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I "discovered" a few minutes ago that every poset can be embedded into a meet-semilattice.

Let $\mathfrak{A}$ be a poset. Then it is embedded into the meet-semilattice generated by sets $\{ x \in \mathfrak{A} \mid x \le a \}$ where $a$ ranges through $\mathfrak{A}$.

I'm sure I am not the first person who discovered this. Which book could you suggest to read about such things?

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    This is a special case of the Yoneda embedding, which you can read about in any category theory textbook.2012-08-11
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    @ZhenLin why don't you post this as an answer.2012-08-11
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    @ZhenLin: Probably really stupid, but to the moment, I don't understand how is this related with Yoneda embedding (I am yet a beginner with CT). Does Yoneda embedding produce a semilattice (the same as mine), lattice, or whatever in this particular case?2012-08-11
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    Section 3.4 of Stanley's Enumerative Combinatorics Vol 1 might be helpful. It focusses on finite posets, but gives some nicer results. I believe your observation is implicit in that section.2012-08-11
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    I don't understand the downvote.2012-08-11
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    @JD It isn't an answer – just a remark. The question deserves an answer phrased in purely order-theoretic terms.2012-08-11
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    I've found that every poset can be embedded into a completely distributive lattice: http://en.wikipedia.org/wiki/Completely_distributive_lattice2012-08-11

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This result is mentioned for example as theorem 1.1 in chapter 1 of J.B. Nation's "Revised Notes on Lattice Theory". See also theorem 2.2 in chapter 2. One advantage of Nation's text is that it is freely available.

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Simply searching for every poset embedded in Google Books returns some reasonably looking references. (Of course, you can try some other similar search queries.)

For example Theorem 1.11 in the book Steven Roman: Lattices and Ordered Sets uses precisely the embedding you suggested to show that every poset $P$ can be order embedded in a powerset $\mathscr P(P)$.