Construction of Inverse Completion/Quotient Structure is Commutative Semigroup

Theorem
Then:
 * $\struct {T', \oplus'}$ is a commutative semigroup.

Proof
The canonical epimorphism from $\struct {S \times C, \oplus}$ onto $\struct {T', \oplus'}$ is given by:


 * $q_\boxtimes: \struct {S \times C, \oplus} \to \struct {T', \oplus'}: \map {q_\boxtimes} {x, y} = \eqclass {\tuple {x, y} } \boxtimes$

where, by definition:

By Morphism Property Preserves Closure, as $\oplus$ is closed, then so is $\oplus'$.

By Epimorphism Preserves Associativity, as $\oplus$ is associative, then so is $\oplus'$.

By Epimorphism Preserves Commutativity, as $\oplus$ is commutative, then so is $\oplus'$.

Thus $\struct {T', \oplus'}$ is closed, associative and commutative, and therefore a commutative semigroup.