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

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.