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Finite sublattices of three-generated lattices

Published online by Cambridge University Press:  09 April 2009

Brian A. Davey  and
Ivan Rival
Brian A. Davey
Affiliation:
Department of MathematicsUniversity of Manitoba Winnipeg, Manitoba, R3T 2N2, Canada.
Ivan Rival
Affiliation:
Department of MathematicsUniversity of Manitoba Winnipeg, Manitoba, R3T 2N2, Canada.
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Abstract

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Every lattice generated by three unordered elements contains a finite sublattice generated by three unordered elements. A list ℒ of twelve finite lattices, each generated by a three-element unordered set, is given. It is proved that every lattice generated by a three-element unordered set contains a sublattice isomorphic to onė of the lattices in ℒ moreover, ℒ is the smallest such list.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 2 , March 1976 , pp. 171 - 178
Copyright
Copyright © Australian Mathematical Society 1976

References

Crawley, P. and Dean, R. A. (1959), ‘Free lattices with infinite operations’, Trans. Amer. Math. Soc. 92, 35–47.CrossRefGoogle Scholar
Crawley, P. and Dilworth, R. P. (1973), Algebraic Theory of Lattices, (Prentice-Hall, Englewood Cliffs, N. J.).Google Scholar
Davey, B. A., Poguntke, W. and Rival, I. (1975), ‘A characterization of semidistributivity’, Alg. Universalis 5, 72–75.CrossRefGoogle Scholar
Dean, R. A. (1956), ‘Component subsets of the free lattice on n generators’, Proc. Amer. Math. Soc. 7, 220–226.Google Scholar
Dean, R. A. (1961), ‘Sublattices of free lattices’, Lattice Theory, Proc. of Symp. in Pure Math., II. (American Mathematical Society, Providence, R. I., 1961), 31–42.CrossRefGoogle Scholar
Sorkin, Yu. I. (1954), ‘On the embedding of latticoids in lattices’, Dokl. Akad. Nauk SSSR (N. S.) 95, 931–934 (Russian).Google Scholar
Whitman, P. (1942), ‘Free lattices. II’, Ann. of Math. 43, 104–115.CrossRefGoogle Scholar
Wille, R. (1974), ‘Jeder endlich erzeugte, modulare Verband endlicher Weite ist endlich’, Mat. Časopis Sloven. Akad. Vied. 24, 77–80.Google Scholar