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On Finitely Generated Lattices of Finite Width

The width of a lattice L is the maximum number of pairwise noncomparable elements in L. It has been known for some time ([5] ; see also [4]) that there is just one subdirectly irreducible lattice of width twro, namely the five-element nonmodular lattice N 5. It follows that every lattice of width two is in the variety of N 5, and that every finitely generated lattice of width two is finite. Beginning a study of lattices of width three, W. Poguntke [6] showed that there are infinitely many finite simple lattices of width three. Further studies on width three lattices were made in [3], where it was asked whether every finitely generated simple lattice of width three is finite. In this paper we will show that, in fact, more is true: THEOREM 1.1. Every finitely generated subdirectly irreducible lattice of width three is finite.