Baker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group G in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support of a fan |S|. A unimodular fan D over |S| determines a Schauder basis of G: its elements are the minimal positive free generators of the pointwise ordered group of D-linear support functions. Conversely, a Schauder basis H of G determines a unimodular fan over |S|: its maximal cones are the domains of linearity of the elements of H. The main purpose of this paper is to give various representation-free characterisations of Schauder bases. The latter, jointly with the De Concini-Procesi starring technique, will be used to give novel characterisations of finitely generated projective Abelian lattice ordered groups. For instance, G is finitely generated projective iff it can be presented by a purely lattice-theoretical word.

Lattice-ordered Abelian groups and Schauder bases of unimodular fans / C. Manara, V. Marra, D. Mundici. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - 359:4(2007), pp. 1593-1604.

Lattice-ordered Abelian groups and Schauder bases of unimodular fans

V. Marra
Secondo
;
2007

Abstract

Baker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group G in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support of a fan |S|. A unimodular fan D over |S| determines a Schauder basis of G: its elements are the minimal positive free generators of the pointwise ordered group of D-linear support functions. Conversely, a Schauder basis H of G determines a unimodular fan over |S|: its maximal cones are the domains of linearity of the elements of H. The main purpose of this paper is to give various representation-free characterisations of Schauder bases. The latter, jointly with the De Concini-Procesi starring technique, will be used to give novel characterisations of finitely generated projective Abelian lattice ordered groups. For instance, G is finitely generated projective iff it can be presented by a purely lattice-theoretical word.
De Concini-Procesi starring; Lattice-ordered Abelian group; Projective ℓ-group; Singular homology group; Unimodular fan
Settore MAT/01 - Logica Matematica
Settore MAT/02 - Algebra
Settore INF/01 - Informatica
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/19764
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