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.
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 . 


As so often happens when you generalize from groups to monoids, it gets more complicated. Conjugacy is, of course, an equivalence relation on any group X. Two elements x, y are conjugate if and only if there exists g in X with gx = yg. This last equation makes sense in a monoid, but does not define an equivalence relation: it's not symmetric. Here's one way to think about it. With every monoid X there is associated a category C(X). The objects of C(X) are the elements of X, and a map x > y in C(X) is an element g of X such that gx = yg. Composition is defined in the obvious way. When X is a group, C(X) is a groupoid (i.e. a category in which every map is invertible). Now, in a category there are at least two natural equivalence relations on the collection of objects: isomorphism, and being in the same connectedcomponent. When the category is a groupoid, these are the same. Apply this observation to C(X), where X is your monoid. It tells us that there are two natural equivalence relations on the elements of X:
When X is a group, these two equivalence relations are the same thingconjugacy. Postscript The category C(X) can be understood abstractly as follows. A oneobject category is the same thing as a monoid. Functors between oneobject categories are the same thing as monoid homomorphims. A homomorphism from N (the additive monoid of natural numbers) to a monoid X is the same thing as an element of X. Thus, an element of X is the same thing as a functor N > X. In other words, the elements of X are the objects of the functor category X^N = [N, X]. And this functor category is exactly C(X). 


This seems to be a relevant paper: 

