Two elements $x$ and $y$ in a group are said to be conjugate if and only if there exists $t$ satisfying $y = txt^{-1}$. This defines an equivalence relation on the group under question. A variant on this definition is sometimes useful in the context of monoids. Let's say that an element $x$ of a monoid is "conjugate-less-than-or-equal-to" $y$ if and only if there exist elements $s$ and $t$ such that $ts=1$ and $y = txs$. This relation is always a preorder.
Question. Does this relation have a name, and/or is it studied anywhere?