Dilworth's Decomposition Theorem for Posets

Decomposition theorem for the cd-index of Gorenstein* posets

We prove a decomposition theorem for the cd-index of a Gorenstein* poset analogous to the decomposition theorem for the intersection cohomology of a toric variety. From this we settle a conjecture of Stanley that the cd-index of Gorenstein* lattices is minimized on Boolean algebras.

Fully Mechanized Proofs of Dilworths Theorem and Mirskys Theorem

We present two fully mechanized proofs of Dilworth’s and Mirsky’s theorems in the Coq proof assistant. Dilworth’s Theorem states that in any finite partially ordered set (poset), the size of a smallest chain cover and a largest antichain are the same. Mirsky’s Theorem is a dual of Dilworth’s Theorem. We formalize the proofs by Perles [2] (for Dilworth’s Theorem) and by Mirsky [5] (for the dual ...

On Quillen's Theorem A for posets

A theorem of McCord of 1966 and Quillen’s Theorem A of 1973 provide sufficient conditions for a map between two posets to be a homotopy equivalence at the level of complexes. We give an alternative elementary proof of this result and we deduce also a stronger statement: under the hypotheses of the theorem, the map is not only a homotopy equivalence but a simple homotopy equivalence. This leads ...

The Hofmann-Mislove Theorem for general posets

In this paper we attempt to find and investigate the most general class of posets which satisfy a properly generalized version of the Hofmann-Mislove theorem. For that purpose, we generalize and study some notions (like compactness, the Scott topology, Scott open filters, prime elements, the spectrum etc.), and adjust them for use in general posets. Then we characterize the posets satisfying th...

Semiprime Ideals and Separation Theorem in Posets