The Chinese Remainder Theorem for the integers says that every system of congruence equations is solvable as long as the system satisfies an obvious necessary condition. This statement can be generalized in a natural way to arbitrary algebraic structures using the language of Universal Algebra. In this context, an algebra is a structure of a first-order language with no relation symbols, and a congruence on an algebra is an equivalence relation on its base set compatible with its fundamental operations. A tuple of congruences of an algebra is called a Chinese Remainder tuple if every system involving them is solvable. In this article we study the complexity of deciding whether a tuple of congruences of a finite algebra is a Chinese Remainder tuple. This problem, which we denote CRT, is easily seen to lie in coNP. We prove that it is actually coNP-complete and also show that it is tractable when restricted to several well-known classes of algebras, such as vector spaces and distributive lattices. The polynomial algorithms we exhibit are made possible by purely algebraic characterizations of Chinese Remainder tuples for algebras in these classes, which constitute interesting results in their own right. Among these, an elegant characterization of Chinese Remainder tuples of finite distributive lattices stands out. Finally, we address the restriction of CRT to an arbitrary equational class $\mathcal{V}$ generated by a two-element algebra. Here we establish an (almost) dichotomy by showing that, unless $\mathcal{V}$ is the class of semilattices, the problem is either coNP-complete or tractable.

翻译：整数上的中国剩余定理指出，只要同余方程组满足一个明显的必要条件，则该方程组必定有解。这一结论可以通过泛代数的语言自然地推广到任意代数结构上。在此语境下，代数是一种不含关系符号的一阶语言结构，而代数上的同余关系是其基集上与其基本运算相容的等价关系。如果一个代数的同余元组能够使得涉及这些同余关系的所有方程组都有解，则称该元组为中国剩余元组。本文研究了判定有限代数的同余元组是否为中国剩余元组的复杂性。该问题（记为CRT）显然属于coNP类。我们证明它实际上是coNP完全的，同时表明当将其限制到若干经典代数类（如向量空间和分配格）时，该问题是可解的。这些多项式时间算法的实现得益于对这些代数类中中国剩余元组的纯代数刻画——这些刻画本身即为具有独立价值的重要结论，其中对有限分配格的中国剩余元组的优雅刻画尤为突出。最后，我们考虑了CRT限制到由两元素代数生成的任意等式类$\mathcal{V}$的情形。我们在此建立了一个（近乎）二分法：除非$\mathcal{V}$是半格类，否则该问题要么是coNP完全的，要么是可解的。