The Number of Proper Minimal Quasivarieties of Groupoids

It is shown that if an algebra has more than one element, is freely generated in some variety by one element and has a cancellative endomorphism semigroup then it generates a minimal quasivariety. This is used to construct uncountably many minimal quasivarieties of groupoids that are not varieties. A quasivariety [l] JÍ of algebras will be called implicationally complete or minimal if K has exactly two subquasivarieties, namely, K itself and the class of all singleton (one element) algebras. By a proper quasivariety we mean a quasivariety which is not a variety. In the case of semigroups there are countably infinitely many minimal quasivarieties, only one of which is proper [3]. For groupoids in general Received by the editors February 5, 1973 and, in revised form, January 28, 1974. AMS (MOS) subject classifications (1970). Primary 08A15.