The isomorphism theorem in compactly generated lattices
Peter Crawley · Bulletin of the American Mathematical Society · 1959
I t is a well known property of a modular lattice L tha t if a, bCzL then the quotient sublattices aKJb/a and b/aC\b are isomorphic. Morgan Ward [3] has proved that if the ascending or descending chain condition holds, then this property is equivalent to modularity. If the chain conditions are dropped, however, then there are simple examples of nonmodular lattices L for which a^Jb/a and b/aC\b are isomorphic for all a> b(~L. We shall show here that the isomorphism of all transposed quotients does in general characterize modularity, provided this condition is applied to the ideals of the lattice. More precisely, this result is the following. If L is an arbitrary lattice, then L is modular if and only if f or every pair of ideals A> B of L, the quotients of ideals A^JB/A and B/AC\B are isomorphic. Actually this result is a corollary of a more general theorem on compactly generated lattices. An element c in a complete lattice L is compact if for every subset SCZL with c ^ U S there exists a finite subset S'QS such that c^US'. A lattice L is said to be compactly generated if L is complete and every element of L is a join of compact elements.