2014/05/28 by Cabrer, Leonardo Manuel
#06D35 #52B20 #55U10. Secondary: 08B30 #FOS: Mathematics #Group Theory (math.GR) #Primary: 06F20 #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1405.7118
A unital ℓ-group is an abelian group equipped with a translation invariant lattice-order and with a distinguished strong unit, i.e. an element whose positive integer multiples eventually dominate every element of G.If X is a compact subset of Rn, the set M(X) of real-valued piecewise linear maps with integer coefficients, whose addition and lattice operations defined pointwise and whose distinguished element is the constant map 1, is a unital ℓ-group. In this paper we provide a geometric decription of finitely generated (regular) projective unital ℓ-groups. We prove that a finitely unital ℓ-group is projective if and only if it is isomorphic to M(P) for some polyhedron P which is rational, contractible, contains an integer point, and satisfies an elementary arithmetical-topological property.