Set of Subsemigroups forms Complete Lattice

Theorem
Let $\struct {S, \circ}$ be a semigroup.

Let $\mathbb S$ be the set of all subsemigroups of $S$.

Then:
 * $\struct {\mathbb S, \subseteq}$ is a complete lattice.

where for every set $\mathbb H$ of subsemigroups of $S$:
 * the infimum of $\mathbb H$ necessarily admitted by $\mathbb H$ is $\ds \bigcap \mathbb H$.

Proof
From Semigroup is Subsemigroup of Itself:
 * $\struct {S, \circ} \in \mathbb S$

Let $\mathbb H$ be a non-empty subset of $\mathbb S$.

Let $T = \bigcap \mathbb H$.

Then:

Hence, from Set of Subsets which contains Set and Intersection of Subsets is Complete Lattice:


 * $\struct {\mathbb S, \subseteq}$ is a complete lattice

where $\ds \bigcap \mathbb H$ is the infimum of $\mathbb H$.