×

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)

MSC:

06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[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
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.