Over the past 15 years, Ramsey theoretic techniques and concepts have been applied with great success to partially ordered sets. In the last year alone, four new applications of Ramsey theory to posets have produced solutions to some challenging combinatorial problems. First, Kierstead and Trotter showed that dimension for interval orders can be characterized by a single ramsey trail by proving...

We show that a minimal triangulation of the associahedron (Stasheff polytope) of dimension n is made of (n+ 1)n−1 simplices. We construct a natural bijection with the set of parking functions from a new interpretation of parking functions in terms of shuffles. Introduction The Stasheff polytope, also known as the associahedron, is a polytope which comes naturally with a poset structure on the s...

In the first part of this article we present a realization of the m-Tamari lattice T (m) n in terms of m-tuples of Dyck paths of height n, equipped with componentwise rotation order. For that, we define the m-cover poset P〈m〉 of an arbitrary bounded poset P , and show that the smallest lattice completion of the m-cover poset of the Tamari lattice Tn is isomorphic to the m-Tamari lattice T (m) n...

The concept of signed poset has recently been introduced by V. Reiner as a generalization of that of ordinary poset (partially ordered set). We consider the problem of finding a minimum-weight ideal of a signed poset. We show a representation theorem that there exists a bijection between the set of all the ideals of a signed poset and the set of all the "reduced ideals" (defined here) of the as...

We introduce a family of posets which generate Lie poset subalgebras $$A_{n-1}=\mathfrak {sl}(n)$$ whose index can be realized topologically. In particular, if $$\mathcal {P}$$ is such toral poset, then it has simplicial realization homotopic to wedge sum d one-spheres, where the corresponding type-A algebra $$\mathfrak {g}_A(\mathcal {P})$$ . Moreover, when Frobenius, its spectrum binary, that...

We determine the Möbius function of a poset of compositions of an integer. In fact we give two proofs of this formula, one using an involution and one involving discrete Morse theory. This composition poset turns out to be intimately connected with subword order, whose Möbius function was determined by Björner. We show that using a generalization of subword order, we can obtain both Björner’s r...

(a) P is a nonempty set; (b) ≤ is a partial ordering of P; i.e., ≤ is transitive, reflexive relation in P (and so p ≤ q ∧ q ≤ p ∧ p 6= q is not forbidden); (c) for all p ∈ P, p ≤ 1. (1 need not be the only such maximum element.) We shall often write P for 〈P,≤,1〉. Let P be a poset. If p is an element of P, then an extension of p is a q ∈ P such that q ≤ p. Two elements of P are compatible if th...

We prove that every semigroup S in which the idempotents form a sub-semigroup has an E-unitary cover with the same property. Furthermore, if S is E-dense or orthodox, then its cover can be chosen with the same property. Then we describe the structure of E-unitary dense semigroups. Our results generalize Fountain's results on semigroups in which the idempotents commute, and are analogous to thos...