Homomorphism with Cancellable Codomain Preserves Identity

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

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

Let $\left({S, \circ}\right)$ have an identity $e_S$.

Let every element of $\left({T, *}\right)$ be cancellable.

Then $\phi \left({e_S}\right)$ is the identity for $*$.

Proof
Let $\left({S, \circ}\right)$ be an algebraic structure in which $\circ$ has an identity $e_S$.

Let $\left({T, *}\right)$ be such that every element is cancellable.

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

Every element of $\left({T, *}\right)$ is cancellable.

Suppose there is an idempotent element in $\left({T, *}\right)$

So from Identity is Only Idempotent Cancellable Element, it must be an identity.

Thus:

So $\phi \left({e_S}\right)$ is idempotent in $\left({T, *}\right)$ and the result follows.

Also see

 * Epimorphism Preserves Identity


 * Homomorphism with Identity Preserves Inverses