A variety (equational class) of lattices is said to be finitely based if there exists a finite set of identities defining the variety. Let M∞ndenote the lattice variety generated by all modular lattices of width not exceeding n. M∞1 and M∞2are both the class of all distributive lattices and consequently finitely based. B. Jónsson has shown that M∞3is also finitely based. On the other hand, K. Baker has shown that M∞nis not finitely based for 5 ≤ n ˂ ω. This thesis settles the finite basis problem for M∞4. M∞4 is shown to be finitely based by proving the stronger result that there exist ten varieties which properly contain M∞4 and such that any variety which properly contains M∞4 contains one of these ten varieties.
The methods developed also yield a characterization of sub-directly irreducible width four modular lattices. From this characterization further results are derived. It is shown that the free M∞4lattice with n generators is finite. A variety with exactly k covers is exhibited for all k ≥ 15. It is further shown that there are 2Ӄo sub- varieties of M∞4.
