Mathematics – Group Theory
Scientific paper
2009-08-14
Mathematics
Group Theory
17 pages
Scientific paper
A unital $\ell$-group $(G,u)$ is an abelian group $G$ equipped with a translation-invariant lattice-order and a distinguished element $u$, called order-unit, whose positive integer multiples eventually dominate each element of $G$. We classify finitely generated unital $\ell$-groups by sequences $\mathcal W = (W_{0},W_{1},...)$ of weighted abstract simplicial complexes, where $W_{t+1}$ is obtained from $W_{t}$ either by the classical Alexander binary stellar operation, or by deleting a maximal simplex of $W_{t}$. A simple criterion is given to recognize when two such sequences classify isomorphic unital $\ell$-groups. Many properties of the unital $\ell$-group $(G,u)$ can be directly read off from its associated sequence: for instance, the properties of being totally ordered, archimedean, finitely presented, simplicial, free.
Busaniche Manuela
Cabrer Leonardo
Mundici Daniele
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