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Fans, decision problems and generators of free abelian \(\ell\)-groups. (English) Zbl 06814715
Summary: Let \({t_{1},\dots,t_{n}}\) be \(\ell\)-group terms in the variables \({X_{1},\dots,X_{m}}\). Let \({\hat{t}_{1},\dots,\hat{t}_{n}}\) be their associated piecewise homogeneous linear functions. Let \(G\) be the \(\ell\)-group generated by \({\hat{t}_{1},\dots,\hat{t}_{n}}\) in the free \(m\)-generator \(\ell\)-group \({\mathcal{A}_{m}}\). We prove: (i) the problem whether \(G\) is \(\ell\)-isomorphic to \({\mathcal{A}_{n}}\) is decidable; (ii) the problem whether \(G\) is \(\ell\)-isomorphic to \({\mathcal{A}_{l}}\) (\(l\) arbitrary) is undecidable; (iii) for \({m=n}\), the problem whether \({\{\hat{t}_{1},\dots,\hat{t}_{n}\}}\) is a free generating set is decidable. In view of the Baker-Beynon duality, these theorems yield recognizability and unrecognizability results for the rational polyhedron associated to the \(\ell\)-group \(G\). We make pervasive use of fans and their stellar subdivisions.

MSC:
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
08B30 Injectives, projectives
06B25 Free lattices, projective lattices, word problems
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52B55 Computational aspects related to convexity
55N10 Singular homology and cohomology theory
57Q05 General topology of complexes
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