An Extension of Dilworth's Theorem

We show that any lattice of finite length is isomorphic to the lattice of fully dependent flats of a rank-finite independence space, which is finite if the given lattice is finite. If a suitable submodular function is given on the lattice, the independence space may be so chosen that this submodular function corresponds to the rank function of the independence space under the constructed isomorphism. The isomorphism also has the property that it constitutes an embedding of the given lattice in the (geometric) lattice of flats of the independence space. If M is an independence space on a set X, a set S £ X is fully dependent if S is a union of circuits of M. An independence space is uniquely determined by specifying its set of fully dependent flats together with their ranks. We show here that any lattice of finite length is isomorphic to the lattice of fully dependent flats of a rankfinite independence space; if the lattice is finite, it is isomorphic to the lattice of fully dependent flats of a matroid. In fact, we model all such lattices using only a special class of independence spaces, namely those in which the intersection of any two fully dependent flats is fully dependent. In this case, the lattice of fully dependent flats is a sublattice of the lattice of all flats of the independence space, and so our proof includes, incidentally, a proof of Dilworth's theorem that any finite lattice can be embedded in a finite geometric lattice (see [1; page 125]). THEOREM 1. Let Lbe any lattice of finite length and a any integer-valued function defined on L such that (a) a 0 = 0 (b) a a > ab whenever a> b (c) a{avb) + o(a Ab) ^ aa+ab for all a,beL. Then there exists a lattice J§? of fully dependent flats of a rank-finite independence space M with rank function r and an isomorphism \j/ from L onto S£ such that r{\j/a) = aa for all aeL. If L is a finite lattice, then M is a matroid. Proof. The proof is divided into four steps A, B, C and D. Step A. There exists a lattice Z£ of subsets of a set X, which is finite if L is finite, such that S£ is closed under set-intersection, 0 and X belong to Jz? and there exists an isomorphism ij/ from L onto 5£. Let / be the set of all non-zero join-irreducible elements of L. For each eel, choose a new set Xe such that \Xe\ = ae-at+l, Received 28 March, 1977. [J. LONDON MATH. SOC. (2), 16 (1977), 393-396]