Cases where string concatenation is commutative

Question

Strings are known to not satisfy the commutative property. pq is not equal qp unless p is empty string. Is there a case where pq can be qp where both are non empty strings and p and q are distinct

Answer 1

Simple solution: $a\cdot aa = aa \cdot a$?

In general there is a characterization due to Lyndon and Schützenberger. For nonempty words $x,y\in \Sigma^+$ the following are equivalent:

• $xy=yx$
• there exists $z\in \Sigma^+$ such that $x=z^k$ and $y=z^\ell$ for some $k,\ell >0$.
• there exist $i,j>0$ such that $x^i = y^j$.

This means that $x$ and $y$ commute ($xy=yx$) iff they are powers of the same string, like $bbabba\cdot bbabbabba = bbabbabba \cdot bbabba$. So, basically the examples will not get more complicated than the proposal $a\cdot aa = aa \cdot a$ above.