Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-27T16:50:26.636Z Has data issue: false hasContentIssue false
  • English
  • Français

The word problem for lattice ordered groups

Published online by Cambridge University Press:  20 January 2009

A. M. W. Glass
A. M. W. Glass
Affiliation:
Bowling Green State University, Bowling Green, Ohio, 43403, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this note is to show the existence of a lattice ordered group G with a finite set of generators and a recursively enumerable set of defining relations such that there is no decision procedure to determine whether or not an arbitrary word in the generators reduces to the identity in G. In addition to the usual group-theoretic words, we may also use the two lattice operations ∨ and ∧ ; for example, a−l(b∨c) is a word in the generators a, b and c. At first sight it might appear that since we have an even greater harvest of words than in group theory and there exist finitely presented groups H (H has a finite number of generators and defining relations) with an insoluble word problem (no decision procedure to determine whether an arbitrary word in the generators reduces to the identity)—see (1), (2), (4) or (6)—the same would be true of lattice ordered groups. Unfortunately, such a naïve approach overlooks two salient points. First, the class of lattice ordered groups is strictly smaller than the class of all groups; second, there are certain relations connecting the lattice operations with the group operations which hold true for all lattice ordered groups. For example, a(b∧c) = ab∨ac and (a∨b)−1 = a−1∧b−1.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 19 , Issue 3 , March 1975 , pp. 217 - 219
Copyright
Copyright © Edinburgh Mathematical Society 1975

References

REFERENCES

(1) Boone, W. W., The word problem, Ann. of Math. 70 (1959), 207–265.CrossRefGoogle Scholar
(2) Brit-Ton, J. L., The word problem, Ann. of Math. 77 (1963), 16–32.Google Scholar
(3) Cohn, P. M., Universal Algebra (Harper and Row, London, 1965).Google Scholar
(4) Higman, G., Subgroups of finitely presented groups, Proc. Royal Soc. London, Series A 262 (1961), 455–475.Google Scholar
(5) Holland, W. C., The lattice group of all automorphisms of an ordered set, Michigan Math. J., 10 (1963), 399–408.CrossRefGoogle Scholar
(6) Novikov, P. S., On the algorithmic unsolvability of the word problem in group theory, Trudy Maths., Inst. Steklova Akad. Nauk SSSR, 44 (1955).Google Scholar
(6) Pierce, K. R., Amalgamations of lattice ordered groups, Trans. Amer. Math. Soc. 172 (1972), 249–260.CrossRefGoogle Scholar