On lattices of convex sets in R^n
Abstract
Properties of several sorts of lattices of convex subsets of R^n are examined. The lattice of convex sets containing the origin turns out, for n>1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of R^n and the lattice of all convex subsets of R^{n-1}. The lattices of arbitrary, of open bounded, and of compact convex sets in R^n all satisfy the same identities, but the last of these is join-semidistributive, while for n>1 the first two are not. The lattice of relatively convex subsets of a fixed set S \subseteq R^n satisfies some, but in general not all of the identities of the lattice of ``genuine'' convex subsets of R^n.
Cite
@article{arxiv.math/0409288,
title = {On lattices of convex sets in R^n},
author = {George M. Bergman},
journal= {arXiv preprint arXiv:math/0409288},
year = {2007}
}
Comments
35 pages, to appear in Algebra Universalis, Ivan Rival memorial issue. See also http://math.berkeley.edu/~gbergman/papers