Is there any concept in monoids that is similar to the concept "conjugate class" in groups? For example, are there any such similar concept in symmetric inverse monoids? Thank you very much.



The concept "conjugate class" in monoids.
There are several different definitions of conjugacy for semigroups. For inverse semigroups the best, in my opinion, definition is this: $a$ is conjugate to $b$ if there exists $t$ such that $t^{-1}at=b$, $tt^{-1}=e, t^{-1}t=f$, $ae=ea=a$, $bf=fb=b$. In this case the conjugacy relation is an equivalence relation and the concept of conjugacy classes is well defined, and easy to describe in the case of symmetric inverse monoids. See, for example, the following article by Akihiro Yamamura, <http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ70_02%2FS1446788700002639a.pdf&code=213e793755d6e9249ea5704c1f57a0b3> .


