Planar sublattices of FM ( 4 )

R. McKenzie, A. Kostinsky and B. J6nsson have proved the remarkable result that the class of finite sublattices of a free lattice and the class of finite projective lattices coincide [6]. I f we restrict ourselves to modular lattices, the above result is no longer true. Examining the map from the free modular lattice on three generators, FM(3), to the free distributive lattice on three generators, FD(3), we see that FD(3) is not a projective modular lattice (projective in the class of modular lattices. 'Onto ' maps are used in the definition of projective). On the other hand any finite distributive lattice is a sublattice of a free modular lattice. In this paper we show that any finite planar modular lattice which does not contain a Sublattice isomorphic to M4 (the six element two-dimensional lattice) is a sublattice of FM(4). Some of these sublattices which are projective were discovered by Alan Day 1"4]. However, infinitely many of these lattices are simple, whereas previously no example of a simple sublattice of FM(4) with more than five elements was known. Finally, the results of this paper are used to settle a question raised by E. T. Schmidt in [7]. Following A. Huhn, let Pn-1 denote a ( n l ) d i amond . That is, P~_~ is a partial lattice consisting of a Boolean algebra of order 2 n with 0 and 1 as least and greatest elegaents and an element x such that x is a relative complement of the atoms of the Boolean algebra in the quotient 1/0. Let FM(Pn_~) denote the modular lattice freely generated by P~_~.