Quotients of partial abelian monoids (Q1966130)

A hierarchy of partial abelian monoids (PAM) is considered. In order of decreasing generality these structures include cancellative PAMs, effect algebras, orthoalgebras, orthomodular posets and orthomodular lattices (OML). The concept of congruence on PAMs is introduced and related with morphisms and factor structures. The authors introduced a concept of Riesz ideal \(I\), and it is shown that \(I\) generates a congruence on cancellatice PAMs. The corresponding quotiens \(P/I\) for structures in the hierarchy are studied. It is shown that a subset \(I\) of OML is a Riesz ideal iff \(I\) is a \(p\)-ideal. For effect algebras, the congruences generated by Riesz ideals are those that are given by perspectivity. The paper includes a large number of examples which illustrate the introduced concepts.