Closed and Open Maps for Partial Frames

Abstract This paper concerns the notions of closed and open maps in setting partial frames, which, contrast to full do not necessarily have all joins. Examples these include bounded distributive lattices, $$\sigma $$ σ - $$\kappa κ -frames frames. We define using geometrically intuitively appealing conditions involving preservation closed, respectively open, congruences under certain maps. then characterize them terms algebraic identities adjoints. note that frame need neither right nor left adjoints whereas course always The embedding a either its free or congruence has proved illuminating useful. consider which embeddings are skeletal. look at reflection functors providing frame. Points arise naturally construction spectrum functor for frames spaces. They may be viewed as from given 2-chain kinds filters; former description we points. Any point extends on frame; closedness openness these.