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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.
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Abstract

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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

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