Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T13:38:12.869Z Has data issue: false hasContentIssue false
  • English
  • Français

Identical relations in loops, I

Published online by Cambridge University Press:  09 April 2009

Trevor Evans
Trevor Evans
Affiliation:
Emory University Atlanta, Georgia, 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 systematic study of identities in groups was begun by B. H. Neumann in 1937, [7]. Although many special loop identities have been extensively investigated, there is not, as yet, a general theory of identical relations in loops. We cannot hope that such a theory will be as rich in results as in the group case (except in the sense that loop theory includes group theory) but there are many interesting problems, including some which have no analogue in the associative case.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 3 , August 1971 , pp. 275 - 286
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Bates, G. E., ‘Free loops and nets and their generalizations’, Am. J. Math. 69 (1947), 499–550.CrossRefGoogle Scholar
[2]Bruck, R. H., A survey of binary systems (Ergebnisse der Mathematik und ihrer Grenzgebiete, 1958).CrossRefGoogle Scholar
[3]Evans, T., ‘On multiplicative systems defined by generators and relations, I. Normal form theorems’, Proc. Cambr. Phil. Soc. 47 (1951), 637–649.CrossRefGoogle Scholar
[4]Evans, T., ‘Embedding incomplete latin squares’, Amer. Math. Monthly 67 (1960), 958–961.CrossRefGoogle Scholar
[5]Neumann, B. H. and Evans, T., ‘On varieties of groupoids and loops’, J. London Math. Soc. 28 (1953), 342–350.Google Scholar
[6]Kalicki, J., ‘The number of equationally complete classes of equations’, Nederl. Akad. Wetensch. Proc. Ser. A. 58 (1955), 660–662.CrossRefGoogle Scholar
[7]Neumann, B. H., ‘Identical relations in groups I’, Math. Ann. 114 (1937), 506–525.CrossRefGoogle Scholar
[8]Neumann, H., Varieties of groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 37, 1967).CrossRefGoogle Scholar