The \(\mathcal A r\)-free products of archimedean \(l\)-groups. (English) Zbl 0952.06025

All groups considered in the paper are abelian. Let \(U\) be a class of \(l\)-groups and \(\{G_{\lambda };\lambda \in \Lambda \}\subseteq U\). The \(U\)-free product of \(G_{\lambda }\) is an \(l\)-group \(G\in U\) together with a family of injective \(l\)-homomorphism \(\alpha _{\lambda } : G_{\lambda } \rightarrow G\) such that (i) \(\bigcup _{\lambda \in \Lambda } \alpha _{\lambda } (G_{\lambda })\) generates \(G\) as an \(l\)-group; (ii) if \(H\in U\) and \(\{\beta _{\lambda }:G_{\lambda } \rightarrow H;\lambda \in \Lambda \}\) is a family of \(l\)-homomorphisms, then there exists a (necessarily) unique \(l\)-homomorphism \(\gamma : G \rightarrow H\) satisfying \(\beta _{\lambda } = \gamma \alpha _{\lambda }\) for all \(\lambda \in \Lambda \). The class of all archimedean \(l\)-groups is denoted by \(\mathcal A r\). Two descriptions of \(\mathcal A r\)-free products are given in 2.2 and 2.3. The author then studies a weak form of the subalgebra property.
Theorem 4.1: \(\mathcal A r\)-free products satisfy the weak subalgebra property.
Reviewer: L.Beran (Praha)


06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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[1] M. Anderson, T. Feil: Lattice-Ordered Groups (Introduction). D. Reidel Publishing Company, 1988. · Zbl 0636.06008
[2] S.J. Bernau: Free abelian lattice groups. Math. Ann. 180 (1969), 48-59. · Zbl 0157.36801 · doi:10.1007/BF01350085
[3] P. Conrad: Lattice-Ordered Groups. Tulane Lecture Notes, Tulane University, 1970. · Zbl 0258.06011
[4] A.M.W. Class, W.C. Holland: Lattice-Ordered Groups (Advances and Techniques). Kluwer Academic Publishers, 1989. · Zbl 0705.06001
[5] G. Gratzer: Universal Algebra, 2 nd ed. Springer-Verlag, New York, 1979.
[6] W.C. Holland, E. Scringer: Free products of lattice-ordered groups. Algebra Universalis 2 (1972), 247-254. · Zbl 0265.06018 · doi:10.1007/BF02945034
[7] J. Martinez: Free products in varieties of lattice ordered groups. Czechoslovak Math. J. 22(97) (1972), 535-553. · Zbl 0247.06022
[8] J. Martinez: Free products of abelian \(l\)-groups. Czechoslovak Math. J. 23(98) (1973), 349-361. · Zbl 0298.06023
[9] J. Martinez (ed): Ordered Algebraic Structure, 11-49. Kluwer Academic Publishers, 1989.
[10] W.B. Powell, C. Tsinakis: Free products of abelian \(l\)-groups are cardinally indecomposable. Proc. Amer. Math. Soc. 86 (1982), 385-390. · Zbl 0516.06012 · doi:10.2307/2044432
[11] W.B. Powell, C. Tsinakis: Free products in the class of abelian \(l\)-groups. Pacific J. Math. 104 (1983), 429-442. · Zbl 0477.06014 · doi:10.2140/pjm.1983.104.429
[12] W.B. Powell, C. Tsinakis: Free products of lattice ordered groups. Algebra Universalis 18 (1984), 178-198. · Zbl 0545.06007 · doi:10.1007/BF01198527
[13] W.B. Powell, C. Tsinakis: Disjointness conditions for free products of \(l\)-groups. Archiv. Math. 46 (1986), 491-498. · Zbl 0578.06012 · doi:10.1007/BF01195016
[14] Dao-Rong Ton: Note on free abelian lattice groups (to appear). · Zbl 0654.06018
[15] Dao-Rong Ton: Free archimedean \(l\)-groups. Mathematica Scandinavica 74 (1994), 34-48. · Zbl 0819.06017
[16] W.C. Weinberg: Free lattice-ordered abelian lattice groups. Math. Ann. 15 (1963), 187-199. · Zbl 0114.25801 · doi:10.1007/BF01398232
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