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I am trying to show that every finite boolean algebra can be embedded into $\mathcal P(n)$ for some large $n$. Any hints?

  • Embedded? How? With what structure? If just as a set, then isn't it obvious? – MPW Jun 23 '15 at 03:44
  • @MPW: As a Boolean algebra, obviously. – Brian M. Scott Jun 23 '15 at 03:45
  • @BrianM.Scott: Sorry for my ignorance, but what is the natural structure on $\mathcal P(n)$? – MPW Jun 23 '15 at 03:47
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    @MPW: If $X$ is any set, $\wp(X)$ is a Boolean algebra with respect to the operations $\cap$ and $\cup$. – Brian M. Scott Jun 23 '15 at 03:48
  • @BrianM.Scott: Hmm...okay, guess I'll invest some time in reading up on Boolean algebraic structures. Never heard of 'em. EDIT: And yes, I probably shouldn't have commented in the first place. – MPW Jun 23 '15 at 03:58
  • @MPW: They’re pretty important algebraic structures, both in theory and in the real world. The Wikipedia articles here and here with their links and references are one quick starting point. – Brian M. Scott Jun 23 '15 at 04:12

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HINT: Let $B$ be a finite Boolean algebra. Let $A$ be the set of atoms of $B$. Show that every $b\in B$ is uniquely determined by $\{a\in A:a\le b\}$. What are the atoms of $\wp(n)$?

Brian M. Scott
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  • Therefore I must set $n=|A|$, let $f: A \rightarrow n$ be a bijection and define $h: B \rightarrow \mathcal P(n)$ be given by $h(b)=f[{a \in A: a \leq b}]$, right? :) –  Jun 23 '15 at 03:55
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    @Vinicius: You’ve got it. (By the way, I just realized that Artur Tomita’s mathematical ‘great-grandmother’ is my Doktormutter.) – Brian M. Scott Jun 23 '15 at 04:03