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What's the name of the property of the functions $f$ and $g$ that lets one do this: $g(f(a),f(b))=f(g(a,b))$

For example, I'm looking for a certain class of functions that do this: $f(a) \oplus f(b)=f(a \oplus b)$. In this case, $g$ is the binary XOR operator which also happens to be symmetric, but symmetry isn't necessary.

Sorry for the simple question that's probably been asked a million times, but it's hard to find the name of something of which one doesn't know the name. Thanks!

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    $\begingroup$ Depends on the context, but you can think of $f$ as a $g$-homomorphism, where you think of $g$ as a binary operation. Or you can similarly think of it as like a distributive law of $f$ over $g$. Not sure it has a name in general, however. $\endgroup$
    – Thomas Andrews
    Commented Jan 16, 2016 at 21:50
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    $\begingroup$ Your particular example is actually much, much nicer than the general case you're asking about. The binary strings form a vector space with XOR as addition (scalar multiplication is by 1 or 0 treated as integers mod 2) and $f$ is a linear transformation, hence representable by a matrix. $\endgroup$
    – Matt Samuel
    Commented Jan 17, 2016 at 0:28

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Even if $g$ has two and not just one inputs, one usually says "$f$ commutes with $g$", but I'd prefer "$f$ is a homomorphism (in fact: endomorphism) of the magma (or whatever structure it is) $(X, g)$" (where $g\colon X\times X\to X$ and $f\colon X\to X$).

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