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1. • CommentRowNumber1.
   • CommentAuthorRodMcGuire
   • CommentTimeJul 4th 2017
   • (edited Jul 4th 2017)
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItexI 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](https://ncatlab.org/nlab/show/partition#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 [[atom|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 $\{a, c\} \in atoms(\pi \vee \rho)$ .

   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 $\{a, c\} \in atoms(\pi \vee \rho)$ .

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 4th 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItex> 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”?

3. • CommentRowNumber3.
   • CommentAuthorRodMcGuire
   • CommentTimeJul 4th 2017
   • PermaLink
   Author: RodMcGuire
   Format: MarkdownItex>> 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 4th 2017
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexQuotient 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorNoam_Zeilberger
   • CommentTimeJul 4th 2017
   • PermaLink
   Author: Noam_Zeilberger
   Format: MarkdownItexRod, 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](https://en.wikipedia.org/wiki/Noncrossing_partition#Definition), but I don't understand it well enough to write anything yet.

   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.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJun 3rd 2021
   • (edited Jun 3rd 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have touched the subsection "Of numbers", making some small cosmetic changes: - renamed to "Of natural numbers" - hyperlinked *[[partition function (number theory)|partition function]]* - made the relation to [[Young diagrams]] more explicit - disentangled the discussion of relation to partitions of sets - made sub-subsections to make all this more readily visible <a href="https://ncatlab.org/nlab/revision/diff/partition/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/partition/13">v13</a>, <a href="https://ncatlab.org/nlab/show/partition">current</a>

   I have touched the subsection “Of numbers”, making some small cosmetic changes:

   • renamed to “Of natural numbers”

   • hyperlinked partition function

   • made the relation to Young diagrams more explicit

   • disentangled the discussion of relation to partitions of sets

   • made sub-subsections to make all this more readily visible

   diff, v13, current

7. • CommentRowNumber7.
   • CommentAuthorJ-B Vienney
   • CommentTimeJul 30th 2022
   • PermaLink
   Author: J-B Vienney
   Format: MarkdownItexAdded redirection from composition <a href="https://ncatlab.org/nlab/revision/diff/partition/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/partition/14">v14</a>, <a href="https://ncatlab.org/nlab/show/partition">current</a>

   Added redirection from composition

   diff, v14, current