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Planar Sublattices of a Free Lattice. I

Published online by Cambridge University Press:  20 November 2018

Ivan Rival  and
Bill Sands
Ivan Rival
Affiliation:
University of Calgary, Calgary, Alberta
Bill Sands
Affiliation:
University of Manitoba, Winnipeg, Manitoba.
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There are three lattice-theoretic properties that are generally used to open a discussion on sublattices of a free lattice:

(W) for all a, b, c, d, a Λ b ≦ c V d implies a A b ≦ c, a Λ b g rf, a ≦cVd,orb ≦ c V d;

(SDv) for all a, b, c, a V b = a V c implies a V b = a V (b Λ c ) ;

(SDΛ) for all a, b, c, a Λ b = a Λ c implies a Λ b = a Λ (b V c).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 6 , 01 December 1978 , pp. 1256 - 1283
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Baker, K. A., Fishburn, P. C., and Roberts, F. S., Partial orders of dimension 2, interval orders, and interval graphs, Networks 2 (1971), 11–28.Google Scholar
2. Dean, R. A., Sublattices of free lattices, in Lattice Theory, Proc. of Symp. in Pure Math., II American Mathematical Society, Providence, (1961), pp. 31–42.Google Scholar
3. Galvin, F. and Jônsson, B., Distributive sublattices of a free lattice, Can. J. Math. 13 (1961), 265–272.Google Scholar
4. Gaskill, H. S., Transferability in lattices and semilattices, Ph.D. Thesis, Simon Fraser University (1972).Google Scholar
5. Gaskill, H. S., On transferable semilattices, Alg. Univ. 2 (1973), 303–316.Google Scholar
6. Gaskill, H. S., G. Gràtzer, and Piatt, C. R., Sharply transferable lattices, Can. J. Math. 27 (1975), 1246–1262.Google Scholar
7. Gaskill, H. S. and Piatt, C. R., Sharp transferability and finite sublattices of free lattices, Can. J. Math. 27 (1975), 1036–1041.Google Scholar
8. Jônsson, B., Sublattices of a free lattice, Can. J. Math. 13 (1961), 256–264.Google Scholar
9. Jônsson, B. and Kiefer, J. E., Finite sublattices of a free lattice, Can. J. Math. 1 (1962), 487–497.Google Scholar
10. Jônsson, B. and Nation, J. B., A report on sublattices of a free lattice, Proceedings of Colloquium on Universal Algebra, Szeged (1975).Google Scholar
11. Kelly, D. and Rival, I., Crowns, fences, and dismantlable lattices, Can. J. Math. 26 (1974), 1257–1271.Google Scholar
12. Kelly, D. and Rival, I., Planar lattices, Can. J. Math. 27 (1975), 636–665.Google Scholar
13. McKenzie, R., Equational bases and nonmodular lattice varieties, Trans. Amer. Math, Soc. 174 (1972), 1–43.Google Scholar
14. Rival, I., Lattices with doubly irreducible elements, Can. Math. Bull. 17 (1974), 91–95.Google Scholar
15. Rival, I. and Sands, B., Planar sublattices of a free lattice, II, Can. J. Math. (1978).Google Scholar
16. Sorkin, Yu. I., Free unions of lattices, Mat. Sb. 30 (72) (1952), 677–694 (Russian).Google Scholar
17. Whitman, P. M., Free lattices, Ann. of Math. 42 (1941), 325–330.Google Scholar
18. Wille, R., On lathees freely generated by a finite partially ordered set, preprint (1975). Coll. Math. Soc. J. Bolyai 17, Contributions to Universal Algebra, Budapest 1977, pp. 581–593.Google Scholar