Lexicographic Effect Algebras

TL;DR

This paper explores a class of effect algebras represented as lexicographic products of po-groups, establishing algebraic conditions, categorical equivalences, and representation theorems including subdirect product representations.

Contribution

It introduces the category of strong (H,u)-perfect effect algebras and proves its categorical equivalence to directed po-groups with interpolation, advancing the structural understanding of effect algebras.

Findings

Characterization of effect algebras as lexicographic products

Categorical equivalence between strong (H,u)-perfect effect algebras and directed po-groups

Representation theorems including subdirect product representations

Abstract

In the paper we investigate a class of effect algebras which can be represented in the form of the lexicographic product $\Gamma(H\lex G,(u,0))$, where $(H,u)$ is an Abelian unital po-group and $G$ is an Abelian directed po-group. We study algebraic conditions when an effect algebra is of this form. Fixing a unital po-group $(H,u)$, the category of strong $(H,u)$-perfect effect algebra is introduced and it is shown that it is categorically equivalent to the category of directed po-group with interpolation. We show some representation theorems including a subdirect product representation by antilattice lexicographic effect algebras.