Weak Commutative Pseudoeffect Algebras

Abstract

As noncommutative generalizations of effect algebras, we introduce weak commutative pseudoeffect algebras. In this paper, we prove that the generalized pseudoeffect algebras can be unitized if and only if they are weak commutative. Then we discuss the relationships between weak commutative pseudoeffect algebras and weak commutative generalized pseudoeffect algebras. We prove that the category of weak commutative pseudoeffect algebras is a reflective subcategory of weak commutative generalized pseudoeffect algebras. Similarly, we introduce weak commutative pseudodifference posets and show the relationships between weak commutative pseudoeffect algebras and weak commutative pseudodifference posets.