More about sharp and meager elements in Archimedean atomic lattice effect algebras

TL;DR

This paper explores the structure of atomic Archimedean lattice effect algebras, focusing on meager elements, centers, and compatibility centers, and establishes a Triple Representation Theorem for sharply dominating cases.

Contribution

It provides new insights into the relationships between centers and compatibility centers and introduces a Triple Representation Theorem for a specific class of effect algebras.

Findings

Center C(E) is bifull iff compatibility center B(E) is bifull in sharply dominating cases.

New description of the smallest sharp element over x in E.

Proved the Triple Representation Theorem for sharply dominating atomic lattice effect algebras.

Abstract

The aim of our paper is twofold. First, we thoroughly study the set of meager elements M(E), the center C(E) and the compatibility center B(E)in the setting of atomic Archimedean lattice effect algebras E. The main result is that in this case the center C(E) is bifull (atomic) iff the compatibility center B(E) is bifull (atomic) whenever E is sharply dominating. As a by-product, we give a new descriciption of the smallest sharp element over x in E via the basic decomposition of x. Second, we prove the Triple Representation Theorem for sharply dominating atomic Archimedean lattice effect algebras.