The Study Of Some N-Fold Filters And States On Semihoops

Various kinds of fuzzy logical algebras as the semantical systems of fuzzy logic systems have been extensively introduced and studied.Semihoops are generalizations of hoops which were originally introduced by Bosbach.The filter theory of the logical algebras plays the essential role in studying algebraic structure.From a logical point of view,various filters correspond to various sets of provable formulas.Also,filter theory plays an important role in studying the existences of states.The aim of this paper is to study n-fold filter theory and states.We will use the research results to perfect the relevant theory of semihoop and lay the theoretical foundation for the study of other algebraic structures.The main contents are as follows:Firstly,we introduce some special types of n-fold filters on semihoops and derive some of their characterizations.The equivalent characterization of these n-fold filters are given and the relationships among them are discussed.The structures and properties of quotient algebras are researched by the n-fold(positive)implicative filters and MV-filters.Next,we introduce the notions of perfect Bosbach(Riecan)states and research the existences of perfect Bosbach(Riecan)states.Furthermore,we discuss the relationships among Bosbach states,extremal states and state-morphisms on a semihoop.Finally,we study the mutual induction about semihoops and partially ordered abelian monoids.In addition,the relations between states on semihoops and states on partially ordered abelian monoids are investigated.We get the following results:(1)A semihoop L is a n-fold(positive)implicative semihoop if and only if any filter of L is consistent with n-fold(positive)implicative filter.(2)For any filter of a semihoop,it is a positive implicative filter if and only if it is an implicative filter and MV-filter.(3)Let L be a semihoop and F be a filter of L.Then F is an MV-filter if and only if L/F is an MV-algebra.(4)A semihoop L admits perfect Bosbach(Riecan)state if and only if L has a perfect filter.(5)State-morphisms,extremal Bosbach states and extremal Riecan states coincide on a semihoop.(6)If s is a Bosbach state on a semihoop,then s can be regarded as a state on a partially ordered abelian monoid.Conversely,if s is a state on a partially ordered abelian monoid,then s can be regarded as a Bosbach state on a semihoop.