Varieties generated by modular lattices of width four

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