We consider the shadow minimization problem (SMP) for cartesian powers Pn of a Macaulay poset P . Our main result is a local-global principle with respect to the lexicographic order Ln. Namely, we show that under certain conditions the shadow of any initial segment of the order Ln for n ≥ 3 is minimal iff it is so for n = 2. These conditions include such poset properties as additivity , shadow ...

Applying a classical theorem of Smith, we show that the poset property of being Gorenstein∗ over Z2 is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. ...

If S, T are stationary subsets of a regular uncountable cardinal κ, we say that S reflects fully in T , S < T , if for almost all α ∈ T (except a nonstationary set) S ∩ α is stationary in α. This relation is known to be a well-founded partial ordering. We say that a given poset P is realized by the reflection ordering if there is a maximal antichain 〈Xp; p ∈ P 〉 of stationary subsets of Reg(κ) ...

We show that the poset of regions (with respect to a canonical base region) of a supersolvable hyperplane arrangement is a congruence normal lattice. Specifically, the poset of regions of a supersolvable arrangement of rank k is obtained via a sequence of doublings from the poset of regions of a supersolvable arrangement of rank k − 1. An explicit description of the doublings leads to a proof t...

A partial order or poset   , P X   on a (finite) base set X determines the set   P  of linear extensions of P . The problem of computing, for a poset P , the cardinality of   P  is #P-complete. A set   1 2 , , , k P P P  of posets on X covers the set of linear orders that is the union of the   i P  . Given linear orders 1 2 , , , m L L L  on X , the Poset Cover problem is to de...

A recent result of Balandraud shows that for every subset S of an abelian group G there exists a non trivial subgroup H such that |TS| ≤ |T |+ |S| − 2 holds only if H ⊂ Stab(TS). Notice that Kneser’s Theorem only gives {0} 6= Stab(TS). This strong form of Kneser’s theorem follows from some nice properties of a certain poset investigated by Balandraud. We consider an analogous poset for nonabeli...

We study the computational problem of checking whether a quantified conjunctive query (a first-order sentence built using only conjunction as Boolean connective) is true in a finite poset (a reflexive, antisymmetric, and transitive directed graph). We prove that the problem is already NP-hard on a certain fixed poset, and investigate structural properties of posets yielding fixedparameter tract...

In an Appalachian Set Theory Workshop [3], Gitik presented some of the details of a simplified version of the poset from his original paper. The discussion there motivates the definition of the forcing by modifying a poset which requires a stronger large cardinal assumption which is sometimes called the long extender forcing. However, the discussion recorded in the Appalachian Set Theory (AST) ...