On hypercomplex number systems. I
Henry Taber · Transactions of the American Mathematical Society · 1904
The method invented by Benjamin Peirce t for treating the problem to determine all hypereomplex number systems (or algebras) in a given number of units depends chiefly, first, upon the classification of hypereomplex number systems into idempotent number systems, containing one or more idempotent numbers, \ and non-idempotent number systems containing no idempotent number ; and, second, upon the regularizaron of idempotent number systems, that is, the classification of each of the units of such a system with respect to one of the idempotent numbers of the system.For the purpose of such classification and regularizaron the following theorems are required : Theorem I. § In any given hypereomplex number system there is an idempotent number (that is, a number I + 0 such that I2 = I), or every number of the system is nilpotent.|| Theorem H.*rj In any hypereomplex number system containing an idempotent number I, the units ex, e2, ■ ■ -, en can be so selected that, with reference