Archimedean Atomic Lattice Effect Algebras with Complete Lattice of Sharp Elements

TL;DR

This paper investigates the structure of Archimedean atomic lattice effect algebras with complete sharp elements, establishing properties of their centers and states under certain conditions.

Contribution

It characterizes properties of centers and states in atomic lattice effect algebras with complete sharp elements, including existence of faithful and subadditive states.

Findings

Existence of faithful states in separable, modular effect algebras.

Presence of (o)-continuous subadditive states when certain conditions hold.

Structural insights into centers and compatibility centers of such algebras.

Abstract

We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect algebras. Moreover, we prove that if such effect algebra $E$ is separable and modular then there exists a faithful state on $E$. Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra $\widehat{E}$ and the compatiblity center of $E$ is not a Boolean algebra then there exists an $(o)$-continuous subadditive state on $E$.