Published by De Gruyter March 7, 2013

Simplicial geometry of unital lattice-ordered abelian groups

Leonardo Manuel Cabrer

https://doi.org/10.1515/forum-2011-0131

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Abstract

By a unital ℓ-group we mean a lattice-ordered abelian group with a distinguished order unit. This paper is concerned with the category 𝖴fp of finitely presented unital ℓ-groups. Using the duality between 𝖴fp and a category of rational polyhedra, we will provide (i) a construction of finite limits and co-limits in 𝖴fp; (ii) a Cantor–Bernstein–Schröder theorem for finitely presented unital ℓ-groups; (iii) a proof that the fibered product of finitely generated projective unital ℓ-groups is projective; (iv) a geometrical characterization of exact unital ℓ-groups.

Keywords: Unital lattice-ordered group; finitely presented; rational polyhedron; exact formula; projective algebra

MSC: 06F20; 52B20; 18B30; 05E45; 52B11; 18A35; 55U05; 55U10; 57Q05

The main results of this paper have been obtained while the author was holding a postdoctoral position in the Mathematical Institute of the University of Bern. We would like to thank Professor Daniele Mundici and Professor Manuela Busaniche for their helpful comments and suggestions on previous drafts of this paper. We are very grateful to the anonymous referee for her/his careful reading of this paper and for all the corrections and suggestions that have improved enormously the present work.

Received: 2011-11-14

Revised: 2012-8-19

Published Online: 2013-3-7

Published in Print: 2015-5-1

Simplicial geometry of unital lattice-ordered abelian groups

Abstract

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