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For any set $X$ we denote by $\text{Part}(X)$ the set of all partitions of $X$, ordered by the refinement ordering. It is well known that this is a complete lattice for all sets $X$.

Let $L$ be a finite lattice. Is there a finite set such that there is an injective lattice homomorphism $\varphi:L\to\text{Part}(S)$ for some finite set $S$?

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Yes, this is apparently a fairly hard theorem of Pudlak and Tuma (or at least I assume it is hard, because it seems to have been an open problem for decades before they finally proved it in 1980).

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Quoting Wehrung and Semenova, they mention "... the result of Ph. M. Whitman [19] published in 1946 that every lattice embeds into the partition lattice of a set." Here reference [19] is

  • Ph. M. Whitman, Lattices, equivalence relations, and subgroups, Bull. Amer. Math. Soc. 52 (1946), 507–522.
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    $\begingroup$ Whitman's result does not guarantee that $S$ is finite, which makes the problem much easier. $\endgroup$
    – Eric Wofsey
    Commented Jul 7, 2015 at 10:57
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    $\begingroup$ I realize. I'll leave my answer in case it's still useful. $\endgroup$
    – Todd Trimble
    Commented Jul 7, 2015 at 11:01

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