Direct limits of generalized pseudo-effect algebras with the Riesz decomposition properties

Abstract

In this paper, we focus on direct limits and inverse limits in the category with generalized pseudo-effect algebras (GPEAs for short) as objects and GPEA-morphisms as morphisms. We show that direct limits exist in the category of GPEAs and direct limits of GPEAs satisfy the Riesz decomposition properties whenever the directed systems of GPEAs satisfy the Riesz decomposition properties. Then, we give a condition under which the quotient of a direct limit of GPEAs is a direct limit of quotients of GPEAs. Moreover, we prove that if inverse systems of GPEAs satisfy the Riesz decomposition properties, then inverse limits also satisfy the Riesz decomposition properties.