Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form “is x < y?”. This application motivates looking for stronger notions of dimensi...

We present a method of computing the generating function fP (x) of P -partitions of a poset P . The idea is to introduce two kinds of transformations on posets and compute fP (x) by recursively applying these transformations. As an application, we consider the partially ordinal sum Pn of n copies of a given poset, which generalizes both the direct sum and the ordinal sum. We show that the seque...

Abstract It is elementary and well known that if an element x of a bounded modular lattice L $ \mathbf L has complement in then relative every interval [ , b ] containing . We show the relatively strong assumption modularity can be replaced by weaker one formulated language so-call...

We study r-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite pro...

We investigate avoidance in (2+2)-free partially ordered sets, posets that do not contain any induced subposet isomorphic to the union of two disjoint chains of length two. In particular, we are interested in enumerating the number of partially ordered sets of size n avoiding both 2+2 and some other poset α. For any α of size 3, the results are already well-known. However, out of the 15 such α ...

this work is devoted to give a charcaterization of a pomonoid $s$such that all cyclic $s$-posets are regular injective.

the notion of a $d$-poset was introduced in a connection withquantum mechanical models. in this paper, we introduce theconditional expectation of  random variables on thek^{o}pka's $d$-poset and prove the basic properties ofconditional expectation on this  structure.

We prove analogues for sub-posets of the Dowling lattices of the results of Calderbank, Hanlon, and Robinson on homology of sub-posets of the partition lattices. The technical tool used is the wreath product analogue of the tensor species of Joyal.

To each labeled rooted tree is associated a hyperplane arrangement, which is free with exponents given by the depths of the vertices of this tree. The intersection lattices of these arrangements are described through posets of forests. These posets are used to define coalgebras, whose dual algebras are shown to have a simple presentation by generators and relations.