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Finite Sublattices of a Free Lattice

Published online by Cambridge University Press:  20 November 2018

Bjarni Jónsson  and
James E. Kiefer
Bjarni Jónsson
Affiliation:
University of Minnesota
James E. Kiefer
Affiliation:
University of Minnesota
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It is known that every sublattice A of a free lattice satisfies the following conditions:

(W) For all a, b, c, d ∈ A, if ab ≤ c + d, then ab ≤ c or ab ≤ d or a ≤ c+dor b ≤ c + d.

(SD) For all u, a, b, c ∈ A, if u = a + b = a + c, then u = a + bc.

(SD′) For all u, a, b, c ∈ A, if u = ab = ac, then u = a(b + c).

In fact, (W) is one of the four conditions used in Whitman (4) to characterize free lattices, and in Jónsson (3) it was shown that (SD) and (SD′) follow from Whitman's canonical representations of elements of a free lattice.

This note is concerned with lattices that satisfy one or more of the above conditions, and especially with finite lattices that satisfy all three conditions.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 487 - 497
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Galvin, F. and Jônsson, B., Distributive sublattices of a free lattice, Can. J. Math., 13 (1961), 265–272.Google Scholar
2. Jónsson, B., Arithmetic properties of freely a-generated lattices, Can. J. Math., 14 (1962), 000–000.Google Scholar
3. Jonsson, B., Sublattices of a free lattice, Can. J. Math., 13 (1961), 256–264.Google Scholar
4. Whitman, P. M., Free lattices I, Ann. Math. (2), 42 (1941), 325–330.Google Scholar