The Universality of the variety of quasigroups

Abstract

AbstractThe 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, …〉wheref0, f1, f2, …is an arbitrary finite or infinite sequence of operations of finite rank, there exists a quasigroup 〈B,.〉 and polynormial operationsF0, 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, thenBcan also be taken to coincide withA. This universal property of quasigroups has a number of consequences for their equational metatheory.