Minimizing the regularity of maximal regular antichains of 2- and 3-sets

Let $n\geqslant 3$ be a natural number. We study the problem to find the smallest $r$ such that there is a family $\mathcal{A}$ of 2-subsets and 3-subsets of $[n]=\{1,2,...,n\}$ with the following properties: (1) $\mathcal{A}$ is an antichain, i.e. no member of $\mathcal A$ is a subset of any other member of $\mathcal A$, (2) $\mathcal A$ is maximal, i.e. for every $X\in 2^{[n]}\setminus\mathcal A$ there is an $A\in\mathcal A$ with $X\subseteq A$ or $A\subseteq X$, and (3) $\mathcal A$ is $r$-regular, i.e. every point $x\in[n]$ is contained in exactly $r$ members of $\mathcal A$. We prove lower bounds on $r$, and we describe constructions for regular maximal antichains with small regularity.Comment: 7 pages, updated reference

Short antichains in root systems, semi-Catalan arrangements, and B-stable subspaces

Let \be be a Borel subalgebra of a complex simple Lie algebra \g. An ideal of \be is called ad-nilpotent, if it is contained in [\be,\be]. The generators of an ad-nilpotent ideal give rise to an antichain in the poset of positive roots, and the whole theory can be expressed in a combinatorial fashion, in terms of antichains. The aim of this paper is to present a refinement of the enumerative theory of ad-nilpotent ideals for the case in which \g has roots of different length. An antichain is called short, if it consists of short roots. We obtain, for short antichains, analogues of all results known for the usual antichains.Comment: LaTeX2e, 20 page

On infinite-finite duality pairs of directed graphs

The (A,D) duality pairs play crucial role in the theory of general relational structures and in the Constraint Satisfaction Problem. The case where both classes are finite is fully characterized. The case when both side are infinite seems to be very complex. It is also known that no finite-infinite duality pair is possible if we make the additional restriction that both classes are antichains. In this paper (which is the first one of a series) we start the detailed study of the infinite-finite case. Here we concentrate on directed graphs. We prove some elementary properties of the infinite-finite duality pairs, including lower and upper bounds on the size of D, and show that the elements of A must be equivalent to forests if A is an antichain. Then we construct instructive examples, where the elements of A are paths or trees. Note that the existence of infinite-finite antichain dualities was not previously known

Ad-nilpotent ideals and The Shi arrangement

We extend the Shi bijection from the Borel subalgebra case to parabolic subalgebras. In the process, the $I$-deleted Shi arrangement $\texttt{Shi}(I)$ naturally emerges. This arrangement interpolates between the Coxeter arrangement $\texttt{Cox}$ and the Shi arrangement $\texttt{Shi}$, and breaks the symmetry of $\texttt{Shi}$ in a certain symmetrical way. Among other things, we determine the characteristic polynomial $\chi(\texttt{Shi}(I), t)$ of $\texttt{Shi}(I)$ explicitly for $A_{n-1}$ and $C_n$. More generally, let $\texttt{Shi}(G)$ be an arbitrary arrangement between $\texttt{Cox}$ and $\texttt{Shi}$. Armstrong and Rhoades recently gave a formula for $\chi(\texttt{Shi}(G), t)$ for $A_{n-1}$. Inspired by their result, we obtain formulae for $\chi(\texttt{Shi}(G), t)$ for $B_n$, $C_n$ and $D_n$.Comment: The third version, quasi-antichains are shown to be in bijection with elements of L(Cox). arXiv admin note: text overlap with arXiv:1009.1655 by other author

On bisequentiality and spaces of strictly decreasing functions on trees

We present a characterization of spaces of strictly decreasing functions on trees in terms of bisequentiality. This characterization answers Questions 6.1 and 6.2 of "A filter on a collection of finite sets and Eberlein compacta" by T. Cie\'sla. Moreover we study the relation between these spaces and the classes of Corson, Eberlein and uniform Eberlein compacta.Comment: 9 page

Finite semilattices with many congruences

For an integer $n\geq 2$, let NCSL$(n)$ denote the set of sizes of congruence lattices of $n$-element semilattices. We find the four largest numbers belonging to NCSL$(n)$, provided that $n$ is large enough to ensure that $|$NCSL$(n)|\geq 4$. Furthermore, we describe the $n$-element semilattices witnessing these numbers.Comment: 14 pages, 4 figure