Adjointness Aspects of the Down-Set Functor

Abstract

The down-set construction, when applied to the category of Boolean frames, can be viewed as a functor into the category of frames with frame homomorphisms subject to various conditions akin to openness. We prove that it has a right adjoint, which is then given by Booleanization, exactly for near openness and one other, closely related property; a similar result is obtained for the finitary case of pseudocomplemented distributive lattices. In addition, we present a characterization of the frames which are down-set frames of Boolean frames.