Research On The Algebraic Directional Limit And Inverse Limit Of Generalized Pseudo-effects

The field of quantum logic is aimed at establishing the corresponding mathe-matical model for quantum mechanics,that is,to provide mathematical foundation of quantum mechanics.This field originates in 1936,when Birkhoff and von Neu-mann started to investigate mathematical models of logic of propositions about quantum mechanical system-Orthomodular lattices,then orthomodular lattices valued logics are called quantum logic.Categorical limits of quantum logics are widely concerned.In this thesis,firstly,we discuss the existence of direct limits and inverse limits in the category of gen-eralized pseudo effect algebras.Then,we study Riesz decomposition properties in the category of generalized pseudo effect algebras.At last,we focus on morphisms of lattice ordered generalized pseudo effect algebras.The following are the main results:1.We introduce a definition of direct systems and direct limits of generalized pseudo effect algebras.We show that direct limits of generalized pseudo effect algebras exist in the category with generalized pseudo effect algebras as objects and generalized pseudo effect algebraic morphisms as morphisms.On this basis,we demonstrate that the direct limits of generalized pseudo effect algebras satisfy the Riesz decomposition properties whenever the directed systems of generalized pseudo effect algebras satisfy the Riesz decomposition properties.Then,we prove that a binary operation of direct limits of generalized pseudo effect algebras satisfy congruence relation whenever the binary operation of directed systems of generalized pseudo effect algebras satisfy congruence relation.Moreover,we give a condition under which the quotient of a direct limit of generalized pseudo effect algebras is a direct limit of quotients of generalized pseudo effect algebras.2.We introduce a definition of inverse systems and inverse limits of generalized pseudo effect algebras.We demonstrate that inverse limits of generalized pseudo effect algebras exist in the category with generalized pseudo effect algebras as objects and generalized pseudo effect algebraic morphisms as morphisms.We prove that if inverse systems of generalized pseudo effect algebras satisfy the Riesz decomposition properties,then inverse limits also satisfy the Riesz decomposition properties.3.We show a correspondence between morphisms of intervals in generalized pseudo effect algebras and morphisms of generalized pseudo effect algebras.We give a necessary and sufficient condition under which a morphism of lattice ordered generalized pseudo effect algebra is also a morphism of underlying lattices.