Some recent developments in abstract algebra
Öystein Ore · Bulletin of the American Mathematical Society · 1931
E I. Determination.For two arbitrary elements, either a = b or a^b.E II.Reflexitivity. a = a.E III.Symmetry.From a -b follows b -a.E IV. Transitivity.From a = 5, b = c follows a -c.Every such definition of equality, which is ordinarily not unique for the given system, constitutes a division of the elements into classes.Usually algebra deals with systems which are closed with respect to one or two operations, addition and multiplication, satisfying all or some of the following axioms : A. AXIOMS OF ADDITION A I. For two arbitrary elements a and b in 5 there exists a sum a + b, which is a uniquely defined element of S. A II. Equality.If a -b and ai=&i, then a-{-ai = b + bi.A 111.A ssociative law.a + (b + c) = (a + b) + c.A IV. Zero-element.There exists an element 0 for which 0+a =a+0-a for all a.A V. Subtraction.To every element a there exists another -a such that a + ( -a) = 0.A VI. Commutative law.a+b = b+a.