Research On Properties Of Involutive Quantale And Its Related Category

Since the concept of quantale was introduced by C. J. Mulvey and J. W. Pelletier in 1986, many mathematicians and logicists have paid attention to the theory of quantale. Based on the theory of quantale and the theory of C~*-algebra, in 1992, C. J. Mulvey and J. W. Pelletier advanced a new concept—involutive quantale. In past more than ten years, many scholars made a great deal of research on the structure to involutive quantale. This text studied several classes of involutive quantale structures, studied carefully and deeply the sub-☉-involutive quantale and the category of☉-involutive quantale based on the concept of quantum frame which J. Rosicky introduced. The arrangement of this paper is as follows:Chapter One Preliminary knowledge. In this chapter, we give the basic concepts and results of the theory of lattice, quantale and that of category which are used in the whole paper.Chapter Two Co-support'quantale. Firstly, the concept of co-support quantale is introduced, a list of properties are given. Then, we introduce the concept of stable co-support quantale, a sufficent and necessary condition that co-support quantale turn into stable co-support quantale is obtained. Also, we prove that stable co-support is involutive quantic nucleus in some conditions.Chapter Three The involutive-ordered relations on involutive quantale. Firstly, the concept of involutive-ordered relations is introduced, some involutiveordered relations are given. We prove that the《-relation and(?)-relation are involutiveordered relations. Secondly, we introuce the concrpt of *-∧-functions and discuss the relations between involutive-ordered relations and *-∧-functions. Then, we study their properties further.Chapter Four☉-Involutive quantale and its categorical properties. Firstly, the concept of☉-involutive quantale,☉-involutive quantale homomorphism and sub-☉-involutive quantale are introduced, a list of sub-☉-involutive quantales are given. Then, we study mainly some special objects in☉-involutive quantale. For example, the special objects of retrations, initial object and terminal object etc. are characterized. According to these, we know that the category of☉-involutive quantale is a connected and pointed category. At last, the stractures of the equalizer and multiple equaizer are given. We prove that the category of☉-involutive quantale has prosucts, pullback, collective pullback and it is a complete category.