1,107 research outputs found
Convex sublattices of a lattice and a fixed point property
The collection CL(T) of nonempty convex sublattices of a lattice T ordered by bi-domination is a lattice. We say that T has the fixed point property for convex sublattices (CLFPP for short) if every order preserving map f from T to CL(T) has a fixed point, that is x > f(x) for some x > T. We examine which lattices may have CLFPP. We introduce the selection property for convex sublattices (CLSP); we observe that a complete lattice with CLSP must have CLFPP, and that this property implies that CL(T) is complete. We show that for a lattice T, the fact that CL(T) is complete is equivalent to the fact that T is complete and the lattice of all subsets of a countable set, ordered by containment, is not order embeddable into T. We show that for the lattice T = I(P) of initial segments of a poset P, the implications above are equivalences and that these properties are equivalent to the fact that P has no infinite antichain. A crucial part of this proof is a straightforward application of a wonderful Hausdor type result due to Abraham, Bonnet, Cummings, Dzamondja and Thompson [2010]
Helly numbers of Algebraic Subsets of
We study -convex sets, which are the geometric objects obtained as the
intersection of the usual convex sets in with a proper subset
. We contribute new results about their -Helly
numbers. We extend prior work for , , and ; we give sharp bounds on the -Helly numbers in
several new cases. We considered the situation for low-dimensional and for
sets that have some algebraic structure, in particular when is an
arbitrary subgroup of or when is the difference between a
lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz
method we obtain colorful versions of many monochromatic Helly-type results,
including several colorful versions of our own results.Comment: 13 pages, 3 figures. This paper is a revised version of what was
originally the first half of arXiv:1504.00076v
On lattices of convex sets in R^n
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.Comment: 35 pages, to appear in Algebra Universalis, Ivan Rival memorial
issue. See also http://math.berkeley.edu/~gbergman/paper
Lattice-like operations and isotone projection sets
By using some lattice-like operations which constitute extensions of ones
introduced by M. S. Gowda, R. Sznajder and J. Tao for self-dual cones, a new
perspective is gained on the subject of isotonicity of the metric projection
onto the closed convex sets. The results of this paper are wide range
generalizations of some results of the authors obtained for self-dual cones.
The aim of the subsequent investigations is to put into evidence some closed
convex sets for which the metric projection is isotonic with respect the order
relation which give rise to the above mentioned lattice-like operations. The
topic is related to variational inequalities where the isotonicity of the
metric projection is an important technical tool. For Euclidean sublattices
this approach was considered by G. Isac and respectively by H. Nishimura and E.
A. Ok.Comment: Proofs of Theorem 1 and Corollary 4 have been corrected. arXiv admin
note: substantial text overlap with arXiv:1210.232
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