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Let $A$ be a finite set and let $P$ be a partition of $A$ into subsets. Let $B\subset A$ be some subset of $A$. I want an adjective that describes partitions $P$ such that, for every $S\in P$, either $S\subset B$ or $S\cap B=\emptyset$. In other words, I want an adjective for the partitions that refine the partition $\{B,A\setminus B\}$.

I am tempted to write for example that $P$ is "$B$-local", since it does not pair any elements of $B$ with any elements outside of $B$. But I wonder if this gives the wrong impression.

I would appreciate any reference to any book or paper that uses this concept and gives it a name.

felipeh
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1 Answers1

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The terminology is that $B$ is a saturated subset of $A$ (with respect to the partition $P$). You can find references to this in topological contexts, where it is used commonly in discussions of the quotient topology and quotient maps. It is used in the general context of a partition of any set, not just finite sets.

Lee Mosher
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