Let $M$ be a reductive linear algebraic monoid with a zero and with unit group $G$ of rank $r$. Fix a maximal torus $T$ in $G$. Then, $M$ can be embedded into a space of matrices $M_n$ for some $n$.
$x \in M$ is regular if its centralizer in $G$ is as small as possible, i.e. $\mathrm{dim} C_G(x) = r.$
$x \in M$ is semisimple if it is $G$-conjugate to an element in $\overline{T}$. If one embeds $M$ into $M_n(k)$, then $x$ would be a diagonalizable matrix.
$x \in M$ is (quasi)-unipotent if $ue$ is unipotent in $H(e)$, the unit group of $eMe$ for some idempotent $e$. If one embeds $M$ into $M_n$, then $x$ would be a matrix whose only eigenvalues are 0 and 1.
My question is how much of the theory of regular elements in algebraic groups carry over? In particular,
- If $x$ is regular semi-simple, is it contained in $|W|$ many Borel submonoids $\overline{B}$, i.e. closure of Borel subgroups $B$ in $M$?
- If $x$ is regular unipotent, is it contained in a unique Borel submonoid $\overline{B}$?
- If $x$ is regular, is it contained in finitely many Borel submonoids?
My general feeling is "yes" to all of these but I'm not so sure of the proofs and I'm interested to hear whatever references are out there on this topic.