Lattices That Are the Join of Two Proper Sublattices

Every lattice is the complete join of all its one-element sublattices. In this paper we address the question: Which lattices L have the property that L is finitely join reducible in SubL? That is, when do there exist proper sublattices A, B such that L = A∨B? In particular, could it be that every nontrivial lattice has this property, in which case every element of SubL would be finitely join reducible? The authors would like to thank David Wasserman and M. E. Adams for bringing this problem to our attention, along with some elementary observations and helpful discussion. Let us mention a related problem. Recall the following result of Tom Whaley [4]. Theorem 0.1. If L is a lattice and κ = | L | is a regular infinite cardinal, then L has a proper sublattice of cardinality κ. The question is: Is this true when | L | is singular? Note that if | L | is uncountable and L = A∨B, then either | A | = | L | or | B | = | L |. So if there is a lattice which has no proper sublattice of the same cardinality, then it must be join irreducible in SubL. If a lattice contains a maximal proper sublattice, then it is join reducible. However, we have the following theorem of M. E. Adams [1].