Given a partition P on a set X, and seen as how every set is a subset of itself which means that every set in a partition is contained in itself does it follow that every partition contains itself as a possible refinement of itself? - It appears to be true with the examples I have tested but I am unsure if it holds for all other nonempty sets. -Thanks.
2026-03-26 12:44:32.1774529072
Is every Partition a refinement of itself
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Yes. The refinement relation on the set of partitions of $X$ is reflexive, and every partition is a refinement of itself. The argument is exactly the one you give.
By "inheriting" the properties from the $\subseteq$ relation, it is not hard to show that this is in fact a partial order on the set of partitions of $X$, using similar arguments.