Splitting Families and Complete Separability

Splitting Families and Complete Separability

The purpose of this short note is to answer a question posed by the second and third authors in [5] and to use this to solve a problem of Shelah [6]. We say that two infinite subsets a and b of ω are almost disjoint or a.d. if a∩ b is finite. We say that a family A of infinite subsets of ω is almost disjoint or a.d. if its members are pairwise almost disjoint. A Maximal Almost Disjoint family, ...

Complete Reducibility and Separability

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre’s concept of Gcomplete reducibility for subgroups of G. A separability hypothesis appears in many general th...

Mad families, splitting families and large continuum

Let κ < λ be regular uncountable cardinals. Using a finite support iteration of ccc posets we obtain the consistency of b = a = κ < s = λ. If μ is a measurable cardinal and μ < κ < λ, then using similar techniques we obtain the consistency of b = κ < a = s = λ.

Complete family of separability criteria

We introduce a family of separability criteria that are based on the existence of extensions of a bipartite quantum state r to a larger number of parties satisfying certain symmetry properties. It can be easily shown that all separable states have the required extensions, so the nonexistence of such an extension for a particular state implies that the state is entangled. One of the main advanta...

Analytic countably splitting families