Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T00:49:09.045Z Has data issue: false hasContentIssue false
  • English
  • Français

Various varieties1

Published online by Cambridge University Press:  09 April 2009

Sheila Oates MacDonald
Sheila Oates MacDonald
Affiliation:
Department of Mathematics University of Queensland Brisbane, Australia
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 study of varieties of universal algebras2 which was initiated by Birkhoff in 1935, [2], has received considerable attention during the past decade; the question of particular interest being: “Which varieties have a finite basis for their laws?” In that paper Birkhoff showed that the laws of a finite algebra which involve a bounded number of variables are finitely based, so it is not altogether surprising that finite algebras have received their share of this attention.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 3 , November 1973 , pp. 363 - 367
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Baker, K. A., ‘Equational bases for finite algebras’ (Preliminary report), Notices Amer. Math. Soc. 19 (1972), 691–08–2.Google Scholar
[2]Birkhoff, G., ‘On the structure of abstract algebras’, Proc. Cambridge Phil. Soc. 31 (1935), 433–454.CrossRefGoogle Scholar
[3]Cohn, P. M., Universal algebra (Harper & Row, New York, 1965).Google Scholar
[4]Day, A., ‘A characterization of modularity for congruence lattices of algebras’, Canad. Math. Bull. 12 (1969), 167–173.CrossRefGoogle Scholar
[5]Evans, T., ‘Identical relations in loops’, J. Austral. Math. Soc. 12 (1971), 275–286.CrossRefGoogle Scholar
[6]Jónsson, B., ‘Algebras whose congruence lattices are distributive’, Math. Scand. 21 (1967), 110–121.CrossRefGoogle Scholar
[7]Kovács, L. G. and Newman', M. F., Cross varieties of groups', Proc. Roy. Soc. (London) A, 292 (1966), 530–536.Google Scholar
[8]Kruse, R. L., ‘Identities satisfied by a finite ring’, J. Algebra 26 (1973), 298–318.CrossRefGoogle Scholar
[9]Lyndon, R. C., ‘Identities in two-valued calculi’, Trans. Amer. Math. Soc. 71 (1951), 457–465.CrossRefGoogle Scholar
[10]Mal'cev, A. I., ‘On the general theory of algebraic systems’, Mat. Sb. (N. S.) 35 (77) (1954), 3–20.Google Scholar
[11]McKenzie, R., ‘Equational bases for lattice theories’, Math. Scand. 27 (1970), 24–38.CrossRefGoogle Scholar
[12]Murskii, V. L., ‘The existence in three-valued logic of a closed class with finite basis, not having a finite complete system of identities’, Soviet Math. Doklady, 6 (1965), 1020–1024.Google Scholar
[13]Neumann, Hanna, Varieties of groups (Ergebaisse der Mathematik und ihrer Grenzgebiete, Bd. 37, Springer-Verlag, Berlin, 1967).CrossRefGoogle Scholar
[14]Oates, Sheila and Powell, M. B., ‘Identical relations in finite groups’, J. Algebra 1 (1964), 11–39.CrossRefGoogle Scholar
[15]Perkins, P., ‘Bases for equational theories of semigroups’, J. Algebra 11 (1969), 298–314.CrossRefGoogle Scholar