Study On Pseudo Strong-BI-algebras And Prequantales

In the research of fuzzy logic,many algebraic structures related to fuzzy implication are introduced,such as residuated lattice,basic implication algebra,non-associative residuated lattice,quantum B-algebra and so on.The study of the structure of prequantale(non-associative quantale)can more deeply characterize the algebraic structures related to quantum logic.Since basic implication algebra is a very extensive algebraic structure,some properties such as(EP)and(PEP)in fuzzy implication are not reflected in basic implication algebra.In order to characterize these special properties of fuzzy implication,based on basic implication algebra,this paper introduces some new concepts such as pseudo strong BI-algebra and residuated pseudo strong BI-algebra,and studies their filters and quotient structures.In this paper,we explore the relations among filters,ideals,congruences and weak congruences on prequantale.The research contents and main conclusions are as follows:(1)Based on the basic implication algebra,some new concepts such as strong BI-algebras,pseudo strong BI-algebras and residuated pseudo strong BI-algebras are introduced by adding(PEP)and residuated conditions.Among them,pseudo strong BI-algebras are the generalization of quantum b-algebras and pseudo BCK/BCI algebras.Meanwhile,the filter theory and quotient structure of pseudo strong BI-algebras are constructed.The congruence and filter of residuated pseudo strong BI-algebras are defined,and the corresponding quotient algebra structure is established.It is proved that the filter of non-associative residuated lattice is a special case of residuated pseudo strong BI-algebras.Furthermore,it is shown that the filter of residual lattice and non-associative residuated lattice can be unified under the general framework of filter theory of residuated pseudo strong BI-algebras.(2)Based on the previous research work of prequantale,the definition of filter on prequantale is given,and the relationship among filter,ideal and congruence of prequantale is deeply analyzed.The internal relationship among prequantale,semi-uninorm and pseudo strong BI-algebra is systematically revealed for the first time.The properties of involutive prequantale are studied.The one-to-one correspondence between congruences and homomorphisms of involutional prequantale is given.Then,some new concepts such as involutional pseudo strong BI-algebras and involutional complete residuated pseudo strong BI-algebras are introduced,and their close relationship with involutional prequantale is studied.It is proved that involutional prequantale and involutional complete residuated pseudo strong BI-algebras can be derived from each other.Thus,the internal relation between pseudo strong BI-algebras and prequantale is revealed.