Lexicographic effect algebras

Abstract

We investigate a class of effect algebras that can be represented in the form $${\Gamma (H \overrightarrow{\times} G}$$ , (u, 0)), where $${H \overrightarrow{\times} G}$$ means the lexicographic product of an Abelian unital po-group (H, u) and an Abelian directed po-group G. We study conditions when an effect algebra is of this form. Fixing a unital po-group (H, u), the category of strongly (H, u)-perfect effect algebras is introduced and it is shown that it is categorically equivalent to the category of directed po-groups with interpolation. We prove some representation theorems of lexicographic effect algebras, including a subdirect product representation by antilattice lexicographic effect algebras.