Standard Examples as Subposets of Posets

We prove that a poset with no induced subposet $S_k$ (for fixed $k\geq 3$) must have dimension that is sublinear in terms of the number of elements

Incidence Categories

Given a family \F of posets closed under disjoint unions and the operation of taking convex subposets, we construct a category \C_{\F} called the \emph{incidence category of \F}. This category is "nearly abelian" in the sense that all morphisms have kernels/cokernels, and possesses a symmetric monoidal structure akin to direct sum. The Ringel-Hall algebra of \C_{\F} is isomorphic to the incidence Hopf algebra of the collection \P(\F) of order ideals of posets in \F. This construction generalizes the categories introduced by K. Kremnizer and the author In the case when \F is the collection of posets coming from rooted forests or Feynman graphs

On cobweb posets and their combinatorially admissible sequences

The main purpose of this article is to pose three problems which are easy to be formulated in an elementary way. These problems which are specifically important also for the new class of partially ordered sets seem to be not yet solved.Comment: 16 pages, 9 figures, affiliated to The Internet Gian Carlo Rota Polish Seminar: 16 pages, 9 figures, affiliated to The Internet Gian Carlo Rota Polish Seminar http://ii.uwb.edu.pl/akk/sem/sem_rota.ht

Large subposets with small dimension

Dorais asked for the maximum guaranteed size of a dimension $d$ subposet of an $n$-element poset. A lower bound of order $\sqrt{n}$ was found by Goodwillie. We provide a sublinear upper bound for each $d$. For $d=2$, our bound is $n^{0.8295}$.Comment: 4 page

Shellability of generalized Dowling posets

A generalization of Dowling lattices was recently introduced by Bibby and Gadish, in a work on orbit configuration spaces. The authors left open the question as to whether these posets are shellable. In this paper we prove EL-shellability and use it to determine the homotopy type. We also show that subposets corresponding to invariant subarrangements are not shellable in general

Cobweb Posets and KoDAG Digraphs are Representing Natural Join of Relations, their diBigraphs and the Corresponding Adjacency Matrices

Natural join of $di-bigraphs$ that is directed biparted graphs and their corresponding adjacency matrices is defined and then applied to investigate the so called cobweb posets and their $Hasse$ digraphs called $KoDAGs$. $KoDAGs$ are special orderable directed acyclic graphs which are cover relation digraphs of cobweb posets introduced by the author few years ago. $KoDAGs$ appear to be distinguished family of $Ferrers$ digraphs which are natural join of a corresponding ordering chain of one direction directed cliques called $di-bicliques$. These digraphs serve to represent faithfully corresponding relations of arbitrary arity so that all relations of arbitrary arity are their subrelations. Being this $chain -way$ complete if compared with kompletne $Kuratowski$ bipartite graphs their $DAG$ denotation is accompanied with the letter $K$ in front of descriptive abbreviation $oDAG$. The way to join bipartite digraphs of binary into $multi-ary$ relations is the natural join operation either on relations or their digraph representatives. This natural join operation is denoted here by \os symbol deliberately referring to the direct sum $\oplus$ of adjacency matrices as it becomes the case for disjoint $di-bigraphs$.Comment: 19 pages, 7 figures,affiliated to The Internet Gian-Carlo Polish Seminar: http://ii.uwb.edu.pl/akk/sem/sem_rota.ht

The cylinder of a relation and generalized versions of the Nerve Theorem

We introduce the notion of cylinder of a relation in the context of posets, extending the construction of the mapping cylinder. We establish a local-to-global result for relations, generalizing Quillen's Theorem A for order preserving maps, and derive novel formulations of the classical Nerve Theorem for posets and simplicial complexes, suitable for covers with not necessarily contractible intersections.Comment: 9 pages, 1 figur

Fibonomial cumulative connection constants

In this note we present examples of cumulative connection constants included new fibonomial ones. All examples posses combinatorial interpretation.Comment: affiliated to The Internet Gian-Carlo Polish Seminar: http://ii.uwb.edu.pl/akk/sem/sem_rota.ht

Topology of eigenspace posets for imprimitive reflection groups

This paper studies the poset of eigenspaces of elements of an imprimitive unitary reflection group, for a fixed eigenvalue, ordered by the reverse of inclusion. The study of this poset is suggested by the eigenspace theory of Springer and Lehrer. The posets are shown to be isomorphic to certain subposets of Dowling lattices (the `d-divisible, k-evenly coloured Dowling lattices'). This enables us to prove that these posets are Cohen-Macaulay, and to determine the dimension of their top homology.Comment: 25 page

Diversity as Width

It is argued that if the population of options is a finite poset, diversity comparisons may be conveniently based on widths i.e. on the respective maximum numbers of pairwise incomparable options included in the relevant subposets. The width-ranking and the undominated width-ranking are introduced and characterized