Kühr, Jan; Mundici, Daniele From free abelian groups to free abelian \(\ell \)-groups. (English) Zbl 1265.06055 Math. Slovaca 61, No. 3, 439-450 (2011). Using the Baer-Specker theorem, A. Glass and V. Marra proved that the underlying group of any finitely generated abelian lattice-ordered group is free. The authors of the present paper give an elementary geometric proof of this theorem without using the Baer-Specker theorem. Moreover, they prove the converse statement: for each natural number \(n = 2, 3,\dots \), the free abelian group \(F\) on countably many free generators can be equipped with a distinguished monoid \(P_n\) making \((F,P_n)\) a free \(n\)-generated abelian lattice-ordered group. Reviewer: Ivan Chajda (Olomouc) Cited in 1 Document MSC: 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 20K99 Abelian groups 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Keywords:free abelian group; free abelian lattice-ordered group; fan; regular stellas subdivision; piecewise linear function; Schauder hat PDF BibTeX XML Cite \textit{J. Kühr} and \textit{D. Mundici}, Math. Slovaca 61, No. 3, 439--450 (2011; Zbl 1265.06055) Full Text: DOI References: [1] BAKER, K. A.: Free vector lattices, Canad. J. Math. 20 (1968), 58–66. · Zbl 0157.43401 · doi:10.4153/CJM-1968-008-x [2] BEYNON, M.: Applications of duality in the theory of finitely generated lattice-ordered abelian groups, Canad. J. Math. 29 (1977), 243–254. · Zbl 0361.06017 · doi:10.4153/CJM-1977-026-4 [3] BIRKHOFF, G.: Lattice Theory (3rd ed.). Amer. Math. Soc. Colloq. Publ. 25, Amer. Math. Soc., Providence, RI, 1967. [4] DE CONCINI, C.– PROCESI, C.: Complete symmetric varieties. II. Intersection theory. In: Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), North-Holland, Amsterdam, 1985, pp. 481–513. [5] EWALD, G.: Combinatorial Convexity and Algebraic Geometry. Grad. Texts in Math. 168, Springer, Berlin, 1996. [6] FUCHS, L.: Infinite Abelian Groups, Vol. I, Academic Press, New York, 1970. · Zbl 0209.05503 [7] GLASS, A. M. W.– MARRA, V.: The underlying group of any finitely generated abelian lattice-ordered group is free, Algebra Universalis 56 (2007), 467–468. · Zbl 1122.06014 · doi:10.1007/s00012-007-2015-3 [8] MUNDICI, D.: Revisiting the free 2-generator abelian -group, J. Pure Appl. Algebra 208 (2007), 549–554. · Zbl 1118.06010 · doi:10.1016/j.jpaa.2006.01.015 [9] ODA, T.: Convex Bodies and Algebraic Geometry, Springer, Berlin, 1988. · Zbl 0628.52002 [10] WEINBERG, E. C. Free lattice-ordered abelian groups, Math. Ann. 151 (1963), 187–199. · Zbl 0114.25801 · doi:10.1007/BF01398232 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.