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Coloring digraphs by iterated antichains

Svatopluk Poljak (1991)

Commentationes Mathematicae Universitatis Carolinae

We show that the minimum chromatic number of a product of two n -chromatic graphs is either bounded by 9, or tends to infinity. The result is obtained by the study of coloring iterated adjoints of a digraph by iterated antichains of a poset.

Compactness properties of weighted summation operators on trees-the critical case

Mikhail Lifshits, Werner Linde (2011)

Studia Mathematica

The aim of this paper is to provide upper bounds for the entropy numbers of summation operators on trees in a critical case. In a recent paper [Studia Math. 202 (2011)] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator to those of the underlying tree equipped with an appropriate metric. However, the results were left incomplete in a critical case of the entropy behavior, because this case requires much more involved techniques....

Compactness properties of weighted summation operators on trees

Mikhail Lifshits, Werner Linde (2011)

Studia Mathematica

We investigate compactness properties of weighted summation operators V α , σ as mappings from ℓ₁(T) into ℓ q ( T ) for some q ∈ (1,∞). Those operators are defined by ( V α , σ x ) ( t ) : = α ( t ) ∑ s ⪰ t σ ( s ) x ( s ) , t ∈ T, where T is a tree with partial order ⪯. Here α and σ are given weights on T. We introduce a metric d on T such that compactness properties of (T,d) imply two-sided estimates for e ₙ ( V α , σ ) , the (dyadic) entropy numbers of V α , σ . The results are applied to concrete trees, e.g. moderately increasing, biased or binary trees and to weights with α(t)σ(t)...

Complemented ordered sets

Ivan Chajda (1992)

Archivum Mathematicum

We introduce the concept of complementary elements in ordered sets. If an ordered set S is a lattice, this concept coincides with that for lattices. The connections between distributivity and the uniqueness of complements are shown and it is also shown that modular complemented ordered sets represents “geometries” which are more general than projective planes.

Congruences in ordered sets

Ivan Chajda, Václav Snášel (1998)

Mathematica Bohemica

A concept of congruence preserving upper and lower bounds in a poset P is introduced. If P is a lattice, this concept coincides with the notion of lattice congruence.

Congruences in ordered sets and LU compatible equivalences

Václav Snášel, Marek Jukl (2009)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

A concept of equivalence preserving upper and lower bounds in a poset P is introduced. If P is a lattice, this concept coincides with the notion of lattice congruence.

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