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This is a purely syntax question. If $A$ is a structure and $\pi$ is a homomorphism sending $A$ to something.

We say that $\pi$ is a homomorphism of $A$.

Similarly we might want to say $A$ is a ____ of $\pi$.

What is that word? I found myself writing "$A$ is a structure for which $\pi$ is a homomorphism" but that really was verbose and convoluted.

What I had considered:

$A$ is a domain, but this doesn't carry the importance that $\pi$ is a homomorphism of $A$, it only says that $\pi$ is a function which maps $A$ to something.

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    $\begingroup$ We say that $\pi:A \rightarrow B$ is a homomorphism. So $B$ is a homomorphic image of $A$. I am not sure if you would say that $A$ is a homomorphic preimage? I don't agree with "$\pi$ is a homomorphism of $A$". It is only a homomorphism from $A$ to $B$. $\endgroup$ – Dietrich Burde Dec 30 '20 at 20:14
  • $\begingroup$ Categorically, $A$ is an object and $\pi$ is morphism, and $\pi$ is a member of $\text{Hom}(A,Z)$ for some object $Z$. $\endgroup$ – Ty Jensen Dec 30 '20 at 20:15
  • $\begingroup$ I usually say "$\pi$ is a homomorphism out of $A$". $\endgroup$ – Kaj Hansen Dec 30 '20 at 20:24
  • $\begingroup$ If $\pi:A\to B$ is a map, then it has no knowledge about the structure on $A$ or on $B$. Thus, $A$ can only be a domain of $\pi$. Similarly to a letter $\pi$ in the post office - you can only have a "sender" $A$ and "recipient" $B$ of the letter, which may be "business partners" (if the letter is a business letter) or "sweethearts" (if the letter is a love letter) but you would never say "$A$ is a business partner of $\pi$" or "$A$ is a sweetheart of $\pi$. (That information is encoded outside of the letter itself. The letter may only say "Yes" - which one is it then?) $\endgroup$ – Stinking Bishop Dec 30 '20 at 20:55
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I'm not aware of any context where we have a fixed map $\pi$ but consider different structures on the domain of $\pi$ with respect to which $\pi$ remains a homomorphism, and this is the only sort of context I can think of where "$A$ is the domain of $\pi$" would seem inappropriate to me. So I don't think such a term exists.

(Contrast this with "$\pi$ is a homomorphism from $A$" - this language is useful since we do often consider multiple homomorphisms out of the same structure.)


EDIT: you clarify that this is in fact the context you're in. I stand by my claim that this context is rare enough that there probably isn't a standard term in the literature. That said, the term which fits the bill most cleanly for me is "compatible," and I would suggest the phrase "$A$ is a compatible domain for $\pi$," or - if you only ever talk about varying domains and not codomains - "$A$ is compatible with $\pi$." But of course unless I'm wrong and there is after all a standard term this should be defined explicitly in your text.

Alternatively, a phrase like "$A$ is a domain object for $\pi$" has a categorical flavor which may be desired (this is motivated by Ty Jensen's comment above). Personally I think this sort of thing is clunkier, but I could see it being preferred.

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  • $\begingroup$ thats exactly the type of situation i'm considering $\endgroup$ – frogeyedpeas Dec 30 '20 at 20:16
  • $\begingroup$ @frogeyedpeas Oh interesting, can you elaborate? $\endgroup$ – Noah Schweber Dec 30 '20 at 20:18
  • $\begingroup$ @frogeyedpeas I still think there's no standard term but I've given a couple suggestions. $\endgroup$ – Noah Schweber Dec 30 '20 at 20:24
  • $\begingroup$ The situation arises when you ask a question of the type "what is this a homomorphism of". My particular use case was extremely specific: I started with a associative trinary function $f(x,y,z)=xy+yz+xz$ and observed $f(x,y,a+b) \ne f(x,y,a) + f(x,y,b)$ so that got me wondering what operation does the aforementioned actually distribute over. So now i thought to ask the question geometrically as, what closed surface with a binary operation * on it, is such that f(x,y,z)=xy+yz+xz is a homomorphism of the surface. $\endgroup$ – frogeyedpeas Dec 30 '20 at 20:24
  • $\begingroup$ at this point I wanted to be more direct with my verbiage so i wanted to say something akin to what are the _____ of xy+yz+xz, perhaps saying "what are the domain objects of xy+yz+xz" is a way to put it. $\endgroup$ – frogeyedpeas Dec 30 '20 at 20:26

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