subsemigroup,, submonoid,, and subgroup

Let $S$ be a semigroup, and let $T$ be a subset of $S$.

$T$ is a subsemigroup of $S$ if $T$ is closed under the operation of $S$; that it if $xy\in T$ for all $x,y\in T$.

$T$ is a submonoid of $S$ if $T$ is a subsemigroup, and $T$ has an identity element.

$T$ is a subgroup of $S$ if $T$ is a submonoid which is a group.

Note that submonoids and subgroups do not have to have the same identity element as $S$ itself (indeed, $S$ may not have an identity element). The identity element may be any idempotent element of $S$.

Let $e\in S$ be an idempotent element. Then there is a maximal subsemigroup of $S$ for which $e$ is the identity:

$$eSe=\{exe\mid x\in S\}.$$

In addition, there is a maximal subgroup for which $e$ is the identity:

$$𝒰(eSe)=\{x\in eSe\mid \exists y\in eSe\text{st}xy=yx=e\}.$$

Subgroups with different identity elements are disjoint. To see this, suppose that $G$ and $H$ are subgroups of a semigroup $S$ with identity elements $e$ and $f$ respectively, and suppose $x\in G\cap H$.Then $x$ has an inverse $y\in G$, and an inverse $z\in H$. We have:

$$e=xy=fxy=fe=zxe=zx=f.$$

Thus intersecting subgroups have the same identity element.