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Distributive Projective Lattices

Published online by Cambridge University Press:  20 November 2018

Kirby A. Baker  and
Alfred W. Hales
Kirby A. Baker
Affiliation:
University of California, Los Angeles, California
Alfred W. Hales
Affiliation:
University of California, Los Angeles, California
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Two basic unsolved problems of lattice theory are (1) the characterization of sublattices of free lattices and (2) the characterization of projective lattices. A solution to an important case of the first problem has been provided by Galvin and Jónsson [3], who characterize distributive sublattices of free lattices. In this paper, we solve the same case of the second problem by characterizing distributive projective lattices (Theorem 4.1). An interesting corollary is the verification for distributive lattices of the conjecture that a, finite lattice is projective if and only if it is a sublattice of a free lattice.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 472 - 475
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Balbes, R. and Horn, A., Order sums of distributive lattices, Pacific J. Math. 21 (1967), 421–435.Google Scholar
2. Dean, R., Sublattices of free lattices, Proc. Sympos. Pure Math., Vol. II, pp. 31–42 (Amer. Math. Soc, Providence, R.I., 1961).Google Scholar
3. Galvin, F. and Jonsson, B., Distributive sublattices of a free lattice, Can. J. Math. 13 (1961), 265–272.Google Scholar
4. McKenzie, R., Equational bases and non-modular lattice varieties (to appear).Google Scholar