Sharply Orthocomplete Effect Algebras

TL;DR

This paper investigates the properties of sharply orthocomplete effect algebras, establishing conditions under which their sharp elements and centers form complete lattices, and demonstrating completeness in certain atomic cases.

Contribution

It proves that sharply orthocomplete S-dominating effect algebras have complete sharp elements and centers, and that Archimedean atomic effect algebras are complete if sharply orthocomplete.

Findings

Sharp elements and centers are complete lattices in sharply orthocomplete S-dominating effect algebras.

Sharply orthocomplete Archimedean atomic effect algebras are complete.

Connections between sharp orthocompleteness, sharp dominancy, and completeness are established.

Abstract

Special types of effect algebras $E$ called sharply dominating and S-dominating were introduced by S. Gudder in \cite{gudder1,gudder2}. We prove statements about connections between sharp orthocompleteness, sharp dominancy and completeness of $E$. Namely we prove that in every sharply orthocomplete S-dominating effect algebra $E$ the set of sharp elements and the center of $E$ are complete lattices bifull in $E$. If an Archimedean atomic lattice effect algebra $E$ is sharply orthocomplete then it is complete.