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A Simple Solution to the Word Problem for Lattices

Published online by Cambridge University Press:  20 November 2018

Alan Day
Alan Day*
Affiliation:
Mcmaster University, Hamilton, Ontario
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Whitman [2] solved the word problem for lattices by giving an explicit construction of the free lattice, FL(X), on a given set of generators X.

The solution is the following:

For x, y ∊ X, and a, b, c, d ∊ FL(X),

(W1)

(W2)

(W3)

(W4)

where [p, q] = {x; p ≤ x ≤ q}.

The purpose of this note is to give a simple nonconstructive proof that the condition (W4) must hold in every projective (hence every free) lattice. Jonsson [1] has shown that in every equational class of lattices (Wl), (W2), and (W3) hold. Therefore the combination of these results gives a complete nonconstructive solution to the word problem for lattices.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 2 , June 1970 , pp. 253 - 254
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Jonsson, , Relatively free lattices, Coll. Math, (to appear).Google Scholar
2. Whitman, , Free lattices, Ann. Math. 42 (1941), 325-330.Google Scholar