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 |Σ| of a fan Σ. A unimodular fan Δ over |Σ| determines a Schauder basis of G: its elements are the minimal positive free generators of the pointwise ordered group of Δ-linear support functions. Conversely, a Schauder basis H of G determines a unimodular fan over |Σ|: 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 / D. MUNDICI; MANARA C; MARRA V. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 359:(2007), pp. 1593-1604. [10.1090/S0002-9947-06-03935-3]
Lattice-ordered Abelian groups and Schauder bases of unimodular fans
MUNDICI, DANIELE;
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 |Σ| of a fan Σ. A unimodular fan Δ over |Σ| determines a Schauder basis of G: its elements are the minimal positive free generators of the pointwise ordered group of Δ-linear support functions. Conversely, a Schauder basis H of G determines a unimodular fan over |Σ|: 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.File | Dimensione | Formato | |
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