Lattices with Large Minimal Extensions

This paper characterizes those finite lattices which are a maximal sublattice of an infinite lattice. There are 145 minimal lattices with this property, and a finite lattice has an infinite minimal extension if and only if it contains one of these 145 as a sublattice. In [12], I. Rival showed that if L is a maximal sublattice of a distributive lattice K with |K| > 2, then |K| ≤ (3/2)|L|. In [1] the authors took up the question for more general varieties. In particular a 14 element lattice was constructed which is a maximal sublattice of an infinite lattice. The question of how small such a lattice could be was raised. In this paper we will use the term big lattice to mean a finite lattice which is a maximal sublattice in an infinite lattice, and small lattice for a finite lattice which is not big. Our main result provides an algorithm for determining whether a given finite lattice is big. This allows us to give many interesting examples of both big and small lattices. We will show that M3 is big but no smaller lattice is, answering the question mentioned above. On the other hand, N5 is small. Using this algorithm, we produce a complete list of all 145 minimal big lattices. The minimal big lattices (up to dual isomorphism) are denoted by Gi for 1 ≤ i ≤ 81, and are drawn in the Figures 30–38 in Section 21. A finite lattice is big if and only if it contains some Gi or Gi as a sublattice.