Talk:Schützenberger group

left and right Schutzenberger groups are isomorphic or antiisomorphic?

"In fact, there are two Schutzenberger groups associated with a given H-class and each is antiisomorphic to the other." I supposte the "two" Schutzenberger groups means the left and right Schutzenberger group for H. Let $\Gamma _{l}(H)=\{y_{s}|sH=H,s\in S^{1}\}$ be the left one, $\Gamma _{r}(H)=\{z_{s}|Hs=H,s\in S^{1}\}$ the right one.

Then, only this natural isomorphism $\varphi :\Gamma _{r}(H)\rightarrow \Gamma _{l}(H)$ comes to my mind: $\varphi (z_{s})=y_{t}\Leftrightarrow ths=h$. If $\varphi (z_{s})=y_{p},\varphi (z_{t})=y_{q}$, then $phs=h,qht=h$, so $qphst=h$ meaning that $\varphi (z_{st})=y_{qp}$.

Under this mapping, left and right Schutzenberger groups are isomorphic, not antiisomorphic. Moreover, the isomorphism can be used for proving that all Green H-classes in a regular D-class are isomorphic groups, while the antiisomorphism is not directly required. So, I think the article introduction is somewhat confusing. Could someone describe the antiisomorphism explicitly in the article, or add a direct reference? Or is the antiisomorphism just $(()^{-1}\circ \varphi )$? — Preceding unsigned comment added by Jsedenka (talk • contribs) 12:28, 9 February 2013 (UTC)