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 dimension of a poset X is the smallest positive integer t for which there exists an embedding of X in the cartesian product of t chains. R. P. Dilworth proved that the dimension of a distributive lattice v L = 2_ is the width of X. In this paper we derive an analogous result for embedding distributive lattices in the cartesian product of chains of bounded length. We prove that for each k > ...

We introduce a partial order structure on the set of interval orders of a given size, and prove that such a structure is in fact a lattice. We also provide a way to compute meet and join inside this lattice. Finally, we show that, if we restrict to series parallel interval order, what we obtain is the classical Tamari poset.

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...

It is shown that Aut(L ~ is naturally isomorphic to Aut(L) x Aut(Q) when L is a directly and exponentially indecomposable lattice, Q a non-empty connected poset, and one of the following holds: Q is arbitrary but L is a jm-lattice, Q is finitely factorable and L is complete with a join-dense subset of completely join-irreducible elements, or L is arbitrary but Q is finite. A problem of J6nsson ...

in this paper, we investigate the l-fuzzy proximities and the relationships betweenl-fuzzy topologies, l-fuzzy topogenous order and l-fuzzy uniformity. first, we show that the category of-fuzzy topological spaces can be embedded in the category of l-fuzzy quasi-proximity spaces as a coreective full subcategory. second, we show that the category of l -fuzzy proximity spaces is isomorphic to the ...

Given a partially order set (poset) $P$, and pair of families ideals $\mathcal{I}$ filters $\mathcal{F}$ in $P$ such that each $(I,F)\in \mathcal{I}\times\mathcal{F}$ has non-empty intersection, the dualization problem over is to check whether there an ideal $X$ which intersects every member does not contain any $\mathcal{I}$. Equivalently, for distributive lattice $L=L(P)$, given by poset its ...

nets, useful topological tools, used to generalize certainconcepts that may only be general enough in the context of metricspaces. in this work we introduce this concept in an $s$-poset, aposet with an action of a posemigroup $s$ on it whichis a very useful structure in computer sciences and interestingfor mathematicians, and give the the concept of $s$-net. using $s$-nets and itsconvergency we...