On partial orderings having precalibre-$\aleph_1$ and fragments of Martin's axiom

TL;DR

This paper introduces a new ccc property for partial orderings weaker than precalibre-\aleph_1, explores its implications for Martin's axiom, and demonstrates how to alter these properties via forcing while preserving cardinals.

Contribution

It defines a weaker ccc property, analyzes its relation to Martin's axiom, and shows how to modify partial orderings' properties through forcing without collapsing cardinals.

Findings

Martin's axiom restricted to certain partial orderings does not imply the full axiom.

A new ccc property weaker than precalibre-\aleph_1 is introduced.

Forcing can destroy precalibre-\aleph_1 while preserving ccc-ness.

Abstract

We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-$\aleph_1$, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for $\sigma$-linked partial orderings. This answers an old question of the first author about the relative strength of Martin's axiom for $\sigma$-centered partial orderings together with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprans and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-$\aleph_1$ property of a partial ordering while preserving its ccc-ness.