Set of Subsemigroups forms Complete Lattice
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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:
| \(\ds a, b\) | \(\in\) | \(\ds T\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \forall K \in \mathbb H: \, \) | \(\ds a, b\) | \(\in\) | \(\ds K\) | Definition of Set Intersection | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds \forall K \in \mathbb H: \, \) | \(\ds a \circ b\) | \(\in\) | \(\ds K\) | Subsemigroups are closed | |||||||||
| \(\ds \leadsto \ \ \) | \(\ds a \circ b\) | \(\in\) | \(\ds T\) | Definition of Set Intersection |
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 $\mathbb H$.
$\blacksquare$