Is there accepted notation to distinguish $A^2 = \{a^2 \mid a \in A\}$ from $A^2 = \{ab \mid a,b \in A\}$?

Question

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?

Answer 1

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.

Answer 2

Without any further warning, I would interpret the notation $A^2$ as the set $\{ab \mid a, b \in A\}$. The reason is that $\mathcal{P}(M)$, the set of subsets of $M$, is a monoid under the multiplication given, for $S, T \in \mathcal{P}(M)$, by $$ ST = \{st \mid s \in S, t \in T\} $$ and thus the notation $S^2$ is just the square of $S$ in this monoid.

That being said, you may introduce a different meaning for $A^2$ if you give a local definition, but I would rather suggest something like $SQ(A)$ for the set of squares of elements of $A$ if there is any risk of ambiguity.