It is known since the late 1960's that the dual of the category of compact
Hausdorff spaces and continuous maps is a variety -- not finitary, but bounded
by ℵ1. In this note we show that the dual of the category of partially
ordered compact spaces and monotone continuous maps is a ℵ1-ary
quasivariety, and describe partially its algebraic theory. Based on this
description, we extend these results to categories of Vietoris coalgebras and
homomorphisms. We also characterise the ℵ1-copresentable partially
ordered compact spaces