On partial orderings having precalibre-${\mathrm{\aleph }}_1$ and fragments of Martin's axiom

We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-${\mathrm{\aleph }}_1$, 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-${\mathrm{\aleph }}_1$ property of a partial ordering while preserving its ccc-ness.