The spectrum of an equational class of groupoids

The spectrum of an equational class % is the set of positive integers Spec(3) = {n 1321 G JC, | 21 | = n}. It is obvious that 1 G Spec($O and JC, y G Spec(3) implies xy G Spec(3O for any equational class %\ i.e. Spec(3 is a multiplicative monoid of positive integers. Conversely, G. Gratzer showed that given any multiplicative monoid of positive integers ff there is an equational class JC such that Sf = Spec(3O. In this paper we show that j/ can be chosen to be an equational class of groupoids.