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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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