We develop some new inequalities for the dimension of a finite poset. These inequalities are then used to bound dimension in terms of the maximum size of matchings. We prove that if the dimension of P is d and d ≥ 3, then there is a matching of size d in the comparability graph of P . There is no analogue of this result for cover graphs, as we show that there is a poset P of dimension d for whi...

A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived equivalent if D(X) and D(Y ) are equivalent as triangulated categories. We give explicit combinatorial properties of X which are invariant under derived equivalence, ...

We investigate approximating posets with projections (approximating pop's). These are triples (D; ; P) consisting of a poset (D;) and a directed set P of projections with sup P = id D. They carry a canonical uniformity and thus a topology. We relate their properties such as completeness and compactness to properties of the poset and the projection set. We show that each monotone net in D is con...

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 ...

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....

In this dissertation, we study the following two topics, i.e., the union closed set conjecture and the maximum edges cut in connected digraphs. The union-closed-set-conjecture-topic goes as follows. A finite family of finite sets is union closed if it contains the union of any two sets in it. Let XF = ∪F∈FF . A union closed family of sets is separating if for any two distinct elements in F , th...

We define posets of types B, C, and D. These encode the matrix forms certain Lie algebras which lie between upper-triangular diagonal matrices. Our primary concern is index spectral theories such type-B, D poset algebras. For an important restricted class, we develop combinatorial formulas and, in particular, characterize corresponding to Frobenius In this latter case show that spectrum binary;...

Telgársky’s conjecture states that for each k ∈ ℕ, there is a topological space Xk such in the Banach-Mazur game on Xk, player nonempty has winning (k + 1)-tactic but no k-tactic. We prove this statement consistently false. More specifically, we prove, assuming GCH+□, if strategy T3 X, then she 2-tactic. The proof uses coding argument due to Galvin, whereby X π-base with certain nice properties...

A poset game is a two-player game played over a partially ordered set (poset) in which the players alternate choosing an element of the poset, removing it and all elements greater than it. The first player unable to select an element of the poset loses. Polynomial time algorithms exist for certain restricted classes of poset games, such as the game of Nim. However, until recently the complexity...