firstly, relationships between quantic lattices and quantales, quantic lattices and distribu-tive lattices are discussed. It is proved that a left-sided quantic lattice Q is a commutative quantale if and only if a→(b→c)=b→(a→c) for all a, b, c∈Q. Secondly, some properties of prime elements are discussed. A characterization for an element to be prime is obtained. Finally, the concept of S-prime elements is given. It is shown that the image of an S-prime element is af (S)-prime element when the map f satisfies certain conditions.Chapter Three:Ideals and fuzzy ideals of quantic lattices. Firstly, the concepts of ideal and ideal extension are introduced by combining order with the&operator in quantic lattices. It is proved that the pre-images of ideals and prime ideals under the quantic lattice homomorphism are also ideals and prime ideals. Secondly, the definition of fuzzy power-set quantic lattice is given. It is obtained that if a quantic lattice is commutative, then its fuzzy power-set quantic lattice is also a commutative quantic lattice. Finally, we give the concepts of fuzzy ideal and fuzzy ideal extension, and some properties of them are discussed.Chapter Four:Upper rough ideals of quantic lattices. Firstly, the concepts of upper approximation and lower approximation of quantic lattices are given by using the rough set theory, and some properties of upper approximation and lower ap-proximation are studied. Secondly, we give the concepts of upper rough subqunatic lattices and upper rough ideals. We prove that a subqunatic lattice must be an upper rough subqunatic lattice and obtain the condition that an ideal is an upper rough ideal. Finally, the relationship between quantic lattice homomorphisms and ideals is discussed. It is shown that in a quantic lattice, the pre-images of upper rough ideals and upper rough subqunatic lattices under the surjective homomorphism are also upper rough ideals and upper rough subqunatic lattices. |
