Fans, decision problems and generators of free abelian \(\ell\)-groups. (English) Zbl 06814715

Mundici, Daniele

Forum Math. 29, No. 6, 1429-1439 (2017).

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

Keywords:

abelian \(\ell\)-group; nonsingular Fan; desingularization; decision problem; Baker-Beynon duality; Markov unrecognizability theorem

References:

[1]	M. Anderson and T. Feil, Lattice-Ordered Groups. An Introduction, D. Reidel, Dordrecht, 1988. · Zbl 0636.06008
[2]	K. A. Baker, Free vector lattices, Canad. J. Math. 20 (1968), 58-66. · Zbl 0157.43401
[3]	W. M. Beynon, Duality theorems for finitely generated vector lattices, Proc. Lond. Math. Soc. (3) 31 (1975), 114-128. · Zbl 0309.06009
[4]	W. M. Beynon, Applications of duality in the theory of finitely generated lattice-ordered abelian groups, Canad. J. Math. 29 (1977), 243-254. · Zbl 0361.06017
[5]	A. Bigard, K. Keimel and S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math 608, Springer, Berlin, 1971.
[6]	A. V. Chernavsky and V. P. Leksine, Unrecognizability of manifolds, Ann. Pure Appl. Logic 141 (2006), 325-335. · Zbl 1115.57014
[7]	T. Evans, Finitely presented loops, lattices, etc. are hopfian, J. Lond. Math. Soc. (2) 44 (1969), 551-552. · Zbl 0177.02702
[8]	G. Ewald, Combinatorial Convexity and Algebraic Geometry, Grad. Texts in Math. 168, Springer, New York, 1996.
[9]	A. M. W. Glass, Partially Ordered Groups, Ser. Algebra 7, World Scientific, Singapore, 1999. · Zbl 0933.06010
[10]	A. M. Glass and W. C. Holland, Lattice-Ordered Groups: Advances and Techniques, Math. Appl. 48, Kluwer Academic, Dordrecht, 1989. · Zbl 0705.06001
[11]	A. M. W. Glass and J. J. Madden, The word problem versus the isomorphism problem, J. Lond. Math. Soc. (2) 30 (1984), 53-61. · Zbl 0551.20018
[12]	G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Clarendon Press, Oxford, 1960. · Zbl 0086.25803
[13]	R. Hirshon, Some Theorems on Hopficity, Trans. Amer. Math. Soc. 141 (1969), 229-244. · Zbl 0186.32002
[14]	A. I. Kostrikin and I. R. Shafarevich, Algebra II: Noncommutative Rings Identities, Springer, New York, 2012.
[15]	A. Mijatović, Simplifying triangulations of S^3, Pacific J. Math. 208 (2003), no. 2, 291-324. · Zbl 1071.52016
[16]	D. Mundici, Simple Bratteli diagrams with a Gödel incomplete C^*-equivalence problem, Trans. Amer. Math. Soc. 356 (2003), no. 5, 1937-1955. · Zbl 1042.46033
[17]	T. Oda, Convex Bodies and Algebraic Geometry, Springer, Berlin, 1988.
[18]	E. Outerelo and J. M. Ruiz, Mapping Degree Theory, Grad. Stud. Math. 108, American Mathematical Society, Providence, 2009. · Zbl 1183.47056
[19]	M. A. Shtan’ko, Markov’s theorem and algorithmically non-recognizable combinatorial manifolds, Izv. Math. 68 (2004), 207-224.
[20]	J. R. Stallings, Lectures on Polyhedral Topology, Tata Institute of Fundamental Research, Mumbay, 1967. · Zbl 0182.26203
[21]	A. Thompson, Thin position and the recognition problem for S^3, Math. Res. Lett. 1 (1994), no. 5, 613-630. · Zbl 0849.57009
[22]	V. Weispfenning, The complexity of the word problem for abelian ℓ-groups, Theoret. Comput. Sci. 48 (1986), 127-132. · Zbl 0633.03036