Abstract. A finite distributive sublattice $T$ of a lattice $L$ is said to have the property $P$ if $T$ is contained in a Boolean sublattice of $L$. We consider the class $\Gamma $ of all lattices $L$ each of whose finite distributive sublattices has the property $P$. We prove that every relatively complemented modular lattice $L$ is in $\Gamma $. If, in addition, $L$ is atomic then we characterize subdirect irreducibility of $L$. A weaker result is given for semimodular lattices.

AMS Subject Classification (1991): 06D05, 06C05, 06C10

Received September 25, 1992 and in revised form December 13, 1993. (Registered under 5571/2009.)