Definitions of a linear associative algebra by independent postulates

Abstract

The term linear associative algebra, introduced by BENJAMIN PEIRCE, has the same significance as the term system of (higher) complex numbers. t In the usual theory of complex numbers, the co6rdinates are either real numbers or else ordinary complex quantities. To avoid the resulting double phraseology and to attain an evident generalization of the theory, I shall here consider systems of complex numbers whose co6rdinates belong to an arbitraryfield F. I first give the usual definition by means of a multiplication table for the n units of the system. It employs three postulates, shown to be independent, relating to n3 elements of the field F. The second definition is of abstract character. It employs four independent postulates which completely define a system of complex numbers. The first definition may also be presented in the abstract form used for the second, namely, without the explicit use of units. The second definition may also be presented by means of units. Even aside from the difference in the form of their presentation, the two definitions are essentially different.