On partial orderings having precalibre-N₁ and fragments of Martin's axiom

We define a countable antichain condition (ccc) property for partial or- derings, weaker than precalibre-N₁, and show that Martin's axiom restricted to the class of partial orderings that have the property does not imply Martin's axiom for σ-linked partial orderings. This yields a new solution to an old question of the first author about the relative strength of Martin's axiom for σ-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-N₁ property of a partial ordering while preserving its ccc-ness.