Fans, decision problems and generators of free abelian \(\ell\)-groups. (English) Zbl 1473.06023
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 |
| 06B25 | Free lattices, projective lattices, word problems |
| 08B30 | Injectives, projectives |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
| 52B55 | Computational aspects related to convexity |
| 57Q05 | General topology of complexes |
Keywords:
abelian \(\ell\)-group; nonsingular fan; desingularization; decision problem; Baker-Beynon duality; Markov unrecognizability theoremReferences:
| [1] | M. Anderson and T. Feil, Lattice-Ordered Groups. An Introduction, D. Reidel, Dordrecht, 1988.; Anderson, M.; Feil, T., Lattice-Ordered Groups. An Introduction (1988) · Zbl 0636.06008 |
| [2] | K. A. Baker, Free vector lattices, Canad. J. Math. 20 (1968), 58-66.; Baker, K. A., Free vector lattices, Canad. J. Math., 20, 58-66 (1968) · Zbl 0157.43401 |
| [3] | W. M. Beynon, Duality theorems for finitely generated vector lattices, Proc. Lond. Math. Soc. (3) 31 (1975), 114-128.; Beynon, W. M., Duality theorems for finitely generated vector lattices, Proc. Lond. Math. Soc. (3), 31, 114-128 (1975) · 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.; Beynon, W. M., Applications of duality in the theory of finitely generated lattice-ordered abelian groups, Canad. J. Math., 29, 243-254 (1977) · Zbl 0361.06017 |
| [5] | A. Bigard, K. Keimel and S. Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Math 608, Springer, Berlin, 1971.; Bigard, A.; Keimel, K.; Wolfenstein, S., Groupes et anneaux réticulés (1971) · Zbl 0384.06022 |
| [6] | A. V. Chernavsky and V. P. Leksine, Unrecognizability of manifolds, Ann. Pure Appl. Logic 141 (2006), 325-335.; Chernavsky, A. V.; Leksine, V. P., Unrecognizability of manifolds, Ann. Pure Appl. Logic, 141, 325-335 (2006) · Zbl 1115.57014 |
| [7] | T. Evans, Finitely presented loops, lattices, etc. are hopfian, J. Lond. Math. Soc. (2) 44 (1969), 551-552.; Evans, T., Finitely presented loops, lattices, etc. are hopfian, J. Lond. Math. Soc. (2), 44, 551-552 (1969) · Zbl 0177.02702 |
| [8] | G. Ewald, Combinatorial Convexity and Algebraic Geometry, Grad. Texts in Math. 168, Springer, New York, 1996.; Ewald, G., Combinatorial Convexity and Algebraic Geometry (1996) · Zbl 0869.52001 |
| [9] | A. M. W. Glass, Partially Ordered Groups, Ser. Algebra 7, World Scientific, Singapore, 1999.; Glass, A. M. W., Partially Ordered Groups (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.; Glass, A. M.; Holland, W. C., Lattice-Ordered Groups: Advances and Techniques (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.; Glass, A. M. W.; Madden, J. J., The word problem versus the isomorphism problem, J. Lond. Math. Soc. (2), 30, 53-61 (1984) · Zbl 0551.20018 |
| [12] | G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Clarendon Press, Oxford, 1960.; Hardy, G. H.; Wright, E. M., An Introduction to the Theory of Numbers (1960) · Zbl 0086.25803 |
| [13] | R. Hirshon, Some Theorems on Hopficity, Trans. Amer. Math. Soc. 141 (1969), 229-244.; Hirshon, R., Some Theorems on Hopficity, Trans. Amer. Math. Soc., 141, 229-244 (1969) · Zbl 0186.32002 |
| [14] | A. I. Kostrikin and I. R. Shafarevich, Algebra II: Noncommutative Rings Identities, Springer, New York, 2012.; Kostrikin, A. I.; Shafarevich, I. R., Algebra II: Noncommutative Rings Identities (2012) |
| [15] | A. Mijatović, Simplifying triangulations of \(S^3\), Pacific J. Math. 208 (2003), no. 2, 291-324.; Mijatović, A., Simplifying triangulations of \(S^3\), Pacific J. Math., 208, 2, 291-324 (2003) · Zbl 1071.52016 |
| [16] | D. Mundici, Simple Bratteli diagrams with a Gödel incomplete \(\text{C}^*\)-equivalence problem, Trans. Amer. Math. Soc. 356 (2003), no. 5, 1937-1955.; Mundici, D., Simple Bratteli diagrams with a Gödel incomplete \(\text{C}^*\)-equivalence problem, Trans. Amer. Math. Soc., 356, 5, 1937-1955 (2003) · Zbl 1042.46033 |
| [17] | T. Oda, Convex Bodies and Algebraic Geometry, Springer, Berlin, 1988.; Oda, T., Convex Bodies and Algebraic Geometry (1988) · Zbl 0628.52002 |
| [18] | E. Outerelo and J. M. Ruiz, Mapping Degree Theory, Grad. Stud. Math. 108, American Mathematical Society, Providence, 2009.; Outerelo, E.; Ruiz, J. M., Mapping Degree Theory (2009) · Zbl 1183.47056 |
| [19] | M. A. Shtan’ko, Markov’s theorem and algorithmically non-recognizable combinatorial manifolds, Izv. Math. 68 (2004), 207-224.; Shtan’ko, M. A., Markov’s theorem and algorithmically non-recognizable combinatorial manifolds, Izv. Math., 68, 207-224 (2004) · Zbl 1069.57013 |
| [20] | J. R. Stallings, Lectures on Polyhedral Topology, Tata Institute of Fundamental Research, Mumbay, 1967.; Stallings, J. R., Lectures on Polyhedral Topology (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.; Thompson, A., Thin position and the recognition problem for \(S^3\), Math. Res. Lett., 1, 5, 613-630 (1994) · Zbl 0849.57009 |
| [22] | V. Weispfenning, The complexity of the word problem for abelian \(\ell \)-groups, Theoret. Comput. Sci. 48 (1986), 127-132.; Weispfenning, V., The complexity of the word problem for abelian \(\ell \)-groups, Theoret. Comput. Sci., 48, 127-132 (1986) · Zbl 0633.03036 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
