1. Introduction. Consider a set P with elements a, b, c,… and a certain distinguished element e. Assume that P satisfies the following conditions.

(I) With each ordered pair (a, b) of distinct elements a and b of P there is associated a unique element of P (to be denoted by ab and called the product of a and b). Further, with the pair (e, e) there is associated a unique element ee of P.

(II)(ab)c = a(bc) for all elements a, b, c in P for which both sides are defined.

(III) ea = a = ae for all elements a ∈ P.

(IV) To each element a ∈ P, there corresponds an element a′ ∈ P such that