Open AccessLattice-ordered Abelian groups and Schauder bases of unimodular fans
Author(s) -
Corrado Manara,
Vincenzo Marra,
Daniele Mundici
Publication year - 2006
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/s0002-9947-06-03935-3
Subject(s) - algorithm , annotation , artificial intelligence , computer science , mathematics
Baker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group G G in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support | Σ | |\Sigma | of a fan Σ \Sigma . A unimodular fan Δ \Delta over | Σ | |\Sigma | determines a Schauder basis of G G : its elements are the minimal positive free generators of the pointwise ordered group of Δ \Delta -linear support functions. Conversely, a Schauder basis H \mathbf {H} of G G determines a unimodular fan over | Σ | |\Sigma | : its maximal cones are the domains of linearity of the elements of H \mathbf {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 G is finitely generated projective iff it can be presented by a purely lattice-theoretical word.