Blocks of homogeneous effect algebras

TL;DR

This paper introduces homogeneous effect algebras, a new class that generalizes several existing structures, and proves they can be decomposed into blocks with specific properties, expanding understanding of effect algebra structures.

Contribution

It defines homogeneous effect algebras, shows they are unions of blocks satisfying Riesz decomposition, and explores their structural properties and examples.

Findings

Homogeneous effect algebras include orthoalgebras and lattice-ordered effect algebras.

Every homogeneous effect algebra is a union of blocks with Riesz decomposition.

Standard effect algebra on a Hilbert space with dimension > 1 is not homogeneous.

Abstract

Effect algebras, introduced by Foulis and Bennett in 1994, are partial algebras which generalize some well known classes of algebraic structures (for example orthomodular lattices, MV algebras, orthoalgebras etc.). In the present paper, we introduce a new class of effect algebras, called {\em homogeneous effect algebras}. This class includes orthoalgebras, lattice ordered effect algebras and effect algebras satisfying Riesz decomposition property. We prove that every homogeneous effect algebra is a union of its blocks, which we define as maximal sub-effect algebras satisfying Riesz decomposition property. This generalizes a recent result by Rie\v{c}anov\'a, in which lattice ordered effect algebras were considered. Moreover, the notion of a block of a homogeneous effect algebra is a generalization of the notion of a block of an orthoalgebra. We prove that the set of all sharp elements in a homogeneous effect algebra $E$ forms an orthoalgebra $E_S$. Every block of $E_S$ is the center of a block of $E$. The set of all sharp elements in the compatibility center of $E$ coincides with the center of $E$. Finally, we present some examples of homogeneous effect algebras and we prove that for a Hilbert space $\mathbb H$ with $dim(\mathbb H)>1$, the standard effect algebra $\mathcal E(\mathbb H)$ of all effects in $\mathbb H$ is not homogeneous.