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 Algebra Logika, 2004, Volume 43, Number 3, Pages 261–290 (Mi al70)

Sublattices of Lattices of Convex Subsets of Vector Spaces

F. Wehrunga, M. V. Semenovab

a Caen University
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Let ${\mathbf{Co}}(V)$ be a lattice of convex subsets of a vector space $V$ over a totally ordered division ring ${\mathbb{F}}$. We state that every lattice $L$ can be embedded into ${\mathbf{Co}}(V)$, for some space $V$ over ${\mathbb{F}}$. Furthermore, if $L$ is finite lower bounded, then $V$ can be taken finite-dimensional; in this case $L$ also embeds into a finite lower bounded lattice of the form ${\mathbf{Co}}(V,\Omega)=\{X\cap\Omega \mid X\in {\mathbf{Co}}(V)\}$, for some finite subset $\Omega$ of $V$. This result yields, in particular, a new universal class of finite lower bounded lattices.

Keywords: lattice of convex subsets of a vector space, finite lower bounded lattice

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English version:
Algebra and Logic, 2004, 43:3, 145–161

Bibliographic databases:

UDC: 512.56
Revised: 11.02.2004

Citation: F. Wehrung, M. V. Semenova, “Sublattices of Lattices of Convex Subsets of Vector Spaces”, Algebra Logika, 43:3 (2004), 261–290; Algebra and Logic, 43:3 (2004), 145–161

Citation in format AMSBIB
\Bibitem{WehSem04} \by F.~Wehrung, M.~V.~Semenova \paper Sublattices of Lattices of Convex Subsets of Vector Spaces \jour Algebra Logika \yr 2004 \vol 43 \issue 3 \pages 261--290 \mathnet{http://mi.mathnet.ru/al70} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2084037} \zmath{https://zbmath.org/?q=an:1115.06011} \transl \jour Algebra and Logic \yr 2004 \vol 43 \issue 3 \pages 145--161 \crossref{https://doi.org/10.1023/B:ALLO.0000028929.28946.d6} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-23944466271} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Roman'kov, N. G. Khisamiev, “Constructible Matrix Groups”, Algebra and Logic, 43:5 (2004), 339–345
2. M. V. Semenova, “Lattices That are Embeddable in Suborder Lattices”, Algebra and Logic, 44:4 (2005), 270–285
3. Wehrung F, “Sublattices of complete lattices with continuity conditions”, Algebra Universalis, 53:2–3 (2005), 149–173
4. Bergman GM, “On lattices of convex sets in R-n”, Algebra Universalis, 53:2–3 (2005), 357–395
5. Adaricheva KV, “Lattices of algebraic subsets”, Algebra Universalis, 52:2–3 (2005), 167–183
6. M. V. Semenova, “Lattices Embeddable in Subsemigroup Lattices. I. Semilattices”, Algebra and Logic, 45:2 (2006), 124–133
7. M. V. Semenova, “Lattices Embeddable in Subsemigroup Lattices. II. Cancellative Semigroups”, Algebra and Logic, 45:4 (2006), 248–253
8. Semenova M.V., “On lattices embeddable into subsemigroup lattices. IV. Free semigroups”, Semigroup Forum, 74:2 (2007), 191–205
9. Adaricheva K., Wild M., “Realization of abstract convex geometries by point configurations”, European Journal of Combinatorics, 31:1 (2010), 379–400
10. M. V. Semenova, “Embedding Lattices into Derived Lattices”, Proc. Steklov Inst. Math., 278, suppl. 1 (2012), S116–S130
11. Adaricheva K., “Representing Finite Convex Geometries by Relatively Convex Sets”, Eur. J. Comb., 37:SI (2014), 68–78
12. Richter M., Rogers L.G., “Embedding Convex Geometries and a Bound on Convex Dimension”, Discrete Math., 340:5 (2017), 1059–1063
13. Adaricheva K., Pouzet M., “On Scattered Convex Geometries”, Order-J. Theory Ordered Sets Appl., 34:3 (2017), 523–550
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