The Variety of Projection of a Tree-Prikry Forcing

TL;DR

This paper investigates the embedding of certain $oldsymbol{ ext{kappa}}$-distributive forcing notions into tree Prikry forcing and explores conditions under which dense open filters extend to ultrafilters, under large cardinal assumptions.

Contribution

It characterizes which $oldsymbol{ ext{kappa}}$-distributive forcings of size $oldsymbol{ ext{kappa}}$ can be embedded into tree Prikry forcing and examines the extension of dense open filters to ultrafilters.

Findings

Certain $oldsymbol{ ext{kappa}}$-distributive forcings embed into tree Prikry forcing.

Conditions under which filters extend to ultrafilters are identified.

Results depend on large cardinal assumptions.

Abstract

We study which $\kappa$-distributive forcing notions of size $\kappa$ can be embedded into tree Prikry forcing notions with $\kappa$-complete ultrafilters under various large cardinal assumptions. An alternative formulation -- can the filter of dense open subsets of a $\kappa$-distributive forcing notion of size $\kappa$ be extended to a $\kappa$-complete ultrafilter.