Vol. 85, No. 1, 1979

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Embedding lattices into lattices of ideals

George Grätzer, Craig Robert Platt and George William Sands

Vol. 85 (1979), No. 1, 65–75
Abstract

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 Iattice 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.

Mathematical Subject Classification 2000
Primary: 06B10
Milestones
Received: 3 May 1978
Revised: 29 December 1978
Published: 1 November 1979
Authors
George Grätzer
Craig Robert Platt
George William Sands