162 research outputs found
Characterizing circles by a convex combinatorial property
Let be a compact convex subset of the plane , and assume
that is similar to , that is, is the
image of with respect to a similarity transformation . Kira Adaricheva and Madina Bolat have recently proved that
if is a disk and both and are included in a triangle with
vertices , , and , then there exist a and a
such that is included in the convex hull of
. Here we prove that this property
characterizes disks among compact convex subsets of the plane. Actually, we
prove even more since we replace "similar" by "isometric" (also called
"congruent"). Circles are the boundaries of disks, so our result also gives a
characterization of circles.Comment: 18 pages, 15 figure
Finite semilattices with many congruences
For an integer , let NCSL denote the set of sizes of congruence
lattices of -element semilattices. We find the four largest numbers
belonging to NCSL, provided that is large enough to ensure that
NCSL. Furthermore, we describe the -element semilattices
witnessing these numbers.Comment: 14 pages, 4 figure
Finite convex geometries of circles
Let F be a finite set of circles in the plane. We point out that the usual
convex closure restricted to F yields a convex geometry, that is, a
combinatorial structure introduced by P. H Edelman in 1980 under the name
"anti-exchange closure system". We prove that if the circles are collinear and
they are arranged in a "concave way", then they determine a convex geometry of
convex dimension at most 2, and each finite convex geometry of convex dimension
at most 2 can be represented this way. The proof uses some recent results from
Lattice Theory, and some of the auxiliary statements on lattices or convex
geometries could be of separate interest. The paper is concluded with some open
problems.Comment: 22 pages, 7 figure
Quasiplanar diagrams and slim semimodular lattices
A (Hasse) diagram of a finite partially ordered set (poset) P will be called
quasiplanar if for any two incomparable elements u and v, either v is on the
left of all maximal chains containing u, or v is on the right of all these
chains. Every planar diagram is quasiplanar, and P has a quasiplanar diagram
iff its order dimension is at most 2. A finite lattice is slim if it is
join-generated by the union of two chains. We are interested in diagrams only
up to similarity. The main result gives a bijection between the set of the
(similarity classes of) finite quasiplanar diagrams and that of the (similarity
classes of) planar diagrams of finite, slim, semimodular lattices. This
bijection allows one to describe finite posets of order dimension at most 2 by
finite, slim, semimodular lattices, and conversely. As a corollary, we obtain
that there are exactly (n-2)! quasiplanar diagrams of size n.Comment: 19 pages, 3 figure
A note on finite lattices with many congruences
By a twenty year old result of Ralph Freese, an -element lattice has
at most congruences. We prove that if has less than
congruences, then it has at most congruences. Also, we describe the
-element lattices with exactly congruences.Comment: 5 pages, 2 figure
Notes on the description of join-distributive lattices by permutations
Let L be a join-distributive lattice with length n and width(Ji L) \leq k.
There are two ways to describe L by k-1 permutations acting on an n-element
set: a combinatorial way given by P.H. Edelman and R.E. Jamison in 1985 and a
recent lattice theoretical way of the second author. We prove that these two
approaches are equivalent. Also, we characterize join-distributive lattices by
trajectories.Comment: 8 pages, 1 figur
Representing convex geometries by almost-circles
Finite convex geometries are combinatorial structures. It follows from a
recent result of M.\ Richter and L.G.\ Rogers that there is an infinite set
of planar convex polygons such that with respect to geometric
convex hulls is a locally convex geometry and every finite convex geometry can
be represented by restricting the structure of to a finite subset in a
natural way. An \emph{almost-circle of accuracy} is a
differentiable convex simple closed curve in the plane having an inscribed
circle of radius and a circumscribed circle of radius such that
the ratio is at least . % Motivated by Richter and
Rogers' result, we construct a set such that (1) contains
all points of the plane as degenerate singleton circles and all of its
non-singleton members are differentiable convex simple closed planar curves;
(2) with respect to the geometric convex hull operator is a locally
convex geometry; (3) as opposed to , is closed with respect
to non-degenerate affine transformations; and (4) for every (small) positive
and for every finite convex geometry, there are continuum
many pairwise affine-disjoint finite subsets of such that each
consists of almost-circles of accuracy and the convex geometry
in question is represented by restricting the convex hull operator to . The
affine-disjointness of and means that, in addition to , even is disjoint from for every
non-degenerate affine transformation .Comment: 18 pages, 6 figure
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