We connect Priestley duality for distributive lattices and its generalization to distributive meet-semilattices to Hofmann-Mislove-Stralka duality for semilattices.Among other things, this involves consideration of various morphisms between algebraic frames.We also show how Stone duality for boolean algebras and generalized boolean algebras fits as a particular case of the general picture we develop.
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Bezhanishvili et al. (2024) studied this question.
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