An inverse semigroup $S$ is a semigroup in which every element $x \in S$ has a unique inverse $y \in S$ such that $x = xyx$ and $y = yxy$. Are there some references characterizing the maximal sub-inverse-semigroups of $M\_n(\mathbb{C})$ and $M\_n(F\_p)$? Here $M\_n(\mathbb{C})$ is the monoid of all $n \times n$ matrices over $\mathbb{C}$ and $M\_n(F\_p)$ is the monoid of all $n \times n$ matrices over the $p$-element field $F\_p$. Thank you very much.



Maximal sub-inverse semigroups of $M_n(\mathbb{C})$ and $M_n(F_p)$
Maybe this article will be useful for you:


Zhu Yong Wen, Inverse Semigroups of Matrices. Journal of Mathematical Research & Exposition, Aug., 2008, Vol. 28, No. 3, pp. 549–557.


