Every semigroup containing an ideal subgroup is called a homogroup. Let $(S,\cdot)$ be homomgroup, hence it contains an ideal $I$ that is also a subgroup. It is easy to see that $I$ is the least ideal, a maximal subgroup of $S$, and its identity (denoted by $e\_I$) is a central idempotent of $S$. Now,


(1) Is the ideal subgroup of $S$ **unique**?


(2) Is $e\_I$ **the only** central idempotent of $S$?


(3) Is $I$ the **largest subgroup** of $S$?


(4) Is $e\_I$ the identity element of the subgroup of all **central idempotents** of $S$?


(5) Is $e\_I$ a zero element of the set of all **idempotents** of $S$?



Some questions about homogroups
The answers to these problems are the following:


(1) Yes: the ideal subgroup $I$ is unique. Indeed, if $H$ is another ideal subgroup, then $HI\subset H\cap I$, $H\cap I$ is a subgroup of $H$ and $I$, so $e\_I=e\_H$ and $HI\subset H\cap I\subset H\cup I\subset HI$ implies $H=I$. 


(2,3,4) No: the semigroup $S=\{0,1\}$ endowed with the operation $\min$ provides a counterexample to questions (2), (3), (4).


(5) Yes: for any idempotent $x$ of $S$ and the unique idempotent $e$ of $I$ we get $xe\in I$ and $xexe=xxee=xe$ as $e$ is the central idempotent in $S$. So, $xe$ is an idempotent in the group $I$ and hence $xe=e$, which means that $e$ is zero in the set of all idempotents of $S$.


