The Research On Some Algebraic Structures Based On BL-algebras

Basic logic is a typical kind of non-classical logic.Based on continuous tri-angle norm,it is the public foundation of all propositional calculus systems.BL-algebra is the algebraic semantic of basic logic,it not only provides an algebraic framework for basic logic,but also offers an algebraic method for studying continu-ous triangle norm on[0,1].In this paper,we study the annihilators in BL-algebras,hyper BL-algebras,and the monadic operators on effect algebras,which are closely related to BL-algebras.We aim to improve the ideal theory of BL-algebras,and of-fer a new method for studying partial algebras.The main contents of the research are as follows:In the second chapter,we study the annihilators in BL-algebras.Firstly,we introduce and investigate the annihilators and generalized annihilators in BL-algebras.We obtain that the ideal lattice(I(L),(?))is pseudo-complemented,and for any ideal I,its pseudo-complement is the annihilator of I.Secondly,we com-bine the homomorphism with annihilators,and give a necessary and sufficient con-dition,under which the homomorphism image and pre-image of an annihilator will be an annihilator.Thirdly,we note E(I)= {x ∈L|(?)i I,i⊥(?)x⊥},and show that(E(I(L)),∧E,∨E,E(0),E(L))is a pseudo-complemented lattice,a com-plete Brouwerian lattice and an algebraic lattice,when L is a finite product of BL-chains.Finally,we introduce the notion of relative involutory ideals and denote the set of all of them by SI(L),then obtain that SI(L)can be made into a complete Boolean lattice and a BL-algebra with respect to the suit operations,respectively.In the third chapter,we introduce and investigate hyper BL-algebras.Firstly,we establish the axiomatic system of hyper BL-algebras,which is a generalization of BL-algebras.And then we give some non-trivial examples and properties of hy-per BL-algebras.Secondly,we introduce hyper filters and hyper deductive systems of hyper BL-algebras and study the relationships between them.Further we prove that every proper hyper filter(hyper deductive system)is contained in a maximal hyper filter(maximal hyper deductive system).Thirdly,we put forth the concept of hyper congruence on hyper BL-algebras.And we obtain that the quotient struc-ture L/θ is a hyper BL-algebra,when θ is a regular compatible hyper congruence,and L/θ is a MV-algebra,when θ is a good regular compatible hyper congruence.Finally,we introduce the notion of sup-Bosbach states and sup-Riecan states on hyper BL-algebras and obtain that every sup-Bosbach state is a sup-Riecan state,when hyper BL-algebra is involutory.In the fourth chapter,we study the monadic operators on effect algebras.Firstly,we design the axiomatic system of existential quantifiers on effect alge-bras,and use it to give the definition of monadic effect algebras.Then define the universal quantifier as the dual of the existential quantifiers.Secondly,we intro-duce the relatively complete subalgebra of an effect algebra and obtain that there exists an one-to-one correspondence between the set of all the existential quanti-fiers and the set of all the relatively complete subalgebras.Thirdly,we introduce the notion of monadic ideals on monadic effect algebras and give the generated for-mula of it.Then we get that there exists a lattice isomorphism between the lattice of all monadic ideals of(E,(?))and the lattice of all ideals of(?)E.Further,we in-troduce the definition of strong existential quantifier and get that(E/I,(?)I)is also a monadic effect algebra,when I is a Riesz ideal and(?)is strong.Finally,we discuss the relationship between monadic BL-algebras and monadic effect algebras.