  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 $\aleph_1$. In this note we show that the dual of the category of partially
ordered compact spaces and monotone continuous maps is a $\aleph_1$-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 $\aleph_1$-copresentable partially
ordered compact spaces.
