We begin our study of algebraic structures by investigating sets associated with single operations that satisfy certain reasonable axioms; that is, we want to define an operation on a set in a way that will generalize such familiar structures as the integers $${\mathbb Z}$$ together with the single operation of addition, or invertible $$2 \times 2$$ matrices together with the single operation of matrix multiplication. The integers and the $$2 \times 2$$ matrices, together with their respective single operations, are examples of algebraic structures known as groups. 1