Alternative analysis generated by a differential equation

Abstract

It was shown in [1] that for a wide class of differential equations there exist infinitely many binary laws of addition of solutions such that every binary law has its conjugate. From this set of operations we extract commutative algebraic object that is a pair of two alternative to each other fields with common identity elements. The goal of the present paper is to detect those mathematical constructions that are related to the existence of alternative fields dictated by differential equations. With this in mind we investigate differential and integral calculus based on the commutative algebra that is generated by a given differential equation. It turns out that along with the standard differential and integral calculus there always exists an isomorphic alternative calculus. Moreover, every system of differential equations generates its own calculus that is isomorphic (or homomorphic) to the standard one. The given system written in its own calculus appears to be linear. It is also shown that there always exist two alternative to each other geometries, and matrix algebra has its alternative isomorphic to the classical one.