Does anyone know of a standard notation for the situation when we want to define the ‘same’ function but on a larger or smaller range.
More precisely, if $$f:A to B$$ is a function and $C$ contains $f(A)$, then we can define a function $$g:A to C$$ in the obvious way (put $g(a)=f(a)$).
These functions are different, but it’s extremely cumbersome to have to write out this difference every time.
(By the way, the context this has come up in has been in thinking about algebraic substructures. For example, if $G$ is a group then there is a binary operation $m$ on $G$ satisfying the usual properties. We’d like to define a subgroup by saying that it is a subset $H$ of $G$ which is a group under the restriction of $m$ to $H times H$, but unfortunately this restriction is not a binary operation (because it’s range is technically still $G$). So we’re forced to make an annoying adjustment as described above…)