Jimmie D. Lawson:
Maximal Ol'shanskii semigroups 
Journal of Lie Theory, vol. 4 (1), p. 17-35 

In this article we show for
simple complex groups $G_\C$ that if a subsemigroup 
$S$ contains a  
real form $G$ such that $G$ is not isolated in 
$S$, then $S$ contains  
a minimal Ol'shanski\u\i\ semigroup and is contained in a maximal  
Ol'shanski\u\i\ semigroup.  It follows in particular that maximal  
Ol'shanski\u\i\ semigroups are maximal semigroups.