Embedding lattices into lattices of ideals
George Grätzer, Craig Platt, George Sands · Pacific Journal of Mathematics · 1979
A lattice L is transferable iff, whenever L can be embedded in the ideal lattice of a lattice M, then L can be embedded in M.This concept was introduced by the first author in 1965 who also proved in 1966 that in a transferable lattice there are no doubly reducible elements.In fact, he proved that every lattice can be embedded in the ideal lattice of a lattice containing no doubly reducible elements.In a recent paper of the first two authors, the idea emerged that one should study transferability via classes K of lattices with the property that every lattice is embeddable in the ideal lattice of a lattice in K.This approach was used to establish that transferable lattices are semi-distributive.This investigation is carried further in this paper.Our main result shows that every lattice can be embedded in the ideal lattice of a lattice satisfying the two semi-distributive properties and two variants of Whitman's condition.