# What is a thing that is homomorphized?

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.

$$A$$ 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.

• 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 $B$. – Dietrich Burde Dec 30 '20 at 20:14
• Categorically, $A$ is an object and $\pi$ is morphism, and $\pi$ is a member of $\text{Hom}(A,Z)$ for some object $Z$. – Ty Jensen Dec 30 '20 at 20:15
• I usually say "$\pi$ is a homomorphism out of $A$". – Kaj Hansen Dec 30 '20 at 20:24
• 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?) – Stinking Bishop Dec 30 '20 at 20:55

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.

• thats exactly the type of situation i'm considering – frogeyedpeas Dec 30 '20 at 20:16
• @frogeyedpeas Oh interesting, can you elaborate? – Noah Schweber Dec 30 '20 at 20:18
• @frogeyedpeas I still think there's no standard term but I've given a couple suggestions. – Noah Schweber Dec 30 '20 at 20:24
• 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. – frogeyedpeas Dec 30 '20 at 20:24
• 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. – frogeyedpeas Dec 30 '20 at 20:26