Fermat实数环上的拓扑和代数结构
Topological and algebraic structures on the ring of Fermat reals
摘要：
费马实数环是包含幂零无穷小的实数域的扩展，代表了古典逻辑中综合微分几何的替代。在本文中，我们的第一个目标是研究这个环从使用标准的拓扑和代数结构。我们提出的费马拓扑，产生一个完整的伪度量，和欧米茄拓扑，产生一个完整的度量。第一个问题与定义在费马实数开集上的（非标准）光滑函数的微分密切相关。第二个问题与定义在无穷小集合上的光滑函数的微分有关。随后，我们证明了每一个（真）理想是一组无穷小，其阶小于或等于某个实数。最后，我们定义并研究了无穷小的根。给出了一种计算机实现方法，并应用于含分数阶导数的无穷小泰勒公式。
英文摘要：
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.
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