Identities in finite partition lattices
David Sachs · Proceedings of the American Mathematical Society · 1961
In his discussion of the problem of imbedding a finite lattice into a finite partition lattice, Birkhoff [1 ] speculates that there are no nontrivial identities satisfied in every finite partition lattice. We give a simple proof of this conjecture based upon Whitman's Theorem [4]. Since a partition lattice P is a complete, meet-continuous lattice in which every element is a join of points [3], P is isomorphic to the lattice of ideals of its sublattice p' of finite-dimensional elements. We prove Birkhoff's conjecture by focusing attention on P' rather than on P directly. We denote the join and meet operations by + and *. A lattice polynomial form will be denoted byf(x1, * . . , x1, . . *, xn, * * *, Xn), where we repeat a variable m times if it appears m times in the form.