If a posets lacks joins of directed subsets, one can pass to its ideal completion. But doing this means also changing the setting: The universal property of ideal completion of posets suggests that it should be regarded as a functor from the category of posets with monotone maps to the category of dcpos with Scott-continuous functions as morphisms. The same applies for the quantitative version ...

The posets of dimension 2 are those posets whose minimal realizations have two elements, that is, which may be obtained as the intersection of two of their linear extensions. Gallai's decomposition of a poset allows for a simple formula to count the number of the distinct minimal realizations of the posets of dimension 2. As an easy consequence, the characterization of M. El-Zahar and of N.W. S...

In the lattice of subsets of an n-set X , every maximal Sperner family S may be partitioned into disjoint subfamilies A and B, such that the union of the upset of A with the downset of B yields the entire lattice: U(A) [ D(B) = 2 X . The result generalizes to certain posets.

Consider the posets PI = Borel(R) \ I where I is a σ-ideal σ-generated by a projective collection of closed sets. Then the PI extension is given by a single real r of an almost minimal degree: every real s ∈ V [r] is Cohen-generic over V or

A poset is (3 + 1)-free if it does not contain the disjoint union of chains of length 3 and 1 as an induced subposet. These posets play a central role in the (3 + 1)-free conjecture of Stanley and Stembridge. Lewis and Zhang have enumerated (3+1)-free posets in the graded case by decomposing them into bipartite graphs, but until now the general enumeration problem has remained open. We give a f...

The dualization problem on arbitrary posets is a crucial step in many applications in logics, databases, artificial intelligence and pattern mining. The objective of this paper is to study reductions of the dualization problem on arbitrary posets to the dualization problem on boolean lattices, for which output quasi-polynomial time algorithms exist. We introduce convex embedding and poset refle...

For a stationary set S ⊆ ω1 and a ladder system C over S, a new type of gaps called C-Hausdorff is introduced and investigated. We describe a forcing model of ZFC in which, for some stationary set S, for every ladder C over S, every gap contains a subgap that is CHausdorff. But for every ladder E over ω1 \ S there exists a gap with no subgap that is E-Hausdorff. A new type of chain condition, c...

We introduce some equivalence relations on graphs and posets and prove that they are closed under the cartesian product operation. These relations concern the edge-isoperimetric problem on graphs and the shadow minimization problems on posets. For a long time these problems have been considered quite independently. We present close connections between them. In particular we show that a number o...

In early beginnings of the past century Felix Hausdorff introduced the concept of a partially ordered set thus extending Richard Dedekind lattice theory which began in the early 1890s. Then the subject lay more or less dormant until Garrett Birkhoff, Oystein Ore and others picked it up in the 1930s. Since then, many noted mathematicians have contributed to the subject, including Garrett Birkhof...