A Development of Associative Algebra and an Algebraic Theory of Numbers, I
Harry S. Vandiver · Mathematics Magazine · 1952
in which if we denote a particular element by Ck, its immediate successor in this CkJ, where k denotes a natural number and k' its immediate successor in the set of natural numbers. We then introduced in addition to these symbols the symbol + (called a plus sign); x (called a multiplication sign); and (, called a left parenthesis symbol; and ), called a right parenthesis symbol. We next set up a system of postulates covering combinations of the symbols just mentioned and the idea of equality (=) between these symbols. We assume nothing concerning equality among the symbols (1) themselves and using the and using the postulate and various assumptions conserning equality and the elements in (1), we obtained elementary arithmetic of the of the natural numbers and certain finite arithmetics. In particular we derived finite arithmetics in which the cancellation law of addition did not hold. Another feature of the development in (I) the fact that we do not use the symbol of equality to mean or the symbol / to mean is not, so that we were obliged to introduce the postulate of substitution to take care of a possible generalization of ordinary equality. Also, let A, B, C, and denote combinations of the type indicated above. Let us then symbolize the statement If A = B, then C = D by