The Research Of States And Internal States On Some Algebraic Structures Based On Equality Algebras

Equality algebras are the algebraic system corresponding to high order fuzzy logics,pseudo equality algebras are the noncommutative generalization of equality algebras,and hyper equality algebras are the lift of equality alge-bras.Since equivalential equality algebras are term equivalent with BCK-meet semilattice,and BCK-algebras are the proper subclass of BCI-algebras,then BCI-algebras can be considered as the generalization of equivalential equal-ity algebras.In this paper,we focus on studying the state theory on three kinds of algebraic structures based on equality algebras,namely,BCI-algebra,pseudo equality algebras and hyper equality algebras.On the one hand,we investigate the structures of logical algebras by means of states and internal states.On the other hand,we further refined the probability problems in fuzzy logics by use of algebraic methods.The main studies of this thesis are as follows:In the second chapter,we study internal states on BCI-algebras.Firstly,we construct the axiomatic system of the internal states on a BCI-algebra,and deliver some nontrivial examples.Also we study state ideals,maximal state ideals and prime state ideals and discuss the relations among them.We obtain that there is a bijection between the set of all state congruence rela-tions and the set of all state closed ideals on a state BCI-algebra,and get the condition that the image?(L)of a nontrivial subdirectly irreducible state BCI-algebra(L,?)is a nontrivial subdirectly irreducible subalgebra of L.Sec-ondly,we introduce internal state-morphisms on BCI-algebras,and by use of internal state-morphisms and internal states,give some characterizations of commutative BCI-algebras,p-semisimple BCI-algebras and(positive)implica-tive BCI-algebras.Finally,we introduce left-right(right-left)state multipliers on BCI-algebras,and discuss the relation between state multipliers and deriva-tions.We get that an internal state?is left-right(right-left)state multiplier if and only if it is a left-right expansion(right-left compression)derivation of L.Moreover,by use of state multipliers,we characterize several types of special BCI-algebras.In the third chapter,we introduce state in a universal algebraic setting,namely,the generalized state maps(simply,GS-maps)including two special classes,that is,generalized states(simply,G-states)and generalized internal states(simply,GI-states).We deliver some examples and related properties of them.Also,we introduce and study Bosbach states,Rie(?)an states on pseudo equality algebras.We study the existence of the two types of states and deliver characterization of Bosbach states.We focus on discussing the relations among Bosbach states,Rie(?)an states and state-morphisms,and get the conclusions that the state-morphisms and Bosbach states are equivalent in linearly ordered pseudo equality algebras and the Rie(?)an states and the Bosbach states coincide in involutive pseudo equality algebras,respectively.Finally,we discuss the relationships among the generalized state maps,the states and the internal states on pseudo equality algebras.We come to the conclusion that one can extend any state s of the image space?(X)into the state s_?of the entire space X by an internal state(or an internal state-morphism)?.In addition,in a sense,generalized state maps can be viewed as a possible united framework of states and internal states,state-morphisms and internal state-morphism on pseudo equality algebras.In the fourth chapter,we study states and internal states on hyper equal-ity algebras.Firstly,we apply the theory of hyper algebras to equality algebras and propose a new structure which is called as hyper equality algebras which are a generalization of equality algebras.We define various hyper filters and hyper deductive systems and discuss the relations of them.Moreover,we de-liver the relations between hyper equality algebras and hyper EQ-algebras,hyper BCK-algebras and weak hyper residuated lattices.We also construct quotient hyper equality algebras via regular hyper congruence relations.Next,we introduce the Bosbach states and the Rie(?)an states on hyper equality alge-bras and find some examples of them.Also using?-invariant Bosbach states,it follows that the induced function s by a a Bosbach state s on H is also a Bosbach state on H/?.At last,we introduce internal states on hyper equality algebras and give the representations of generated state strong hyper deduc-tive systems.We study images and inverse images under the action of internal states,and in the meantime,we prove that maximal state strong hyper deduc-tive systems are prime state strong hyper deductive systems in lattice-ordered separated good state hyper equality algebras.Moreover,by means of internal states on hyper equality algebras,we induced the internal states on quotient hyper equality algebras.