TITLE: Is there accepted notation to distinguish $A^2 = \{a^2 \mid a \in A\}$ from $A^2 = \{ab \mid a,b \in A\}$?
QUESTION [6 upvotes]: Let $M$ denote a monoid. Then given a subset $A$ of $M$, we may be interested in one or both of the following subsets of $M$:
$$\{a^2 \mid a \in A\}, \qquad \{ab \mid a,b \in A\}$$
Both could reasonably be denoted $A^2$.

Question. Has anyone ever proposed notation to distinguish these two entities, especially by decorating the notation $A^2$ with further symbols?

REPLY [3 votes]: You could use $A^{\square}$ for the set of squares.  This is common when you want to refer to the set of squares in a finite field.