$${f}$$ f -Algebra Structure on Hyperbolic Numbers

Abstract

The algebra of hyperbolic numbers is endowed with a partial order structure. We show that this system of numbers is the only (natural) generalization of real numbers into Archimedean \({f}\)-algebra of dimension two. We establish various properties of hyperbolic numbers related to the \({f}\)-algebra structure. In particular, we generalize fundamental properties of real numbers and give some order interpretations for the two dimensional space-time geometry.