Effect Algebra And Pseudo Bl-algebra Structure

It is well-known that quantum mechanics is a set of rules to construct physical theories. In 1936, G.Birkhoff and J.von Neumann proposed the concept which is the mathematical foundation of quantum logics. In algebra, quantum logics is a ortho-modular lattice(the lattice of all closed subspaces of a separable infinite dimensional complete Hilbert space is an orthomodular lattice). However, at the sixties of the last century, some new kinds of algebraic structures, such as orthoalgebras, weak orthoalgebras, appeared in the field of quantum logics with its development. In the nineties of the 20th century, Slovak and Italian schools introduced the concept of a difference poset. Later the American school came up with the concept of an effect algebra which is equivalent to difference poset.In 1996, BL-algebra was introduced by P.Hajek as an algebraic structure of "Basic Logic"(BL, for short). MV algebras, product algebras, Godle algebras are purticular cases of BL-algebras.In order to describe series or sequential measurements of effects, Gudder gave the notion of sequential product in an effect algebra in 2001, and researched the properties of sequential effect algebras.In this thesis we mainly foucs on three kinds of algebraic structures, i.e., pseudo effect algebras, sequential effect algebras ang pseudo BL-algebras. The main contri-butions in this thesis are listed as follows:(1) The structures of perfect pseudo effect algebras are studied.We know that there exists a very important category equivalence theory, i.e., for every perfect pseudo effect algebra E, there is a unique(up to isomorphism) directed Abelian po-group G with interpolation such that E is an interval in the lexicographical product of the group of all integers with G, and there is a categorical equivalence between the category (?) of directed Abelian po-groups with interpola-tion and the category (?) of perfect effect algebras. we give the notion of perfect pseudo effect algebras, and show that there are similar results in perfect pseudo effect algebras. It is proved that there exists a faithful and full functor from the category of directed interpolation po-groups (?) into the category of perfect pseudo effect algebras (?) and every perfect pseudo effect algebra is an interval in the lexicographical product of the group of all integers with a directed interpolation po-group.(2) The ideal, congruence and subdirect product representation in pseudo weak effect algebras are studied.We introducte the concept of pseudo weak effect algebras and give the concept of pseudo weak difference poset. If we define two different operations/,\in a pseudo weak effect algebra, then the pseudo weak effect algebra also is a pseudo weak difference poset:conversely, if we define a non-commutative operation⊕in a pseudo weak difference poset, then the pseudo weak difference poset also is a pseudo weak effect algebra, thus we prove that the pseudo weak effect algebras and the pseudo weak difference posets are the same thing. We give the exact conditions that an equivalence relation does not only allow the formation of a quotient algebra, but also that this quotient is also again a pseudo weak effect algebra. Last, we show that pseudo BL-effect algebras are subdirect products of linearly ordered ones.(3)The ideal, congruence and Holland theorey in sequential effect algebras are studied.A distributive sequential effect algebra(DSEA) is introduced, and left ideal, right ideal, ideal, prime ideal and congruence are given in a distributive sequen-tial effect algebra. It is proved that every distributive sequential effect algebra (E;⊕,o,0,1) with the RDP having 1 as a product unity is a subdirect product of antilattice distributive sequential effect algebra with the RDP. For the special com-mutative sequential effect algebras, we research the Holland theory in it. It is shown that such a sequential effect algebra is isomorphic to a sequential effect algebra of automorphisms of aΛ-antilattice(4)The connection of pseudo BL-algebras and pseudo MV-algebras and the fuzzy fiters of pseudo BL-algebras are studied.We introduce the notions of local pseudo BL-algebra and symmetrical pseudo BL-algebra and show that a symmetrical pseudo BL-algebra L is local iff ord(x)<∞or ord(x*)<∞for every x∈L. we also show some properties of symmetrical pseudo BL-algebras. Further, we research the connection of local pseudo BL-algebras and local pseudo MV-algebras.We introduce the notions of fuzzy filters, fuzzy normal filters and fuzzy prime fiters in pseudo BL-algebras and investigate some of their properties. We discuss the fuzzy filter generated by a fuzzy set.