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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 $S$?

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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 $S$.

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  • $\begingroup$ In fact the map $\phi: x\mapsto e_Ix$ is a (retracting) homomorphism from $S$ onto $I$ which implies (1) and (5). $\endgroup$
    – user6976
    Commented Jan 14, 2018 at 19:12

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