# Every poset is embedded into a meet-semilattice

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. –  Zhen Lin Aug 11 '12 at 18:13
@ZhenLin why don't you post this as an answer. –  user2468 Aug 11 '12 at 18:17
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. –  Jack Schmidt Aug 11 '12 at 18:22
I don't understand the downvote. –  anon Aug 11 '12 at 18:24
@JD It isn't an answer – just a remark. The question deserves an answer phrased in purely order-theoretic terms. –  Zhen Lin Aug 11 '12 at 18:25

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 $\mathscr P(P)$.