Definition of Substructures According to Wikipedia

Question

I have a quick question about the definition for substructures in mathematics from its Wikipedia page:

What does it mean by functions and relations are "traces" of the functions and relations of the bigger structure? What does traces mean?

Answer 1

If $X\subseteq Y$ and $f:Y^n\rightarrow Y$, we can look at how $f$ behaves on $X$: that is, look at the function $$f_X: X^n\rightarrow Y.$$ Note that a priori this isn't a function from $X^n$ to $X$: $f$ might "go outside" of $X$, even if we just feed it things from $X$.

This $f_X$ is the restriction of $f$ to $X$ (I've never heard it called the "trace" before, but I wouldn't be too surprised if someone calls it that). If $ran(f_X)\subseteq X$, then $(X;f_X$) is indeed a structure, and "lives inside" $(Y; f)$ in an obvious way.

We can also take restrictions of relations as well: for $R\subseteq Y^n$ an $n$-ary relation on $Y$, its restriction to $X$ is $$R_X=\{(x_1,..., x_n)\in X^n: (x_1,...,x_n)\in R\}.$$ Note that this can be more snappily represented as $R_X=R\cap X^n$. Similarly, conflating a function with its graph we have $f_X=f\cap X^{n+1}$.