Let SN(P) be the poset obtained by adding a dummy vertex on each diagonal edge of the N’s of a finite poset P. We show that SN(SN(P)) is N-free. It follows that this poset is the smallest N-free barycentric subdivision of the diagram of P, poset whose existence was proved by P.A. Grillet. This is also the poset obtained by the algorithm starting with P0 := P and consisting at step m of adding a...

Proof. We denote V -[p] = { 1 , . . . , p}, A(G) is the set of acyclic orientations of G and a(G) = IA(G)I is their number. An n-coloring of G, c: V---> [n] induces an acyclic orientation DceA(G) as follows: If [x,y]eE is an edge, where c(x) > c(y) then in Dc this edge is oriented from x to y. Every acyclic orientation D ~ A(G) defines a partial order on V, which we denote by i>o. If D e A(G), ...

Given a finite irreducible Coxeter group W , a positive integer d, and types T1, T2, . . . , Td (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions c = σ1σ2 · · ·σd of a Coxeter element c of W , such that σi is a Coxeter element in a subgroup of type Ti in W , i = 1, 2, . . . , d, and such that the factorisation is “minimal” in the sense that t...

Let V be an n-dimensional vector space over a finite field Fq and P = {1, 2, . . . , n} a poset. We consider on V the poset-metric dP . In this paper, we give a complete description of groups of linear isometries of the metric space (V, dP ), for any poset-metric dP . We show that a linear isometry induces an automorphism of order in poset P , and consequently we show the existence of a pair of...

We generalize the notion of the rank-generating function of a graded poset. Namely, by enumerating different chains in a poset, we can assign a quasi-symmetric function to the poset. This map is a Hopf algebra homomorphism between the reduced incidence Hopf algebra of posets and the Hopf algebra of quasi-symmetric functions. This work implies that the zeta polynomial of a poset may be viewed in...

We study properties of a poset generating a Cohen-Macaulay algebra with straightening laws (ASL for short). We show that if a poset P generates a Cohen-Macaulay ASL, then P is pure and, if P is moreover Buchsbaum, then P is Cohen-Macaulay. Some results concerning a Rees algebra of an ASL defined by a straightening closed ideal are also established. And it is shown that if P is a Cohen-Macaulay ...

We outline a method for practical use of an interactive system (APL) to compute the homology of finite partially ordered sets. 1. Prerequisites. All partially ordered sets (posets) are assumed finite. Given a poset , we say that b covers a if b > a and a g c ^ h implies a = c or b = c. Since we deal with finite posets, the order relation can be obtained as the reflexive, transitive clos...

In this paper, we investigate the notion of partition of a finite partially ordered set (poset, for short). We will define three different notions of partition of a poset, namely, monotone, regular, and open partition. For each of these notions we will find three equivalent definitions, that will be shown to be equivalent. We start by defining partitions of a poset in terms of fibres of some su...

A poset has the non-messing-up property if it has two covering sets of disjoint saturated chains so that for any labeling of the poset, sorting the labels along one set of chains and then sorting the labels along the other set yields a linear extension of the poset. The linear extension yielded by thus twice sorting a labeled nonmessing-up poset may be independent of which sort was performed fi...

The Bruhat order gives a poset structure to any Coxeter group. The ideal of elements in this poset having boolean principal order ideals forms a simplicial poset. This simplicial poset defines the boolean complex for the group. In a Coxeter system of rank n, we show that the boolean complex is homotopy equivalent to a wedge of (n− 1)-dimensional spheres. The number of these spheres is the boole...