MacNeille完备化和profinite完备化在模代数上是一致的
MacNeille completion and profinite completion can coincide on finitely generated modal algebras
摘要：
在Bezhanishvili和Vosmaer之后，我们通过拼凑来自不同作者的结果来证实Yde Venema的一个猜想。具体地说，我们表明，如果$\mathbb{A}$是一个剩余有限的，生成的模态代数，使$\operatorname{HSP}(\mathbb{A})$方程定义的主同余，那么profinite完成$\mathbb{A}$是同构的MacNeille完成，和$\Diamond$是光滑的。这种模态代数的具体例子是自由$\mathbf{K4}$ -代数和自由$\mathbf{PDL}$ -代数。
英文摘要：
Following Bezhanishvili & Vosmaer, we confirm a conjecture of Yde Venema by piecing together results from various authors. Specifically, we show that if $\mathbb{A}$ is a residually finite, finitely generated modal algebra such that $\operatorname{HSP}(\mathbb{A})$ has equationally definable principal congruences, then the profinite completion of $\mathbb{A}$ is isomorphic to its MacNeille completion, and $\Diamond$ is smooth. Specific examples of such modal algebras are the free $\mathbf{K4}$-algebra and the free $\mathbf{PDL}$-algebra.
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