Topological and algebraic structures on the ring of Fermat reals

TL;DR

This paper explores the topological and algebraic properties of the ring of Fermat reals, introducing new topologies, analyzing ideals, and applying these concepts to infinitesimal calculus and Taylor formulas.

Contribution

It introduces the Fermat topology and omega topology on Fermat reals, studies ideals and roots of infinitesimals, and applies these to infinitesimal calculus and fractional derivatives.

Findings

Fermat topology is generated by a complete pseudo-metric.

Omega topology is generated by a complete metric.

Every proper ideal consists of infinitesimals with bounded order.

Abstract

The ring of Fermat reals is an extension of the real field containing nilpotent infinitesimals, and represents an alternative to Synthetic Differential Geometry in classical logic. In the present paper, our first aim is to study this ring from using standard topological and algebraic structures. We present the Fermat topology, generated by a complete pseudo-metric, and the omega topology, generated by a complete metric. The first one is closely related to the differentiation of (non standard) smooth functions defined on open sets of Fermat reals. The second one is connected to the differentiation of smooth functions defined on infinitesimal sets. Subsequently, we prove that every (proper) ideal is a set of infinitesimals whose order is less than or equal to some real number. Finally, we define and study roots of infinitesimals. A computer implementation as well as an application to infinitesimal Taylor formulas with fractional derivatives are presented.