Intersection of Submagmas is Largest Submagma
Theorem
Let $\struct {S, \odot}$ be a magma.
Let $\HH$ be a set of submagmas of $\struct {S, \odot}$, where $\HH \ne \O$.
Then the intersection $\bigcap \HH$ of the elements of $\HH$ is the largest submagma of $\struct {S, \odot}$ contained in each element of $\HH$.
Proof
Let $K = \bigcap \HH$.
Let $K_i$ be an arbitrary element of $\HH$.
Then:
\(\ds a, b\) | \(\in\) | \(\ds K\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall i: \, \) | \(\ds a, b\) | \(\in\) | \(\ds K_i\) | Definition of Intersection of Set of Sets | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall i: \, \) | \(\ds a \odot b\) | \(\in\) | \(\ds K_i\) | as $K_i$ is itself a magma, therefore $\odot$ is closed | |||||||||
\(\ds \leadsto \ \ \) | \(\ds a \odot b\) | \(\in\) | \(\ds K\) | Definition of Intersection of Set of Sets |
That is, $\odot$ is closed in $K$.
Hence by definition $\struct {K, \odot}$ is a submagma of $\struct {S, \odot}$.
$\Box$
Now to show that $\struct {K, \odot}$ is the largest submagma of $\struct {S, \odot}$ contained in each element of $\HH$.
Let $N$ be a submagma of $\struct {S, \odot}$ such that:
- $\forall H \in \HH: N \subseteq H$
Then by definition $N \subseteq K$.
Let $x, y \in N$.
Then:
- $x \odot y \in N \implies x \odot y \in K$
Thus any submagma of all elements of $\HH$ is also a submagma of $K$ and so no larger than $K$.
Thus $K = \bigcap \HH$ is the largest submagma of $H$ contained in each element of $\HH$.
$\blacksquare$