Epimorphism Preserves Semigroups

Theorem
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures.

Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an epimorphism.

Let $\left({S, \circ}\right)$ be a semigroup.

Then $\left({T, *}\right)$ is also a semigroup.

Proof
As $\left({S, \circ}\right)$ is a semigroup, then by definition it is closed.

As $\phi$ is an epimorphism, it is by definition surjective.

That is:
 * $T = \phi \left[{S}\right]$

where $\phi \left[{S}\right]$ denotes the image of $S$ under $\phi$.

From Morphism Property Preserves Closure it follows that $\left({T, *}\right)$ is closed.

As $\left({S, \circ}\right)$ is a semigroup, then by definition $\circ$ is associative.

From Epimorphism Preserves Associativity, $*$ is therefore also associative.

So:
 * $\left({T, *}\right)$ is closed

and:
 * $*$ is associative.

Therefore, by definition, $\left({T, *}\right)$ is a semigroup.

Also see

 * Isomorphism Preserves Semigroups


 * Epimorphism Preserves Associativity
 * Epimorphism Preserves Commutativity
 * Epimorphism Preserves Identity
 * Epimorphism Preserves Inverses


 * Epimorphism Preserves Groups


 * Epimorphism Preserves Distributivity