Isomorphism Preserves Semigroups

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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 isomorphism.


If $\left({S, \circ}\right)$ is a semigroup, then so is $\left({T, *}\right)$.


Proof 1

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

From Morphism Property Preserves Closure, $\left({T, *}\right)$ is therefore also closed.


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

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


So $\left({T, *}\right)$ is closed, and $*$ is associative, and therefore by definition, $\left({T, *}\right)$ is a semigroup.

$\blacksquare$


Proof 2

An isomorphism is an epimorphism.

The result follows as a direct corollary of Epimorphism Preserves Semigroups.

$\blacksquare$