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On Finitely Generated Lattices of Finite Width

Published online by Cambridge University Press:  20 November 2018

W. Poguntke  and
B. Sands
W. Poguntke
Affiliation:
Lakehead University, Thunder Bay, Ontario
B. Sands
Affiliation:
Lakehead University, Thunder Bay, Ontario
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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 N5. It follows that every lattice of width two is in the variety of N5, 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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 1 , 01 February 1981 , pp. 28 - 48
Copyright
Copyright © Canadian Mathematical Society 1981

References

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