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

Abstract

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.