TITLE: The concept "conjugate class" in monoids.
QUESTION [11 upvotes]: 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.

REPLY [3 votes]: This seems to be a relevant paper: Kudryavtseva and Mazorchuk - On three approaches to conjugacy in semigroups.