| As the semantic system of non-classical logic,various fuzzy logic algebras have been widely introduced and studied by many scholars.As the algebraic structure of truth value of higher order fuzzy logic,pseudo EQ-algebras not only provide a more extensive truth algebraic structure for fuzzy type theory,but also are the generalization of EQ-algebra.The theories of states and internal states play an important role in the study of logical algebraic structures.In this paper,we study the internal state operators on pseudo EQ-algebras to improve the internal state theory of pseudo EQ-algebras,and provide a more general algebraic method to describe the truth average degree of propositions in higher order fuzzy logic.First,we introduce the concept of internal states of pseudo EQ-algebras and study its properties.We call the pair of(E,σ)is a pseudo EQ-algebrs,with internal state(denoted as ISPEQ-algebra,where a is an internal state on E).In addition,we also study the relation between states and internal states on pseudo EQ-algebras.Next,we study the properties of the state filters(denoted as S-filters)and state prefilters(denoted as S-prefilters)of ISPEQ-algebras.Meanwhile,we introduce the concept of congruence θ on ISPEQ-algebras.We get an one-to-one correspondence between the set of all strong normal S-filters of(E,σ)(SNSF(E,σ))and the set of congruences(induced by strong normal filters)on(E,σ)(SNCon(E,σ)).Finally,we obtain the generating formula of the prefilters on the residuated pseudo EQ-algebras,and we get the generating formula of the S-prefilters on the residuated ISPEQ-algebras.The results of the paper include:(1)Let(E,σ)be an ISPEQ-algebra.Then,there is a bijective correspondence between Scom(E,σ)and S(σ(E)),where Scom(E,σ)is the set of all acompatible Bosbach states on(E,σ)and S(σ(E))is the set of all Bosbach states on the pseudo EQ-algebra σ(E).(2)Let(E,σ)be an ISPEQ-algebra and F be a S-filter of(E,σ),thenσ(F)=F∩σ(E).(3)Let X be a nonempty subset of a residuated pseudo EQ-algebra E.Then,... |
