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Normal and Canonical Representations in Free Products of Lattices†

Published online by Cambridge University Press:  20 November 2018

H. Lakser
H. Lakser*
Affiliation:
The University of Manitoba, Winnipeg, Manitoba
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In solving the word problem for free lattices, Whitman [4] showed that free lattices admit canonical representations, that is, of all polynomials over the generating set representing an element of the lattice, the polynomial of shortest length is unique up to commutativity and associativity. These well-defined shortest polynomials have proved very important in analyzing the internal structure of free lattices in detail; see, e.g., [5].

Sorkin [3] proved that the free product of chains also admits canonical representations; these were exploited by Rolf [2]. In the above-mentioned paper, Sorkin also suggested that the free product of two copies of 22 does not admit canonical representations.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 394 - 402
Copyright
Copyright © Canadian Mathematical Society 1970

Footnotes

†

Theorem 4 was first announced in Notices Amer. Math. Soc. 15 (1968), 383.

This research was supported by the National Research Council of Canada.

References

1. Grâtzer, G., Lakser, H., and Piatt, C. R., Free products of lattices (to appear in Fund Math.).Google Scholar
2. Rolf, H. L., The free lattice generated by a set of chains, Pacific J. Math. 8 (1958), 585–595.Google Scholar
3. Sorkin, Yu. I., Free unions of lattices, Mat. Sb. (N.S.) 30 (1952), 677–694.Google Scholar
4. Whitman, P. M., Free lattices. I, Ann. of Math. (2) 42 (1941), 325–330.Google Scholar
5. Whitman, P. M., free lattices. II, Ann. of Math. (2) 43 (1942), 104–115.Google Scholar