Modularity, Atomicity and States in Archimedean Lattice Effect Algebras

TL;DR

This paper investigates the properties of modular Archimedean atomic lattice effect algebras, demonstrating the existence of a specific continuous state and analyzing finite and compact elements within these structures.

Contribution

It establishes the existence of an $(o)$-continuous subadditive state on certain non-orthomodular lattice effect algebras and explores their finite and compact elements.

Findings

Existence of an $(o)$-continuous subadditive state on non-orthomodular lattice effect algebras.

Characterization of finite and compact elements in these algebras.

Insights into the structure of modular Archimedean atomic lattice effect algebras.

Abstract

Effect algebras are a generalization of many structures which arise in quantum physics and in mathematical economics. We show that, in every modular Archimedean atomic lattice effect algebra $E$ that is not an orthomodular lattice there exists an $(o)$-continuous state $\omega$ on $E$, which is subadditive. Moreover, we show properties of finite and compact elements of such lattice effect algebras.