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I see Noam Zeilberger has been updating partition.
That article yet has to mention how a partition is just a quotient object but I’m not sure how to integrate this.
At the section the_lattice_of_partitions_of_a_finite_set I would like to add something like the following but I hesitate because I might conflict with Noam’s further intents and it really needs to be better worded and stated more precisely. Also something should probably be said about coatomisticy.
The lattice of partions of a set $S$ of size $n$, $\Pi(S)$, is atomistic. An atom corresponds to a single equality. It contain a two element set as one block while all other blocks are singletons. Thus $\Pi(S)$ has $n * (n - 1) / 2$ atoms.
Being atomistic means that any partition $\pi$ is the meet of its set of atoms: $\pi = \bigvee atoms(\pi)$.
In terms of atoms, the meet of partitions corresponds to the intersection of their atoms: $atoms(\pi \wedge \rho) = atoms(\pi) \cap atoms(\rho)$,
For the join of partitions new atoms may emerge through transitive closure and thus we have
$atoms(\pi \vee \rho) = transClos(atoms(\pi) \cup atoms(\rho))$where if $\{a, b\} \in atoms(\pi$) and $\{b, c\} \in atoms(\rho)$ then $\{a, c\} \in atoms(\pi \vee \rho)$ .
That article yet has to mention how a partition is just a quotient object
Isn’t it said right there under “Of sets”?
That article yet has to mention how a partition is just a quotient object
Isn’t it said right there under “Of sets”?
Huh? Where is quotient object mentioned or linked to that article?
With my “integration” comment I was more concerned with whether all non-set use of “partition” also correspond to quotient objects.
Quotient objects of $S$ are given by surjections out of $S$. (I didn’t know you were being so literal.)
To be on the safe side: the quotient objects are more accurately described as regular quotient objects (they’re the same thing in $Set$ of course). For categories of algebras over $Set$, there is again a bijective correspondence between congruence equivalence relations and regular quotients. The concept of exact category is a useful context where this is generalized.
Rod, I did not have any big plans, so feel free to add/edit however you see fit. I started thinking a bit about the lattice of noncrossing partitions, but I don’t understand it well enough to write anything yet.
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