Embedding Theorem

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Context

The following is a frequently occurring circumstance in the field of abstract algebra.

We have a magma $\left({T_1, \circ}\right)$.
$\left({T_1, \circ}\right)$ is isomorphic to another magma $\left({T_2, *}\right)$.
$\left({T_2, *}\right)$ is embedded in a magma $\left({S_2, *}\right)$.
We want to embed $\left({T_1, \circ}\right)$ in its own magma $\left({S_1, \circ}\right)$ such that $\left({S_1, \circ}\right) \cong \left({S_2, *}\right)$.


This can always be done, as this theorem shows.


Theorem

Let:

$(1): \quad \left({T_2, \oplus_2}\right)$ be a submagma of $\left({S_2, *_2}\right)$
$(2): \quad f: \left({T_1, \oplus_1}\right) \to \left({T_2, \oplus_2}\right)$ be an isomorphism

then there exists:

$(1): \quad$ a magma $\left({S_1, *_1}\right)$ which algebraically contains $\left({T_1, \oplus_1}\right)$
$(2): \quad g: \left({S_1, *_1}\right) \to \left({S_2, *_2}\right)$ where $g$ is an isomorphism which extends $f$.


Proof

In order to reduce notational confusion, the composition of two mappings $f$ and $g$ is denoted $f \bullet g$ instead of the more usual $f \circ g$.


There are two cases to consider: when $T_1$ and $S_2$ are disjoint, and when they are not.


Suppose $T_1$ and $S_2$ are disjoint.

Let $S_1$ be the set $T_1 \cup \left({S_2 \setminus T_2}\right)$.

Then we can define the mapping $h: S_2 \to S_1$ as:

$\forall x \in S_2: h \left({x}\right) = \begin{cases} x & : x \in S_2 \setminus T_2 \\ f^{-1} \left({x}\right) & : x \in T_2 \end{cases}$

Because $T_1$ and $S_2$ are disjoint, $h$ can be seen to be a bijection.

What $h$ is doing is that it effectively "slots" $T_1$ into the gap in $S_2$ that was taken up by the removed $T_2$.


We are going to show that $\left({T_1, \oplus_1}\right)$ is embedded in $\left({S_1, *_1}\right)$.

The operation $*_1$ is defined as the transplant of $*_2$ under $h$.

Thus by the Transplanting Theorem:

$\forall x, y \in S_1: x *_1 y = h \left({h^{-1} \left({x}\right) *_2 h^{-1} \left({y}\right)}\right)$


Let $x, y \in T_1$.

Then, from the definition of $h$ above:

$h^{-1} \left({x}\right) = \left({f^{-1} \left({x}\right)}\right)^{-1} = f \left({x}\right)$

and similarly:

$h^{-1} \left({y}\right) = f \left({y}\right)$


So:

\(\displaystyle x *_1 y\) \(=\) \(\displaystyle h \left({f \left({x}\right) *_2 f \left({y}\right)}\right)\) Transplanting Theorem
\(\displaystyle \) \(=\) \(\displaystyle h \left({f \left({x}\right) \oplus_2 f \left({y}\right)}\right)\) as $*_2$ is embedded in $\oplus_2$
\(\displaystyle \) \(=\) \(\displaystyle h \left({f \left({x \oplus_1 y}\right)}\right)\) Morphism Property of $\oplus_1$ under $f$
\(\displaystyle \) \(=\) \(\displaystyle f^{-1} \left({f \left({x \oplus_1 y}\right)}\right)\) Definition of $h$ on elements of $T_1$
\(\displaystyle \) \(=\) \(\displaystyle x \oplus_1 y\) Inverse of Inverse of Bijection


proving that $x \oplus_1 y$ is closed.

Thus:

$\left({T_1, \oplus_1}\right) \subseteq \left({S_1, *_1}\right)$

that is, $\left({T_1, \oplus_1}\right)$ is embedded in $\left({S_1, *_1}\right)$.


Now, let $g = h^{-1}$.

By the definition of $*_1$, $g$ is an isomorphism from $\left({S_1, *_1}\right)$ onto $\left({S_2, *_2}\right)$.

Then:

\(\displaystyle x \in T_1\) \(\implies\) \(\displaystyle f \left({x}\right) \in T_2\) by definition of $f$
\(\displaystyle \) \(\implies\) \(\displaystyle h \left({f \left({x}\right)}\right) = f^{-1} \left({f \left({x}\right)}\right) = x\) by definition of $h$


Thus:

\(\displaystyle g \left({x}\right)\) \(=\) \(\displaystyle g \left({h \left({f \left({x}\right)}\right)}\right)\) from above
\(\displaystyle \) \(=\) \(\displaystyle h^{-1} \left({h \left({f \left({x}\right)}\right)}\right)\) by definition of $g$
\(\displaystyle \) \(=\) \(\displaystyle f \left({x}\right)\) Inverse of Inverse of Bijection


So $g$ is an extension of $f$.

We have therefore proved that the embedding theorem holds when $T_1$ and $S_2$ are disjoint.

$\Box$


Next, suppose $T_1$ and $S_2$ are not disjoint.

We can use Exists Bijection to a Disjoint Set to assume the existence of a bijection $k: S_2 \to S_3$ such that $S_3 \cap T = \varnothing$.


Let $*'$ be the transplant of $*$ under $k$.

Let:

$T_3 = \left\{{k \left({x}\right): x \in T_2}\right\}$


Then:

\(\displaystyle x, y \in T_3\) \(\implies\) \(\displaystyle k^{-1} \left({x}\right), k^{-1} \left({y}\right) \in T_2\) by definition of $k$
\(\displaystyle \) \(\implies\) \(\displaystyle k^{-1} \left({x}\right) * k^{-1} \left({y}\right) \in T_2\) by definition of $*$ in $T_2$
\(\displaystyle \) \(\implies\) \(\displaystyle x *' y = k \left({k^{-1} \left({x}\right) * k^{-1} \left({y}\right)}\right) \in T_3\) Transplanting Theorem


So $T_3$ is closed under $*'$.


Let $*' {\restriction_{T_3} }$ be the operation on $T_3$ induced by $*'$.

Then $\left({T_3, *' {\restriction_{T_3} } }\right)$ is embedded in $\left({S_3, *'}\right)$.

Thus $k \circ f$ is an isomorphism from $\left({T_1, \oplus_1}\right)$ onto $\left({T_3, *' {\restriction_{T_3} } }\right)$.

$S_3$ has been constructed so as to be disjoint from $T_1$.

In such a case it has already been shown that the embedding theorem holds.

That is:

$(1): \quad$ there exists a magma $\left({S_1, *_1}\right)$ containing $\left({T_1, \oplus_1}\right)$ algebraically
$(2): \quad$ there exists an isomorphism $g_1$ from $\left({S_1, *_1}\right)$ to $\left({S_3, \oplus_1}\right)$ extending $k \bullet f$.


Let $g = k^{-1} \bullet g_1$.

Let $x \in T$.

Then:

\(\displaystyle g \left({x}\right)\) \(=\) \(\displaystyle k^{-1} \left({g_1 \left({x}\right)}\right)\) definition of $g$
\(\displaystyle \) \(=\) \(\displaystyle k^{-1} \left({k \left({f \left({x}\right)}\right)}\right)\) as $g_1$ extends $k \bullet f$
\(\displaystyle \) \(=\) \(\displaystyle f \left({x}\right)\) Inverse of Inverse of Bijection

and so $g$ extends $f$.


As $k$ is an isomorphism from $\left({S_2, *}\right)$ onto $\left({S_3, *'}\right)$, then:

$(1): \quad g$ is an isomorphism from $\left({S_1, *_1}\right)$ onto $\left({S_2, *}\right)$
$(2): \quad g$ extends $f$.


Hence the result.

$\blacksquare$


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