In New Structures from Old in Seth Werner's Modern Algebra, the author discusses the closure of a composition $\Delta$ on a set $E$ in $A\subset E:$
Let $\Delta$ be a composition on $E\,.$ A subset $A$ of $E$ is stable for $\Delta,$ or closed under $\Delta,$ if $x\Delta y\in A$ for all $x, y\in A\,.$
If $A$ is stable for a composition $\Delta$ on $E,$ we shall denote the restriction of $\Delta$ to $A\times A$ by $\Delta_A$ when it is necessary to emphasize that it is not the same as the given composition on $E\,.$
He then discusses few examples like the set of integral multiples of a positive integer $m$ is stable for the addition and multiplication composition in $\mathbb Z$; any subset of $\mathbb N$ is stable for the composition $\boldsymbol\vee$ defined on $\mathbb N$ as $x\boldsymbol \vee y~=~\textrm{max}\{x,y\}\,.$
So, all the restricted compositions in the examples above are basically the same composition as defined for the superset viz. $\Delta_A ~=~ \Delta$ (although the domains and ranges are different.)
However, is it possible that the restricted composition $\Delta_A$ is different from $\Delta$ viz. $\Delta_E\ne \Delta$?
Is there any example for such case if it is possible?
No, the very definition of the restricted composition $\Delta_A$ is that $\Delta_A(a,b)=\Delta(a,b)$ for each $a,b\in A$. That is what the phrase "we shall denote the restriction of $\Delta$ to $A\times A$ by $\Delta_A$" means. If $f$ is a function on a set $X$ and $Y\subseteq X$, the restriction of $f$ to $Y$ is the function $g$ on $Y$ defined by $g(y)=f(y)$ for each $y\in Y$. In this case $X=E\times E$, $f=\Delta$, and $Y=A\times A$. So "the restriction of $\Delta$ to $A\times A$" refers to the function $\Delta_A(a,b)=\Delta(a,b)$ with domain $A\times A$.
(Note that it's not really correct to write $\Delta_A=\Delta$, since they are not the same function since they have different domain. What is true is that these two functions take the same values at all points where both of them are defined.)