Free lattice

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 DOI 10.2307/1969001 Zbl 67.0085.01 Zbl 0024.24501
[1b]	P.M. Whitman, "Free lattices, II" Ann. of Math. , 43 (1942) pp. 104–115 DOI 10.2307/1968883 Zbl 68.0047.02 Zbl 0063.08232

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