Lattice-ordered Abelian groups and Schauder bases of unimodular fans

Manara, Corrado; Marra, Vincenzo; Mundici, Daniele

doi:10.1090/S0002-9947-06-03935-3

Lattice-ordered Abelian groups and Schauder bases of unimodular fans
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by Corrado Manara, Vincenzo Marra and Daniele Mundici PDF

Trans. Amer. Math. Soc. 359 (2007), 1593-1604 Request permission

Abstract:

Baker-Beynon duality theory yields a concrete representation of any finitely generated projective Abelian lattice-ordered group $G$ in terms of piecewise linear homogeneous functions with integer coefficients, defined over the support $|\Sigma |$ of a fan $\Sigma$. A unimodular fan $\Delta$ over $|\Sigma |$ determines a Schauder basis of $G$: its elements are the minimal positive free generators of the pointwise ordered group of $\Delta$-linear support functions. Conversely, a Schauder basis $\mathbf {H}$ of $G$ determines a unimodular fan over $|\Sigma |$: its maximal cones are the domains of linearity of the elements of $\mathbf {H}$. The main purpose of this paper is to give various representation-free characterisations of Schauder bases. The latter, jointly with the De Concini-Procesi starring technique, will be used to give novel characterisations of finitely generated projective Abelian lattice ordered groups. For instance, $G$ is finitely generated projective iff it can be presented by a purely lattice-theoretical word.

References

Kirby A. Baker, Free vector lattices, Canadian J. Math. 20 (1968), 58–66. MR 224524, DOI 10.4153/CJM-1968-008-x
W. M. Beynon, Duality theorems for finitely generated vector lattices, Proc. London Math. Soc. (3) 31 (197), 114–128. MR 376480, DOI 10.1112/plms/s3-31.1.114
W. M. Beynon, Applications of duality in the theory of finitely generated lattice-ordered abelian groups, Canadian J. Math. 29 (1977), no. 2, 243–254. MR 437420, DOI 10.4153/CJM-1977-026-4
Alain Bigard, Klaus Keimel, and Samuel Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 (French). MR 0552653, DOI 10.1007/BFb0067004
C. De Concini and C. Procesi, Complete symmetric varieties. II. Intersection theory, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 481–513. MR 803344, DOI 10.2969/aspm/00610481
Günter Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, vol. 168, Springer-Verlag, New York, 1996. MR 1418400, DOI 10.1007/978-1-4612-4044-0
A. M. W. Glass, Partially ordered groups, Series in Algebra, vol. 7, World Scientific Publishing Co., Inc., River Edge, NJ, 1999. MR 1791008, DOI 10.1142/3811
A. M. W. Glass and Vincenzo Marra, Embedding finitely generated abelian lattice-ordered groups: Higman’s theorem and a realisation of $\pi$, J. London Math. Soc. (2) 68 (2003), no. 3, 545–562. MR 2009436, DOI 10.1112/S002461070300468X
C. Manara. Relating the theory of non-Boolean partitions to the design of interpretable control systems. Ph.D. thesis, Università degli Studi di Milano, 2003.
V. Marra. Non-Boolean partitions. A mathematical investigation through lattice-ordered Abelian groups and MV algebras. Ph.D. thesis, Università degli Studi di Milano, 2002.
William S. Massey, Singular homology theory, Graduate Texts in Mathematics, vol. 70, Springer-Verlag, New York-Berlin, 1980. MR 569059, DOI 10.1007/978-1-4684-9231-6
Daniele Mundici, Farey stellar subdivisions, ultrasimplicial groups, and $K_0$ of AF $C^*$-algebras, Adv. in Math. 68 (1988), no. 1, 23–39. MR 931170, DOI 10.1016/0001-8708(88)90006-0
Daniele Mundici, Simple Bratteli diagrams with a Gödel-incomplete $C^\ast$-equivalence problem, Trans. Amer. Math. Soc. 356 (2004), no. 5, 1937–1955. MR 2031047, DOI 10.1090/S0002-9947-03-03353-1
D. Mundici. A characterization of free $n$-generated MV-algebras. Archive Mathematical Logic 45:239–247, 2006.
Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894
Giovanni Panti, A geometric proof of the completeness of the Łukasiewicz calculus, J. Symbolic Logic 60 (1995), no. 2, 563–578. MR 1335137, DOI 10.2307/2275851
Zbigniew Semadeni, Schauder bases in Banach spaces of continuous functions, Lecture Notes in Mathematics, vol. 918, Springer-Verlag, Berlin-New York, 1982. MR 653986, DOI 10.1007/BFb0094629