I'm reading the beginning of https://link.springer.com/chapter/10.1007/3-540-60217-8_13, the authors state that :
"$ \pi = axby$ is a regular pattern but $\tau = axbx$ is not a regular pattern, and their languages are $L(\pi) = \{aubu | \ u \in E^+\}$ and $L(\tau) = \{aubv| \ u,v \in E^+\}$, respectively"
Shouldn't it be :
their languages are $L(\pi) = \{aubv | \ u,v \in E^+\}$ and $L(\tau) = \{aubu| \ u \in E^+\}$, respectively?
I'm asking that because I don't see the point of making regular pattern if in order to constructs their languages we substitute different occurrences of variables by the same symbol of constant…
It is most definitely a typo. Generally mathematicians can correct errors like these in their heads when they read it – they rely on referees or others catching the "smaller" errors during a review stage.