Confluence and combinatorics in finitely generated unital lattice-ordered abelian groups

Source abstract

Abstract. A unital $\ell$ -group ( G , u ) $(G,u)$ is an abelian group G $G$ equipped with a translation-invariant lattice-order and a distinguished element u $u$ , called order-unit, whose positive integer multiples eventually dominate each element of G $G$ . It is shown that, for direct systems $\mathcal {S}$ and $\mathcal {T}$ of finitely presented unital $\ell$ -groups, confluence is a necessary condition for lim lim $\lim \mathcal {S} \cong \lim \mathcal {T}$ . (Sufficiency is an easy byproduct of a general result). When ( G , u ) $(G,u)$ is finitely generated we equip it with a sequence ( G , u ) = ( W 0 , W 1 , ... ) $\mathcal {W}_{(G,u)} = (W_{0},W_{1},\ldots )$ of weighted abstract simplicial complexes, where W t + 1 $W_{t+1}$ is obtained from W t $W_{t}$ either by the classical Alexander binary stellar operation, or by deleting a maximal simplex of W t $W_{t}$ . We show that the map ( G , u ) ( G , u ) $(G,u)\mapsto \mathcal {W}_{(G,u)}$ has an inverse. A confluence criterion is given to recognize when two sequences arise from isomorphic unital $\ell$ -groups.