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A set $P \subseteq \mathcal{P}(X)$ is a partition of $X$ if and only if all of the following conditions hold:

  1. $\emptyset \notin P$
  2. For all $x,y \in P$, if $x \neq y$ then $x \cap y = \emptyset$.
  3. $\bigcup P = X$

I have read many times that the partitions of a set form a lattice, but never really considered the idea in great detail. Where can I learn the major results about such lattices? An article recommendation would be nice.

I'm also interested in the generalization where condition 3 is disregarded.

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  • $\begingroup$ the generalization you are referring to is sometimes called the "lattice of subpartitions" $\endgroup$
    – Hans
    Feb 27, 2023 at 16:48

3 Answers 3

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G. Birkhoff, Lattice Theory. Providence, Rhode Island, 1967,

Chapt.4, sec.9.

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George Grätzers book General Lattice Theory has a section IV.4 on partition lattices, see page 250 of this result of Google books search. A more recent version of the book is called Lattice Theory: Foundation.

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The Logic of Partitions: Introduction to the Dual of the Logic of Subsets

I just stumbled upon it a couple of days ago, but it looks to be a great reference.

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