Two Kinds Of Logic Operators On Hoop Algebras

The notion of Hoop algebras is naturally ordered commutative residuated integral monoids,which is introduced by B.Bosbach in the 1960s.Modal logic is an important branch of classical logic category and modal operators is intu-itionistic propositional logic algebraic semantics.The thought of the derivation is derived from analysis.The monadic operators are the algebraization of the existential quantifier and universal quantifier in predicate logic.In this paper,we will concentrate on the theory of modal operators on Hoop algebras and monadic derivational operators on W-Hoop algebras.The main results we have done are as follows:Firstly,we define the notion of modal operators on Hoop algebras and dis-cuss some related properties of it.Moreover,three special mappings are in-troduced and investigated on Hoop algebras and equivalent characterizations of these mappings becoming modal operators are got.By use of closure operators,we make a further study about representable modal operators again.In addition,modal filter on modal Hoop algebras are studied.Then,we define modal congru-ences and modal homomorphisms on modal Hoop algebras,and prove that there is a surjective modal homomorphisms between(H,f)and(I(a),fa).Finally,we introduce the dual operators of modal operators.Based on the above,we define and research dual modal filters.At last,we obtain the relations between modal operators and dual operators.Secondly,we define and study monadic derivational operators combining monadic operators and derivational operators on W-Hoop algebras.Specifically,we define the notion of M-derivations on monadic W-Hoop algebras(M,3)and discuss some properties of it.Based on it we introduce the notions of the strong M-derivations,regular M-derivation and additive M-derivation.By use of these three kinds of special derivations,we give some equivalent conditions in which a W-Hoop algebras becomes a boolean algebra and some characterizations about the isotone M-derivations in monadic W-Hoop algebras are provided by regular M-derivations.In the end,monadic derivational ideals of monadic W-Hoop algebras are studied.In particular,algebraic structures of the set ID(M)of all monadic derivational ideals on regular monadic W-Hoop algebras are researched.Finally,we discuss the relations between modal operators and monadic dif-ferential operators on W-Hoop algebras.