Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-27T07:58:15.972Z Has data issue: false hasContentIssue false
  • English
  • Français

The Universality of the variety of quasigroups

Published online by Cambridge University Press:  09 April 2009

Don Pigozzi
Don Pigozzi
Affiliation:
Department of MathematicsIowa State University Ames, Iowa 50010, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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 variety of quasigroups is universal for varieties of algebras of the most general kind in the sense that each such variety can be interpreted in a natural way in a suitably chosen subvariety of quasigroups. More precisely, for any algebra 〈A, f0, f1, f2, …〉 where f0, f1, f2, … is an arbitrary finite or infinite sequence of operations of finite rank, there exists a quasigroup 〈B,.〉 and polynormial operations F0, F1, F2, … over 〈B,·〉 such that 〈A, f0, f1, …〉 is a subalgebra of 〈B, F0, F1, …〉 satisfying exactly the same identities. Moreover, if there are only finitely many f0, f1, …, then 〈B1〉 can be taken so that its identities are recursive in those of 〈A, f0, f1, …〉, If 〈A, f0, f1, …〉 is a free algebra with an infinite number of free generators, then B can also be taken to coincide with A. This universal property of quasigroups has a number of consequences for their equational metatheory.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 2 , March 1976 , pp. 194 - 219
Copyright
Copyright © Australian Mathematical Society 1976

References

Bol'bot, A. D. (1967), ‘Equationally complete varieties of totally symmetric quasigroups’, (In Russian.) Algebra i Logika 6, 13–19.Google Scholar
Bol'bot, A. D. (1972), ‘Varieties of quasigroups’, (In Russian.) Sibirsk. Maz. Z. 13, 252–271.Google Scholar
[English Translation. Siberian Math. J. 13 (1972), 173–186.]CrossRefGoogle Scholar
Boone, W. W. and Rogers, H. Jr (1966), ‘On a problem of J. H. C. Whitehead and a problem of Alonzo Church’, Math. Scand. 19, 185–192.CrossRefGoogle Scholar
Evans, T. (1951), ‘On multiplicative systems defined by generators and relations. I. Normal form theorems’, Proc. Cambridge Philos. Soc. 47, 637–649.CrossRefGoogle Scholar
Evans, T. (1971), ‘Identical relations in loops, I’, J. Austral. Math. Soc. 12, 275–286.CrossRefGoogle Scholar
Mal'cev, A. I. (1939), ‘On the embedding of associative systems into groups. I’, (In Russian.) Mat. Sb. 6, 331–336.Google Scholar
Mal'cev, A. I. (1966), ‘Identical relations in varieties of quasigroups’, (In Russian.) Mat. Sb. 69, 3–12.Google Scholar
[English translation. Amer. Math. Soc. Transl. (2), 82 (1969), 225–235.Google Scholar
Also in: Mal'cev, A. I., The metamathematics of algebraic systems (North-Holland Publishing Co., Amsterdam, 1971).]Google Scholar
McNulty, G. F., ‘The decision problem for equational bases of algebras’, Ann. Math. Logic (to appear).Google Scholar
Neumann, H. (1967), Varieties of groups (Ergebnisse Der Mathematik und Ihrer Grenzgebiete, Bd. 37, Springer-Verlag, Berlin).CrossRefGoogle Scholar
Neumann, P. M. and Wiegold, J. (1964), ‘Schreier varieties of groups’, Math. Z. 85, 392–400.CrossRefGoogle Scholar
Pigozzi, D. (1973), ‘On the decision problem for equational theories of quasi-groups’, Notices Amer. Math. Soc. 20, A-462.Google Scholar
Pigozzi, D. (a), ‘Universal equational theories and varieties of algebras’, (to appear).Google Scholar
Pigozzi, D. (b), ‘Base-undecidable properties of universal varieties’, (to appear).Google Scholar
Tarski, A. (1968), ‘Equational logic and equational theories of algebras’, in: Schmidt, H. A., Schutte, K., Thiele, H.-J., eds., Contributions to mathematical logic (North-Holland Publishing Co., Amsterdam).Google Scholar