Completely distributive complete lattices
George N. Raney · Proceedings of the American Mathematical Society · 1952
This paper is concerned with the representation theory for completely distributive complete lattices.The results obtained are extensions of similar results of Tarski on distributive lattices and on completely distributive Boolean algebras.1The following definitions and notations are used.If £ is a family of subsets of a set, the intersection of F is denoted by Yl_F, and the union of F is denoted by 22 F-Definition 1.A family R of subsets of a set is called a complete ring of sets if for every FQR, IJ£G£ and J]£G£.If £ is a complete lattice, then every subset K of L has a meet, which is denoted by C\K, and a join, which is denoted by UK.Definition 2. Let L and £' be complete lattices.A mapping 6 of L into L' is called a complete homomorphism if for every subset K of L,