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I was just reading a text on first-order modal logic and found it very interesting to think of the distinction between rigid and non-rigid designators as a failure of associativity between the modal operator of, for instance $\mathcal{M},\Gamma\Vdash_v \Box F(c)$ and the valuation function. If the valuation function is taken as primary then we get rigid designation, and otherwise we get non-rigid designation.

But this led me to wonder, are there any interesting examples of two distinct operations that associate? I know the identity operation would associate with any other operation, and the "sends to 0" constant operation would associate with multiplication. But these are both fairly uninteresting examples--are there others where you really learn something by noticing that the operations associate? I suppose we could regard the constant multiple rule for derivatives and integrals as such an example, yes?

Addem
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  • Can you state a bit more formally what you mean by "two operations associating"? The standard meaning of "associative" doesn't easily generalize to multiple operations that I can see... – Malice Vidrine May 12 '16 at 17:13
  • I mean probably what you have in mind, that for operations $\ast$ and $\star$ and any $x,y,z$, we get $(x\ast y)\star z = x\ast (y\star z)$. To make it more precise and general enough perhaps I can't phrase it in terms of operations but would have to instead phrase it in terms of function composition. But I'm leaving the precise meaning somewhat open-ended intentionally, to leave room for all possible interesting answers. – Addem May 12 '16 at 17:26

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