Derivations On FI-algebras And Filters In Pseudo-CI-algebras

The concept of derivations is derived from the analytic theory,in which it corresponds to the differential.Using derivations to study the logic algebras is a popular research method in recent years.FI-algebras were introduced by Wu with the purpose of studying the common natures of fuzzy implication operators in logic algebraic systems.So the question how to use derivations to depict the nature of FI-algebras is worth considering.In addition,the non-communicative logic,put forward by Abrusci and Ruet and unifying the commutative linear logic and cyclic linear logic,is the standard conservative expansion of Lambek calculus.As the logic algebraic systems matching up with non-commutative logic,non-commutative logic algebras have also attracted many scholars' attention recently.CI-algebras,a new class of logic algebras,were proposed as the generalization of BE-algebras.At present,the research content of pseudo BE-algebras has been very rich.So,being the non-communicative generation of CI-algebras,what kind of properties and structures will be presented in pseudo CI-algebras,it is the part of the dissertation.Specifically speaking,this dissertation is divided into the following several parts:Chapter One:Preliminaries.In this chapter,some basic concepts and relevant conclusions used in this paper are given.Chapter Two:On derivations of FI-algebras.Firstly,we introduce the notions of left-right derivation,right-left derivation and derivation of FI-algebras,and discuss their related properties.Secondly,the identity derivation is propsed to depict the idempotence of FI-algebras,which comes to describe Boole algebras?HFI-algebras?distributive Fl-algebras.At last,we study the derivations of idempotent Fl-algebras as well as the special properties of their fixed point set,we show that all the derivations of idempotent FI-algebras can constitute an idempotent HFI-algebra if and only if it is a HFI-algebra.Chapter Three:Filters in pseudo CI-algebras.Firstly,we put forward a new class of logic algebras-pseudo CI-algbras,and study their properties,meanwhile,we preliminarily explore their relationships with some other logic algebras.Secondly,we introduce the concepts of filters and closed filters in pseudo CI-algebras,study their equivalent characterizations,and give the structure of generated filter and generated closed filter in transitive pseudo CI-algebras.Then,we present the notion of normal closed filters and regular congruences,and we prove that the normal closed filters and regular congruences are one-to-one in transitive pseudo CI-algebras.At last,we show the isomorphism theorems in transitive pseudo GI-algebras.