Consistent lattices and Steinitz spaces (Q1966132)

Denote by \(J(L)\) the set of all join-irreducible elements of a lattice \(L\). \(L\) is consistent if for each \(u\in J(L)\) and \(a\in L\) there holds \(a\vee u \in J([a,a\vee u])\). Geometric lattices (i.e.\ semimodular atomistic lattices) and modular lattices are consistent. It is well known that a finite lattice is isometric to the lattice of flats of a matroid iff \(L\) is geometric. An analogous result is stated for the so-called Steinitz spaces: a closure space \(S\) is a Steinitz space if for all \(A\subseteq S\) and bases \(B_1,B_2\) for \(A\), if \(b_1\in B_1\), then there is \(b_2 \in B_2\) such that \((B_1-b_1)\cup \{b_2\}\) is a basis for \(A\). Theorem: A finite lattice \(L\) is isomorphic to the lattice of all closed sets of a Steinitz space iff \(L\) is consistent.