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

BUSANICHE, MANUELA; CABRER, LEONARDO; MUNDICI, DANIELE

FORUM MATHEMATICUM

WALTER DE GRUYTER & CO

Año: 2012 vol. 24 p. 253 - 271

0933-7741

A unital l-group (G; u) is an abelian group G equipped with a translation-invariant 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 fi nitely presented unital l- groups, confluence is a necessary condition for lim S = lim T : (Suficiency is an easy byproduct of a general result). When (G; u) is fi nitely generated we equip it with a sequence W(G;u) = (W0;W1;...) of weighted abstract simplicial complexes, where Wt+1 is obtained from Wt either by the classical Alexander binary stellar operation, or by deleting a maximal simplex of Wt. We show that the map (G; u) into W(G;u) has an inverse. A confluence criterion is given to recognize when two sequences arise from isomorphic unital l-groups.