Confluence and combinatorics in finitely generated unital lattice-ordered abelian groups. (English) Zbl 1277.06007

Busaniche, Manuela; Cabrer, Leonardo; Mundici, Daniele

Forum Math. 24, No. 2, 253-271 (2012).

In this paper, the authors are interested in the category whose objects are unital \(l\)-groups (i.e. abelian groups \((G,u)\) equipped with a translation-invariant lattice-order and a distinguished order-unit \(u\)) and whose morphisms are unital \(l\)-homomorphisms (i.e. morphisms which preserve the lattice and the group structure and map order-units into order-units).
They use the categorical equivalence of this category with another one to define free unital \(l\)-groups as finitely presented unital \(l\)-groups. Which allows them to infer that finitely generated unital \(l\)-groups are the direct limits of countable direct systems of finitely presented unital \(l\)-groups with surjective connecting unital \(l\)-homomorphisms. Also, they define the confluence of two sequences of unital \(l\)-groups.
After having reported that in general categories confluence is not a necessary condition for direct limits to be isomorphic, the authors show in Theorems 3.1 and 3.3 that direct systems of unital \(l\)-groups and unital \(l\)-homomorphisms with isomorphic limits are necessarily confluent.
Next, they apply Alexander stellar operations to introduce suitable sequences of weighted abstract simplicial complexes, called stellar sequences, and construct a map which associates to each stellar sequence a unital \(l\)-group. They establish that, up to isomorphism, each finitely unital \(l\)-group comes from some stellar sequence.
Finally, they provide a necessary and sufficient conditions under which two stellar sequences can be represented by isomorphic unital \(l\)-groups.

Reviewer: Belmesnaoui Aqzzouz (Salajadida)

MSC:

06F20	Ordered abelian groups, Riesz groups, ordered linear spaces
52B11	\(n\)-dimensional polytopes
52B20	Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
57Q15	Triangulating manifolds
46L05	General theory of \(C^*\)-algebras

Keywords:

lattice-ordered abelian group; rational polyhedron; order-unit; confluence; direct system; abstract simplicial complex; stellar subdivision; Alexander starring; regular fan; De Concini-Procesi theorem; piecewise linear function; Elliott classification; AF \(C^*\)-algebra

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