Abstract
It is known since the late 1960's that the dual of the category of compact Hausdoroff 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.

Abstract
The recently introduced notions of guarded traced (monoidal) category and guarded (pre-)iterative monad aim at unifying di erent instances of partial iteration whilst keeping in touch with the established theory of total iteration and preserving its merits. In this paper we use these notions and the corresponding stock of results to examine di erent types of iteration for hybrid computations. As a starting point we use an available notion of hybrid monad restricted to the category of sets, and modify it in order to obtain a suitable notion of guarded iteration with guardedness interpreted as progressiveness in time – we motivate this modification by our intention to capture Zeno behaviour in an arguably general and feasible way. We illustrate our results with a simple programming language for hybrid computations and interpret it over the developed semantic foundations. © Sergey Goncharov, Julian Jakob, and Renato Neves.

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