In process algebras, a congruence is an equivalence that remains valid when any subsystem is replaced by an equivalent one. Whether or not an equivalence is a congruence depends on the set of operators used in building systems from subsystems. Numerous congruences have been found, differing from each other in fine details, major ideas, or both, and none of them is good for all situations. The world of congruences seems thus chaotic, which is unpleasant, because the notion of congruence is at the heart of process algebras. This study continues attempts to clarify the big picture by proving that in certain sub-areas, there are no other congruences than those that are already known or found in the study. First, the region below stability-preserving fair testing equivalence is surveyed using an exceptionally small set of operators. The region contains few congruences, which is in sharp contrast with an earlier result on the region below Chaos-Free Failures Divergences (CFFD) equivalence, which contains 40 well-known and not well-known congruences. Second, steps are taken towards a general theory of dealing with initial stability, which is a small but popular detail. This theory is applied to the region below CFFD.

中文翻译：

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低于保持稳定的公平测试或 CFFD 的所有同余

在过程代数中，同余是当任何子系统被等效子系统替换时仍然有效的等效项。等价是否是同余取决于从子系统构建系统时使用的运算符集。已经发现了许多一致性，它们在细节、主要思想或两者上都各不相同，而且没有一种适用于所有情况。同余的世界因此看起来很混乱，令人不快，因为同余的概念是过程代数的核心。本研究继续尝试通过证明在某些子领域中，除了研究中已知或发现的一致性之外，没有其他一致性，从而阐明全局。首先，使用极少量的操作员对低于保持稳定性的公平测试等效值的区域进行了调查。该区域包含很少的同余，这与混沌无故障发散 (CFFD) 等价下方区域的早期结果形成鲜明对比，后者包含 40 个众所周知和不知名的同余。其次，朝着处理初始稳定性的一般理论采取步骤，这是一个小而流行的细节。这个理论适用于CFFD以下的区域。