362 research outputs found
The size of maximal systems of brick islands
For integers and a cuboid , a brick of is a closed cuboid whose vertices
have integer coordinates. A set of bricks in is a system of brick
islands if for each pair of bricks in one contains the other or they are
disjoint. Such a system is maximal if it cannot be extended to a larger system
of brick islands. Extending the work of Lengv\'{a}rszky, we show that the
minimum size of a maximal system of brick islands in is . Also, in a cube we define the corresponding notion of a
system of cubic islands, and prove bounds on the sizes of maximal systems of
cubic islands.Comment: 12 page
CD-independent subsets in meet-distributive lattices
A subset of a finite lattice is CD-independent if the meet of any two
incomparable elements of equals 0. In 2009, Cz\'edli, Hartmann and Schmidt
proved that any two maximal CD-independent subsets of a finite distributive
lattice have the same number of elements. In this paper, we prove that if
is a finite meet-distributive lattice, then the size of every CD-independent
subset of is at most the number of atoms of plus the length of . If,
in addition, there is no three-element antichain of meet-irreducible elements,
then we give a recursive description of maximal CD-independent subsets.
Finally, to give an application of CD-independent subsets, we give a new
approach to count islands on a rectangular board.Comment: 14 pages, 4 figure
Spectra of Tukey types of ultrafilters on Boolean algebras
Extending recent investigations on the structure of Tukey types of
ultrafilters on to Boolean algebras in general, we
classify the spectra of Tukey types of ultrafilters for several classes of
Boolean algebras, including interval algebras, tree algebras, and pseudo-tree
algebras.Comment: 18 page
Capacity and coding in digital communications
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