Abstract.
A hierarchy of partial abelian structures is considered. In an order of decreasing generality, these structures include partial abelian monoids (PAM), cancellative PAMs (CPAM), effect algebras (or D-posets), orthoalgebras, orthomodular posets (OMP) and orthomodular lattices (OML). If P is a PAM, the concepts of a congruence \( \sim \) on P and a quotient P \(// \sim \) are defined. Similar definitions are given for quotients of higher level categories in the hierarchy. The notion of a Riesz ideal I on a CPAM P is defined and it is shown that I generates a congruence on P. The corresponding quotients P/I for categories in the hierarchy are studied. It is shown that a subset I of an OML is a Riesz ideal if and only if I is a p-ideal. Moreover, for effect algebras, we show that congruences generated by Riesz ideals are precisely those that are given by a perspectivity. The paper includes a large number of counterexamples and examples that illustrate various concepts.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received April 14, 1997; accepted in final form January 19, 1998.
Rights and permissions
About this article
Cite this article
Gudder, S., Pulmannová, S. Quotients of partial abelian monoids. Algebra univers. 38, 395–421 (1997). https://doi.org/10.1007/s000120050061
Issue Date:
DOI: https://doi.org/10.1007/s000120050061