Color the elements of a finite set S with two colors. A collection of subsets of S is called a 2-part Sperner family if whenever for two distinct sets A and B in this collection we have A ⊂ B then B − A has elements of S of both colors. All 2-part Sperner families of maximum size were characterized in Erdös and Katona [5]. In this paper we provide a different, and quite elementary proof of the ...

For von Neumann algebras M,N not isomorphic to C C and without type I2 summands, we show that for an order-isomorphism f : AbSub M! AbSub N between the posets of abelian von Neumann subalgebras of M and N , there is a unique Jordan ⇤-isomorphism g : M! N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse also holds. This shows the Jordan structure of ...

The dimension of a poset (X, P), denoted dim (A", P), is the minimum number of linear extensions of P whose intersection is P. It follows from Dilworth's decomposition theorem that dim (X, P)& width (X, P). Hiraguchi showed that dim(X, P)s \X\/Z In this paper, A denotes an antichain of (A", P) and E the set of maximal elements. We then prove that dim {X, P) s |X A\; dim(X, P) < 1 + width (X E);...

A special case of the Generalized Baues Conjecture states that the order complex of the Baues poset of an acyclic vector connguration A (the Baues complex of A) is homotopy equivalent to a sphere of dimension equal to the corank of A minus one. The Baues poset of A is the set of proper polyhedral subdivisions of A ordered by reenement. Recently, Santos has found a counterexample in corank 317 t...

We show that the order complex of poset nonempty intervals, ordered by inclusion, is a Tchebyshev triangulation original poset. Besides studying properties this transformation, we dual type B permutohedron combinatorially equivalent to suspension intervals Boolean algebra (with minimum and maximum elements removed).

New q-analogs of Stirling numbers of the second kind(and the first kined) are derived from a poset on [2k] using Stanley’s P -partition theory [?]. We also generalize to the poset on the set [rk].