Generating the algebraic theory of C(X): the case of partially ordered compact spaces

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 an $\aleph_1$-ary quasivariety, and describe partially its algebraic theory. Based on this description, we extend these results to categories of Vietoris coalgebras and homomorphisms on ordered compact spaces. We also characterise the $\aleph_1$-copresentable partially ordered compact spaces.