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

Manuela Busaniche; Leonardo Cabrer; Daniele Mundici

doi:10.1515/form.2011.059

Article

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

Manuela Busaniche , Leonardo Cabrer and Daniele Mundici

Published/Copyright: February 25, 2012

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From the journal Volume 24 Issue 2

Abstract.

A unital -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 and of finitely presented unital -groups, confluence is a necessary condition for limlim. (Sufficiency is an easy byproduct of a general result). When (G,u) is finitely generated we equip it with a sequence (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)(G,u) has an inverse. A confluence criterion is given to recognize when two sequences arise from isomorphic unital -groups.

Keywords.: Lattice-ordered abelian group; rational polyhedron; order-unit; confluence; direct system; confluent direct system; simplicial complex; abstract simplicial complex; weighted abstract simplicial complex; stellar subdivision; Alexander starring; regular fan; De Concini–Procesi theorem; piecewise linear function; Elliott classification; AF

Received: 2009-08-19

Revised: 2010-02-19

Published Online: 2012-02-25

Published in Print: 2012-March

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https://doi.org/10.1515/form.2011.059

Keywords for this article

Lattice-ordered abelian group; rational polyhedron; order-unit; confluence; direct system; confluent direct system; simplicial complex; abstract simplicial complex; weighted abstract simplicial complex; stellar subdivision; Alexander starring; regular fan; De Concini–Procesi theorem; piecewise linear function; Elliott classification; AF