Base Tree Property

TL;DR

This paper characterizes $\sigma$-closed partial orders of size continuum, showing they have a base tree structure and relate to regular suborders of a collapsing algebra, with applications to natural examples.

Contribution

It provides an internal and external characterization of $\sigma$-closed partial orders of size continuum, linking them to base trees and collapsing algebras.

Findings

Every $\sigma$-closed partial order of size continuum has a base tree.

$\sigma$-closed forcing notions of density continuum correspond to regular suborders of $Coll(\omega_1, 2^\omega)$.

Examples of such partial orders are analyzed.

Abstract

Building on previous work of [BPS] we investigate $\sigma$-closed partial orders of size continuum. We provide both an internal and external characterization of such partial orders by showing that (1) every $\sigma$-closed partial order of size continuum has a base tree and that (2) $\sigma$-closed forcing notions of density $\mathfrak c$ correspond exactly to regular suborders of the collapsing algebra $Coll(\omega_1, 2^\omega)$. We further study some naturally ocurring examples of such partial orders.