Finite retracts of Priestley spaces and sectional coproductivity

Abstract

Let Y and P be posets, let P be finite and connected, and let f : Y → P be a surjective monotone map. The map f can be naturally extended to a Priestley surjection ${\widehat{f} : \widehat{Y} \to P}$ which can turn out to be a retraction even if f is not. We characterize those maps f whose Priestley extensions ${\widehat{f}}$ are retractions.

We then use this characterization to contrast, yet again, the behavior of cyclic and acyclic posets P insofar as their appearances in Priestley spaces are concerned. We say that P is sectionally coproductive if the Priestley surjection ${f : {\coprod} X_{i} \to P}$, induced by Priestley surjections f _i : X _i → P, is a retraction only when at least one of the f _i ’s is a retraction. We then prove that P is sectionally coproductive exactly when it is acyclic.