Bands of groups with universal properties

For a nonempty setX, a bandB, and a mapping :XB, we construct a band of groups, here called a cryptogroup,F(X,,B) which exhibits some remarkable properties. The first of these is a universal property relative to the classCG of all cryptogroups. In fact,CG is a variety with the operations of multiplication and inversion. For a varietyV of bands, we find a varietyV ⁰ of cryptogroups such that wheneverB is a band free inV ⁰ on the setX with embedding :XB, F(X,,B) is free inV ⁰. IfB is a normal band given as a strong semilattice of rectangular bands, we construct an isomorphic copy ofF(X,, B) which is a strong semilattice of completely simple semigroups. The objectsX, , B) admit the structure of a category, which is then related to the category of cryptogroups and their homomorphisms.This research was supported, in part by, NSERC Grant A4044.