Standard Notation for Binary Operator Applied to Each Element in a Set (Against a Fixed Element)

Question

Suppose that I have a binary operator $\oplus$. Normally, one uses the notation $x \oplus y$ to denote the result of apply $\oplus$ to $x$ and $y$. Suppose that I had a set of elements $S$ and a fixed element $y$, is there standard notation to denote the set $\{ x \oplus y : x \in S\}$ (i.e the set obtained by applying the operator to each element in $S$ with $y$)? I was thinking perhaps $\oplus_y(S)$ (hoping to mimic the abuse of notation used for functions on one variables), but that looks really strange to me. Is there anything more standard?

Answer 1

It should be $S \oplus y$. If your operator is commutative, it would be even better to write $y \oplus S$, as it is more common to have the element on the left and the set on the right. However, if $x \oplus y \neq y \oplus x$, then also the first one should be ok.