Several Classes Of Quantale Structures And The Category Of Quantum Frames

The concept of Quantale was introduced by C.J.Mulvey in 1986 with the purpose of studying the spectrum of C~*—algebras, as well as constructive foundations for quantum mechanics. Because Quantale can be regarded as the generalization of frame, there are abundant contents in the structure of quantales. That is why many mathematicians and logicists pay close attention to the theory of quantales. Basing on the theory of quantale and the theory of C~*—algebras, C.J.Mulvey and J.W.Pelletier advanced a new concept - involutive Quantale, J.Rosicky introduced the concept of Quantum frame. In this paper, the quotient objects and subobjects of involutive quantales are studied, the relation of involutive quantale quotients and congruence is studied. Ideals of regular quantales and some properties of the category of quantum frames are discussed. The main content is as follows:Chapter One Preliminary knowledge. In this chapter, we give the basic concepts and results of lattices, quantales and the theory of category which were used in this paper.Chapter Two Quotient objects and subobjects of involutive quantales. Firstly, we give the concrete characterization of quotient involutive quantales, the relation of quotient involutive quantales and involutive nucleis is discussed, so is the relation of special quotient involutive quantles and special elements. Then, the concept of involutive conu-cleis is introduced, corresponding of sub-involutive quantales and involutive conucleis is built up. At last, the relation of quotient involutive quantales and congruence is discussed.Chapter Three Regular quantales. Firstly, we give the concept of ideal in a regular quantale, and obtain some properties of ideals and the characterization of prime ideals, and the concrete construction of the ideal generated by an arbitary subset is discussed. Then, we discuss congruence decided by ideals. At last, some basic concepts are introduced in quantales, such as complement elements , quantales generated by the elements which have complement elements , compact quantales, and obtain some new properties of regular quantales.Chapter Four Quantum frame and its categorical properties. Firstly, we give some concepts: strong homemorphisms , sub-quantum frames , quotient quantum frames and congruence relation, and proved that the quotient set of a quantum frame on con-...