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I define a join-semilattice as a poset, every two elements of which have a supremum.

What is the shortest proof of the fact that supremum binary operation is associative?

I am writing a math book and wish to put there the shortest proof not an arbitrary proof.

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Perhaps universal properties and logical $\&$'s associativity: $$\begin{array}{cl} x\ge (a\wedge b)\wedge c & \iff (x\ge a\wedge b)~\&~(x\ge c) \\ & \iff \big((x\ge a)~\&~(x\ge b)\big)~\&~(x\ge c) \\ & \iff (x\ge a)~\&~\big((x\ge b)~\&~(x\ge c)\big) \\ & \iff (x\ge a)~\&~(x\ge b\wedge c) \\ & \iff x\ge a\wedge (b\wedge c). \end{array}$$ –  anon Oct 17 '12 at 20:02
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@anon: You've confused meets and joins –  porton Oct 17 '12 at 20:07
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So I have. Then replace $\wedge$ with $\vee$. –  anon Oct 17 '12 at 20:12
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Brevity is not always a good thing. In general I’d go for an argument that’s maximally easy to understand, one that is most easily adapted or generalized for later use in related settings, or some compromise between these two criteria. –  Brian M. Scott Oct 17 '12 at 20:13

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