Associative Hypercomplex Algebras Arise over a Basic Set of Subgeometric One-Dimensional Elements

An abstract set of one-dimensional (spinor-type) elements randomly oriented on a plane is introduced as a basic subgeometric object. Endowing the set with the binary operations of multiplication and invertible addition sequentially yields a specific semi-group (for which an original Cayley table is given) and a generic algebraic system which is shown to generate, apart from algebras of real and complex numbers, the associative hypercomplex algebras of dual numbers, split-complex numbers, and quaternions. The units of all these algebras turn out to be composed of basic 1D elements, thus ensuring the automatic fulfillment of multiplication rules (once postulated). From the standpoint of a three-dimensional space defined by a vector quaternion triad, the condition of a standard (unit) length of 1D basis elements is considered; it is shown that fulfillment of this condition provides an equation mathematically equivalent to the main equation of quantum mechanics. The similarities and differences of the proposed logical scheme with other approaches that involve abstract subgeometric objects are discussed.