With a nite graph G V E we associate a partially ordered set P X P with X V E and x e in P if and only if x is an endpoint of e in G This poset is called the incidence poset of G In this paper we consider the function M p d de ned for p d as the maximum number of edges a graph G can have when it has p vertices and the dimension of its incidence poset is at most d It is easy to see that M p p as...

In recent years, researchers have shown renewed interest in combinatorial properties of posets determined by geometric properties of its order diagram and topological properties of its cover graph. In most cases, the roots for the problems being studied today can be traced back to the 1970’s, and sometimes even earlier. In this paper, we study the problem of bounding the dimension of a planar p...

We give a new order-theoretic characterization of a complete Heyting and co-Heyting algebra C. This result provides an unexpected relationship with the field of Nash equilibria, being based on the so-called Veinott ordering relation on subcomplete sublattices of C, which is crucially used in Topkis’ theorem for studying the order-theoretic stucture of Nash equilibria of supermodular games. Intr...

This work-in-progress is intended as an exposition of the background material, results, and open problems of a particular poset theoretic study of Weyl characters and semisimple Lie algebra representations begun in the late 1970’s and early 1980’s in the work of Richard P. Stanley and Robert A. Proctor.

In this paper all poset structures are classi£ed which admit the MacWilliams identity, and the MacWilliams identities for poset weight enumerators corresponding to such posets are derived. We prove that being a hierarchical poset is a necessary and suf£cient condition for a poset to admit MacWilliams identity. An explicit relation is also derived between P-weight distribution of a hierarchical ...

Let R be a non empty poset and let A, B be many sorted sets indexed by R. One can check that every many sorted relation between A and B which is os-compatible is also order-sorted. Let R be a non empty poset and let A be a many sorted set indexed by the carrier of R. An order sorted relation of A is an order sorted relation of A, A. Let S be an order sorted signature and let U1 be an order sort...

We introduce an infinitesimal Hopf algebra of planar trees, generalising the construction of the non-commutative Connes-Kreimer Hopf algebra. A nondegenerate pairing and a dual basis are defined, and a combinatorial interpretation of the pairing in terms of orders on the vertices of planar forests is given. Moreover, the coproduct and the pairing can also be described with the help of a partial...

In Harding (1996; see also Harding, 1998, 1999), a method was given to construct an orthomodular poset Fact X from the direct product decompositions of a set. As this forms the basis of our study we briefly review the pertinent facts. For a set X let Eq X denote the set of equivalence relations on X, use ◦ for relational product, 1 for the least equivalence relation on X, and ∇ for the largest....

The quadratic Tits form, introduced by P. Gabriel [1] for quivers, Yu.A. Drozd [2] for posets, and M.M. Kleiner, A.V. Roiter [3] and Yu.A. Drozd [4] for a wide class of classification problems, plays an important role in representation theory. In particular, there are many results on connections between representation types of various objects and properties of the Tits forms. The reader interes...