In combinatorial mathematics, an Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerian poset which is a lattice is an Eulerian lattice. These objects are named after Leonhard Euler. Eulerian lattices generalize face lattices of convex polytopes and much recent research has been devoted to extending known results from polyhedral combinatorics, such as various restrictions on f-vectors of convex simplicial polytopes, to this more general setting.
an Eulerianposet is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerianposet which...
mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank function ρ from P to the set N...
graded posets with unique minimal and maximal elements, preserving the property that the posets are Eulerian. The star product of two graded posets ( P ...
that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P = ( X , ≤ ) {\displaystyle P=(X,\leq )}...
distributivity laws of order theory preservation properties of functions between posets. In the following, partial orders will usually just be denoted by their...
groups act transitively on the set of flags of the polytope. Eulerianposet Graded poset Regular polytope McMullen & Schulte 2002, p. 31 McMullen & Schulte...
189.2.4. S2CID 119158401. Stanley, Richard P. (1994), "A survey of Eulerianposets", in Bisztriczky, T.; McMullen, P.; Schneider, R.; Weiss, A. Ivić...
(transitivity). A set with a partial order on it is called a partially ordered set, poset, or just ordered set if the intended meaning is clear. By checking these...
introducing Zeta polynomials, for explicitly defining Eulerianposets, developing the theory of binomial posets along with Rota and Peter Doubilet, and more....
to mean strong antichain, a subset such that there is no element of the poset smaller than two distinct elements of the antichain.) A maximal antichain...
)-\ell (\pi )}} , and thus this poset is Eulerian, meaning its Möbius function is produced by the rank function on the poset. Kazhdan–Lusztig polynomial Björner...
linear time, if such a diagram exists. In particular, if the input poset is a graded poset, it is possible to determine in linear time whether there is a...
S2CID 38115497. Ganapathy, Jayanthi (1992). "Maximal Elements and Upper Bounds in Posets". Pi Mu Epsilon Journal. 9 (7): 462–464. ISSN 0031-952X. JSTOR 24340068...
element with another element is the other element. Thus every pair in this poset has both a meet and a join and the poset can be classified as a lattice....
generated from a given poset used in place of the set of generators considered above, then one speaks of a completion of the poset. The definition of the...
in terms of category theory. Formally, given two partially ordered sets (posets) ( S , ≤ ) {\displaystyle (S,\leq )} and ( T , ⪯ ) {\displaystyle (T,\preceq...
a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially...
to this poset. Zorn's lemma states that a partial order in which every chain has an upper bound has a maximal element. A chain in this poset is a set...
associated root system, it is the number of anti-chains (or order ideals) in the poset of positive roots. The classical Catalan number C n {\displaystyle C_{n}}...
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