RELATIVELY COMPLEMENTED DISTRIBUTIVE LATTICE
Robert Xin Dong · 1991
In this paper, we have proved that for a relatively complemented distributive lattice L,there exists one and only one completion ? of L such that for every ξ∈?, there exists afamily {S_α}_(α∈△) of non-vacuous subsets of L satisfying ξ= ∨[∧f(S_α)]. Such a completion? is called the entire completion of L. We have in this paper extended the homomorphic extension theorem of a generalizedBoolean lattice to that of a relatively complemented distributive lattice and proved that the lat-tice of the congruences on a relatively complemented distributive lattice is isomorphicwith the lattice of the convex sublattices containing a fixed element and that the entire comple-tion of a relatively complemented distributive lattice L, the lattice of the completable con-gruences on L and the lattice of the completable convex sublattices of L containing a fixed ele-ment are isomorphic.