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Characterizations of Finite Lattices that are Bounded-Homomqrphic Images or Sublattices of Free Lattices

Published online by Cambridge University Press:  20 November 2018

Alan Day
Alan Day*
Affiliation:
Lakehead University, Thunder Bay, Ontario
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In [8], McKenzie introduced the notion of a bounded homomorphism between lattices, and, using this concept, proved several deep results in lattice theory. Some of these results were intimately connected with the work of Jónsson and Kiefer in [6] where an attempt was made to characterize finite sublattices of free lattices. McKenzie's characterization and others that followed (see [7] and [5]) still have not answered the (now) celebrated Jônsson conjecture:

A finite lattice is a sublattice of a free lattice if and only if it satisfies (SD∨), (SD∨) and (W).

(The properties mentioned here are defined in the text.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 1 , 01 February 1979 , pp. 69 - 78
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Davey, B. and Sands, B., An application of Whitman''s Condition to lattices with no infinite chains, Alg. Univ. 7 (1977), 171–178.Google Scholar
2. Day, A., A simple solution to the word problem for lattices, Can. Math. Bull. 13 (1970), 253–254.Google Scholar
3. Day, A., Splitting lattices generate all lattices, Alg. Univ. 7 (1977), 163–169.Google Scholar
4. Day, A., Finite sublattices of free lattices, Notices Amer. Math. Soc. 23 (1976).Google Scholar
5. Gaskill, H., Gratzer, G., and Piatt, C., Sharply transferable lattices, Can. J. Math. 27 (1975), 1246–1262.Google Scholar
6. Jônsson, B. and Kiefer, J., Finite sublattices of a free lattice, Can. J. Math. 14 (1962), 487–497.Google Scholar
7. Jônsson, B. and Nation, J. B., A report on sublattices of a free lattice, Colloq. Math. Soc. Jânos Bolyai: Contributions to Universal Algebra, Szeged (1975), 223–257, (North Holland, Amsterdam).Google Scholar
8. McKenzie, R., Equatorial bases and non-modular lattice varieties, Trans. Amer. Math. Soc. 174 (1972), 1–43.Google Scholar
9. Pudlâk, P. and Tuma, J., Yeast graphs and fermentation of algebraic lattices Colloq. Math. Soc. Jânos Bolyai: Lattice Theory, Szeged (1974), 301–341, (North Holland, Amsterdam).Google Scholar
10. Sivâk, B., Representation of finite lattices by orders on finite sets, prepublication manuscript.Google Scholar