Research On Related Problems In Q-algebras And Binary Q-algebras

Since Mulvey proposed the concept of Quantale in 1986, the research of quantale theory has been concerned by many scholars. Solovyov proposed the concept of Quantale algebra(referred to as Q-algebra) based on Quantale module. Involutive quantale as an important class of Quantale structure, also got the attention of many scholars in recent years. In 2012, Wang Kaiyun introduced the definition of fuzzy Quantale in his PhD thesis, extending the classical Quantale theory to the fuzzy case. This paper introduces definitions of Q-algebra ideal conuclei, involutive Q-algebras and fuzzy involutive quantalcs firstly. Then, the relationship between Q-algebra nuclei and ideal conuclei is discussed and the congruence relation on involutive Q-algebras is studyed. Finally, it is proved that the category of fuzzy involutive quantales is isomorphic to the category of involutive L-algebras.The construction of chapters and the concrete contents of this paper are as follows:Chapter 1:Preliminaries. In this chapter, we give some basic concepts and results of the quantale theory, Q-algebras, fuzzy quantales and the category theory which will be used throughout the thesis.Chapter 2:Ideal conuclei on Q-algebras and pre-girard Q-algebras. In order to further discuss the relationship between the internal structure of Q-algebra. Firstly, we introduce the concept of Q-algebra ideal conuclei and then get a map that is the equivalent characterization of the Q-algebra ideal conuclei. Secondly, the definition of Dual Q-algebra, Girard Q-algebra, pre-dual Q-algebra,and pre-girard Q-algebra. It is proved that every Q-algebra can be embedded into a pre-girard Q-algebra. Finally, it is proved that nuclei and ideal conuclei on pre-girard Q-algebras are in one to one correspondence.Chapter 3:Congruence relations on involutive Q-algebras. The concepts of involutive Q-algebras and its nuclei, conuclei, congruence relations are introduced. It is proved that there is a one-to-one correspondence between the congruence relations and the nuclei on every involutive Q-algebra. Then the relationship between the congruences relation and the homomorphisms is discussed. At last, the congruence relations on the product of involutive Q-algebras is given.Chapter 4:Involutive Q-algebras and fuzzy involutive quantale. First, we intro- duce the concept of a fuzzy involutive quantale. Secondly, the categorical property of a fuzzy involutive quantale is discussed. Finally, it is shown that the category of fuzzy involutive quantales is isomorphic to the category of involutive L-algebras.