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I am introducing a new operation for countable sets, say $\mathcal{X}(\cdot)$ is the operation. I have proved that for two sets $S_1, S_2$ we have: $$\mathcal X(S_1 \cup S_1) = \mathcal{X}(S_1) \cup \mathcal{X}(S_2).$$

How can I name this property in set-theory language?

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    $\begingroup$ Something like linearity or linear invariance, but for sets $\endgroup$ May 25, 2020 at 17:32
  • $\begingroup$ I don't know why, but it's not uncommon for people to tag questions literally about terminology under "notation". $\endgroup$
    – Asaf Karagila
    May 25, 2020 at 17:37
  • $\begingroup$ it looks like a homomorphism $\endgroup$ May 25, 2020 at 17:59
  • $\begingroup$ @AsafKaragila That's mainly for people whose primary language isn't English, couldn't find anything under 'naming' etc. and went for 'notation' mistakenly (I guess) $\endgroup$ May 25, 2020 at 21:34

1 Answer 1

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It is distributive over unions.

Much like how $a\times(b+c)=a\times b+a\times c$ is the distributivity of $\times$ over $+$.

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  • $\begingroup$ ah, thanks ! that helps $\endgroup$ May 25, 2020 at 17:38
  • $\begingroup$ note that $\times$ is a binary operation and $\mathcal X$ is not $\endgroup$ May 25, 2020 at 17:51
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    $\begingroup$ @J.W.Tanner: Do you prefer the case of scalar multiplication in vector spaces? (Which is an unary operator on the vector space.) $\endgroup$
    – Asaf Karagila
    May 25, 2020 at 17:52
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    $\begingroup$ that's a good point, @AsafKaragila, but I will note that the definition of distributive property in Wikipedia says it's for binary operators $\endgroup$ May 25, 2020 at 17:53
  • $\begingroup$ For me, distributivity of an unary operation over a binary operation is (a kind of) linearity. $\endgroup$ May 25, 2020 at 17:55

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