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 $\HH$.

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: \, $

$\ds a, b$

$\in$

$\ds K_i$

$\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 \ \ $

$\ds a \odot b$

$\in$

$\ds K$

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 $\HH$.

$\blacksquare$