For a poset P and a distributive 〈∨, 0〉-semilattice S, a S-valued poset measure on P is a map μ : P×P → S such that μ(x, z) ≤ μ(x, y)∨μ(y, z), and x ≤ y implies that μ(x, y) = 0, for all x, y, z ∈ P . In relation with congruence lattice representation problems, we consider the problem whether such a measure can be extended to a poset measure μ : P ×P → S, for a larger poset P , such that for al...

Given an affine surjection of polytopes : P ! Q, the Generalized Baues Problem asks whether the poset of all proper polyhedral subdivisions of Q which are induced by the map has the homotopy type of a sphere. We extend earlier work of the last two authors on subdivisions of cyclic polytopes to give an affirmative answer to the problem for the natural surjections between cyclic polytopes : C(n; ...

In this paper X denotes a set. Let L be a lattice. Note that Poset(L) has l.u.b.’s and g.l.b.’s. Let L be an upper-bounded lattice. One can verify that Poset(L) is upper-bounded. Let L be a lower-bounded lattice. One can verify that Poset(L) is lower-bounded. Let L be a complete lattice. Note that Poset(L) is complete. Let X be a set. Then X is an order in X . Let X be a set. The functor 〈X ,⊆〉...

in this paper the notion of rees short exact sequence for s-posets is introduced, and we investigate the conditions for which these sequences are left or right split. unlike the case for s-acts, being right split does not imply left split. furthermore, we present equivalent conditions of a right s-poset p for the functor hom(p;-) to be exact.

A poset is (r+s)-free if it does not contain two incomparable chains of size r and s, respectively. We prove that when r and s are at least 2, the First-Fit algorithm partitions every (r + s)-free poset P into at most 8(r− 1)(s− 1)w chains, where w is the width of P . This solves an open problem of Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992–1999, 2010).

‎‎‎in this paper‎, ‎the‎ notion of injectivity with respect to order dense embeddings in ‎‎the category of $s$-posets‎, ‎posets with a monotone action of a‎ pomonoid $s$ on them‎, ‎is studied‎. ‎we give a criterion‎, ‎like the baer condition for injectivity of modules‎, ‎or skornjakov criterion for injectivity of $s$-sets‎, ‎for the order dense injectivity‎. ‎also‎, ‎we consider such injectivit...

Let (W,S) be a finite Weyl group and let w ∈ W . It is widely appreciated that the descent set D(w) = {s ∈ S | l(ws) < l(w)} determines a very large and important chapter in the study of Coxeter groups. In this paper we generalize some of those results to the situation of the Bruhat poset W J where J ⊂ S. Our main results here include the identification of a certain subset S ⊂ W J that convinci...

The Tamari lattice, thought as a poset on the set of triangulations of a convex polygon with n vertices, generalizes to the higher Stasheff-Tamari orders on the set of triangulations of a cyclic d-dimensional polytope having n vertices. This survey discusses what is known about these orders, and what one would like to know about them.

The cyclic zonotope …n; d† is the zonotope in R d generated by any n distinct vectors of the form …1; t; t 2 ;. .. ; t dÀ1 †. It is proved that the re¢nement poset of all proper zonotopal subdivisions of …n; d† which are induced by the canonical projection p: …n; d H † 3 …n; d†, in the sense of Billera and Sturmfels, is homotopy equivalent to a sphere and that any zonotopal subdivision of ...

this paper is devoted to the study of products of classes of right $s$-posets possessing one of the flatness properties and preservation of such properties under products. specifically, we characterize a pomonoid $s$ over which its nonempty products as right $s$-posets satisfy some known flatness properties. generalizing this results, we investigate products of right $s$-posets satisfying condi...