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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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1435-5337
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Volume 24, Issue 2

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

Manuela Busaniche
/ Leonardo Cabrer
• CONICET, Dep. de Matemáticas – Facultad de Ciencias Exactas, Universidad Nacional del Centro de la Provincia de Buenos Aires, Pinto 399 – Tandil (7000), Argentina
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/ Daniele Mundici
• Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
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Published Online: 2012-02-25 | DOI: https://doi.org/10.1515/form.2011.059

## 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.

Revised: 2010-02-19

Published Online: 2012-02-25

Published in Print: 2012-03-01

Citation Information: Forum Mathematicum, Volume 24, Issue 2, Pages 253–271, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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© 2012 by Walter de Gruyter Berlin Boston.