Complete lattices and the generalized Cantor theorem

Cantor's Theorem is generalized to a theorem on partially ordered sets. We shall show that every monotone mapping of a complete lattice into itself has a point of left continuity and a point of right continu- ity. From this result we derive an extension of a theorem of Gleason and Dilworth (2) which in turn can be regarded as a generalization of the classical theorem of Cantor stating that the cardinal of a set is less than the cardinal of its power-set. As a corollary it follows that if E and F are partially ordered sets then the cardinal power FE is not a homomorphic image of E unless \F\ =1. This result answers a question of F. W. Lawvere which provided the stimulus for our investigation. 1. A continuity theorem for complete lattices. If E and F are partially ordered sets then a mapping (a) = V (a) = A d>(x).