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Free lattice

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A free algebra in the variety of all lattices. In a free lattice the problems of the identity of words and of the canonical representation of a word have been solved .

References

[1a] P.M. Whitman, "Free lattices" Ann. of Math. , 42 (1941) pp. 325–330
[1b] P.M. Whitman, "Free lattices, II" Ann. of Math. , 43 (1942) pp. 104–115


Comments

The solution of the word problem for free lattices enables one to show that the free lattice on three generators is infinite, although the free modular lattice on three generators is finite (it has 28 elements — a fact already known to R. Dedekind [a1]), and the free distributive lattice on any finite set of generators is finite. Extending this result, A.W. Hales [a2] has shown that the free complete lattice on three generators does not exist (i.e. there is a proper class of distinct words which can be formed from three generators using the complete-lattice operations).

References

[a1] R. Dedekind, "Ueber die von drei Moduln erzeugte Dualgruppe" Math. Ann. , 53 (1900) pp. 371–403
[a2] A.W. Hales, "On the non-existence of free complete algebras" Fund. Math. , 54 (1964) pp. 45–66
How to Cite This Entry:
Free lattice. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_lattice&oldid=37258
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article