In this paper, we use the language of noncommutative differential geometry to formalise discrete differential calculus. We begin with a brief review of inverse limit of posets as an approximation of topological spaces. We then show how to associate a $C^*$-algebra over a poset, giving it a piecewise-linear structure. Furthermore, we explain how dually the algebra of continuous function $C(M)$ over a manifold $M$ can be approximated by a direct limit of $C^*$-algebras over posets. Finally, in the spirit of noncommutative differential geometry, we define a finite dimensional spectral triple on each poset. We show how the usual finite difference calculus is recovered as the eigenvalues of the commutator with the Dirac operator. We prove a convergence result in the case of the $d$-lattice in $\mathbb{R}^d$ and for the torus $\mathbb{T}^d$.

翻译：本文运用非交换微分几何的语言来形式化离散微分学。我们首先简要回顾偏序集的反向极限作为拓扑空间的逼近。接着展示如何将一个$C^*$-代数与偏序集相结合，赋予其分段线性结构。此外，我们解释对偶地，流形$M$上的连续函数代数$C(M)$如何通过偏序集上$C^*$-代数的正向极限来逼近。最后，遵循非交换微分几何的精神，我们在每个偏序集上定义一个有限维谱三元组。我们展示通常的有限差分学如何作为与Dirac算子对易子的特征值被恢复。我们在$\mathbb{R}^d$中的$d$-格点以及环面$\mathbb{T}^d$的情形下证明了一个收敛性结果。