Abstract The notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4]. An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant. As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

中文翻译：

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主 MS 代数和完美扩展的同余对

摘要 引入了主 MS 代数的同余对概念，比 Beazer 为 K2 代数 [6] 给出的概念更简单。已证明主 MS-代数 L 的同余对应于 [4] 中与 L 相关联的 L 的更简单子结构 L°° 和 D(L) 上的 MS-同余对。一个著名的 Grätzer 问题 [11: 问题 57] 的类比为分布 p-代数制定，它要求根据同余对来表征同余格，这里为主要的 MS-代数（问题 1 ）。与最近对 [2] 中主要 p-代数问题的解决方案不同，这里在主要 MS-代数类上证明了该问题的可能解决方案，虽然不是很具有描述性，但可以简单而优雅. 作为对问题 1 的更具描述性的解决方案的一个步骤，当主 MS 代数 L 是其最大斯通子代数 LS 的完美扩展时，将考虑一个特殊情况。证明这正是当 de Morgan 子代数 L°° of L 是布尔代数 B(L) 的完美扩展时。两个例子说明了这种特殊情况何时发生以及何时不出现。