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

This article is cited in 13 scientific papers (total in 13 papers)

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
Received: 23.09.2002
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
\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
\jour Algebra and Logic
\yr 2004
\vol 43
\issue 3
\pages 145--161

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