Some Researches On Quantale

The concept of quantale were introduced by C.J.Mulvey in 1986 with the purpose of studying the spectrum of c*-algebra, as well as constructive foundations for quantum mechanics. There are abundant contents in the structure of quantales,because quantale can be regard as the generalization of frame .The research of these subjects has related to several research area such as non-commutative C*-algebra,the ideal theory of rings,linear logic and theoretic computer science. This paper analyses and systematically studies the structure of some special quantales (such as characterization of idempotent left- side spatial quantales, characterization of simple quantales); relations among elements of quantale; inner links between prequantales and quantales; relations among nuclei and quotient quantales and congruences of quantale jrelations between closed filters and closed maps ;categorical properties of category Quant. The main content is as follows.1. Some relations among elements of quantale were discussed, and some important results about elements of quantale were given.Sufficent and necessary conditions for prequantale turning into quantale or commutative quantale are obtained, as well as any product of spatial quantales is spatial and any subquantale of a spatial quantale is also spatial. By the aid of factor idempotent left-sided quantales to be spatial quantale, we obtained that an idempotent left-sided quantale is spatial iff it is a subquantale of product of some factor idempotent left-side quantales, as well as sufficent and necessary condition for quantale turning into simple quantale.2. Some concepts in quantale,such as c-nuclei,r-nuclei,closed map closed filter were introduced and Some properties of these new nuclei are discussed, we abtain the characterizations of c-nuclei in different quantales and the equivalance between c-nuclei and r-nuclei. In the category Quant,it is proved that quotient object of any object of Quant can be induced by nuclei in the sence of isomorphism, we dicuss the relationships between closed maps and closed filters which is closely related with Gabriel topologies of quantales. It is proved that the set of all closed filters F(Q) on a coherent left-sided quantale Q ordered under inclusion is a frame and the set of all compact closed filters FC(Q) is a subframe of F(Q) ,as well as ,that the set of allclosed filters on a coherent two-sided commutative quantale ordered under inclusion is a coherent frame .3. Special morphisms such as monomorphisms, epimorphisms,sections,retrac-tions, extremal monomorphisms; extremal epimorphisms, constant morphisms,coco-nstant morphisms, zero morphisms and special objects such as initial objects, terminal objects and zero objects are studied and their characterization is obtained. We proved that category of quantale is well-powered, co-well-powered and is a pointed category and obtained the structure of equalizer and coequalizer, as well as,obtained that category of quantale is complete category. The free objects in category Quant were discussed, and the concrete structure of free objects was obtained, in addition it is proved that Quant is algebraic category. Moreover, we constructed the limit and inverse limit sturcture in the category of quantale. By the aid of defineding an equivalence relation on the disjoint union of a family of quantales, the direct limit structure in the category of quantale was obtained.