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Right orders and amalgamation for lattice-ordered groups

In: Mathematica Slovaca, vol. 61, no. 3
V. V. Bludov - Andrew M. W. Glass
Detaily:
Rok, strany: 2011, 355 - 372
Kľúčové slová:
totally ordered set, permutation group, representation, lattice-ordered group, $\ell$-permutation group, amalgamation, free product with amalgamated subgroup, right-orderable group, convex sublattice subgroups
O článku:
Let $Hi$ be a sublattice subgroup of a lattice-ordered group $Gi$ (${i=1,2}$). Suppose that $H1$ and $H2$ are isomorphic as lattice-ordered groups, say by $φ$. In general, there is no lattice-ordered group in which $G1$ and $G2$ can be embedded (as lattice-order\underline{ed} groups) so that the embeddings agree on the images of $H1$ and $H1φ$. In this article we prove that the group free product of $G1$ and $G2$ amalgamating $H1$ and $H1φ$ is right orderable and so embeddable (as a group) in a lattice-order\underline{able} group. To obtain this, we use our necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable [BLUDOV, V. V.—GLASS, A. M. W.: \textit{Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup}, Proc. London Math. Soc. (3) \textbf{99} (2009), 585–608]. We also provide new limiting examples to show that amalgamation can fail in the category of lattice-ordered groups even when the amalgamating sublattice subgroups are convex and normal ($\ell$-ideals) and solve of Problem 1.42 from [KOPYTOV, V. M.—MEDVEDEV, N. YA.: \textit{Ordered groups}. In: Selected Problems in Algebra. Collection of Works Dedicated to the Memory of N. Ya. Medvedev, Altaii State University, Barnaul, 2007, pp. 15–112 (Russian)].
Ako citovať:
ISO 690:
Bludov, V., Glass, A. 2011. Right orders and amalgamation for lattice-ordered groups. In Mathematica Slovaca, vol. 61, no.3, pp. 355-372. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0017-2

APA:
Bludov, V., Glass, A. (2011). Right orders and amalgamation for lattice-ordered groups. Mathematica Slovaca, 61(3), 355-372. 0139-9918. DOI: https://doi.org/10.2478/s12175-011-0017-2
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