Embedding Theorem/Motivation

Motivation for Embedding Theorem

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

We have a magma $\struct {T_1, \circ}$.

$\struct {T_1, \circ}$ is isomorphic to another magma $\struct {T_2, *}$.

$\struct {T_2, *}$ is embedded in a magma $\struct {S_2, *}$.

We want to embed $\struct {T_1, \circ}$ in its own magma $\struct {S_1, \circ}$ such that $\struct {S_1, \circ} \cong \struct {S_2, *}$.

This can always be done, as the Embedding Theorem theorem shows.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Theorem $8.1$