Imbedding partly ordered sets into infinitely distributive complete lattices
Nenosuke Funayama · Tohoku Mathematical Journal · 1956
A partly ordered set P can be imbedded into complete lattices in various ways.The first fundamental theorem (Theorem 2) asserts that if P is imbedded in a complete lattice L J-densely and M-isomorphically (see the definitions (β) and ( 7) in § 1) then L is completely isomorphic to the lattice completed by some imbedding operator on P. Imbedding operators on lattices have been discussed by several authors (see reference in [2]).Our definition of imbedding operators on partly ordered sets is a generalization of that on lattices.The second fundamental theorem (Theorem 3) gives a necessary and sufficient condition for the infinite distributivity of the lattice completed by an imbedding operator, which is a generalization of Dilworth and McLaughlin's theorem ([2] Theorem 3).Then we introduce a weak imbedding operator on partly ordered set, and give a necessary and sufficient condition for the infinite distributivity of the lattice completed by the induced imbedding operator.Using these theorems we obtain some theorems.Among them the followings are typical: Any partly ordered set can be imbedded in an infinitely distributive complete lattice preserving all gib.and all distributive lub.Any infinitely distributive (non-complete) lattice can be imbedded in an infinitely distributive complete lattice preserving all lub.and all gib.(it was noted in [2] that the normal completion does not give the answer).Any upper continuous lattice can be imbedded in an infinitely distributive complete lattice preserving all gib.and all upper continuous limits.In this paper U (Π) is used for lub.(gib.) in partly ordered sets.While V(Λ) is used for set union (intersection).The other notations are the same as in {1].