Categorical abstract algebraic logic: skywatching in semilattice systems

Abstract

Font and Moraschini established a bijective correspondence between congruences of semilattices with sectionally finite height and certain special subsets of their universes, called clouds. They provided a characterization of clouds and showed that the correspondence is given by the Leibniz operator of abstract algebraic logic. We extend the bijection to one between congruence systems on the semilattice systems of categorical abstract algebraic logic and what we call cloud families. In this context, the categorical analog of the Leibniz operator plays a similar role. In addition, we show that, even though the exact analogue of the Font–Moraschini condition fails in general, a more complex variant provides an analogous characterization of cloud families.