Find a signature for vector spaces such that the homomorphisms correspond to linear maps.

My attempt: Let $F$ denote the corresponding field and consider the structure $\{0;+,-,c_F\}$, where $0$ is a constant, $+$ is a binary function symbol and $-$ and $c_F$ are unary function symbols (for each $c\in F$).

But I don't know how to show the following:

Homomorphisms correspond to linear maps.

I think I understand it intuitively but since I'm new to model theory, I don't know how to precisely write it.


1 Answer 1


A homomorphism between structures $X,Y$ of the given signature would be a function $f:X\to Y$ which satisfies $$f(0)=0,\quad f(x_1+x_2)=f(x_1)+f(x_2),\quad f(-x)=-f(x),\quad f(c_Fx)=c_F\,f(x)$$ for all $x,x_1,x_2\in X$ and $c\in F$.

If $X,Y$ happens to be vector spaces (with the usual interpretation of the operation symbols) then this amounts exactly to $f$ being a linear map.
By the way, in that case it suffices that $f$ preserves addition and scalar multiplication, then the vector space axioms ensure that it preserves zero and subtraction as well.


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