Confluence and combinatorics in finitely generated unital lattice-ordered abelian groups

Abstract

A unital ℓ-group (G; u) is an abelian group G equipped with a translationinvariant lattice-order and a distinguished element u, called order-unit, whose positive integer multiples eventually dominate each element of G. It is shown that, for direct systems S and T of finitely presented unital ℓ-groups, confluence is a necessary condition for lim S ≅ lim T . (Sufficiency is an easy byproduct of a general result). When (G; u) is finitely generated we equip it with a sequence W (G;u) = (W 0;W 1; : : ) of weighted abstract simplicial complexes, where W t+1 is obtained from W t either by the classical Alexander binary stellar operation, or by deleting a maximal simplex of W t. We show that the map (G; u) → W (G;u) has an inverse. A confluence criterion is given to recognize when two sequences arise from isomorphic unital ℓ-groups.