Currently working on the following problem, need a little help with the solution to the last part, any hints?
Q: Let S be a semigroup, and let ρ be a congruence on S. Prove that if e ∈ S is an idempotent, then its equivalence class e/ρ is a subsemigroup of S, and is an idempotent in the quotient S/ρ. Also, prove that if S is finite and x/ρ is an idempotent of S/ρ then x/ρ contains an idempotent.
A: e\ $\rho$ = $ x \in S : (e,x) \in \rho $ ie the set of all things related to e. This is a subset of S and thus still associative so to show it is a subsemigroup we just need to show closure which is easy using the fact $e^{2} = e$ and that the relation is a congruence.
Showing $e / \rho$ is an idempotent in the quotient $S / \rho$ is also fine by considering e/$\rho$ * e/ $\rho$ = $e^{2} / \rho$ = $e / \rho$ as e is an idempotent so thats dandy.
However the last line is the bit thats confusing me "prove that if S is finite and $x/\rho$ is an idempotent of $S/\rho$ then $x/ \rho$ contains an idempotent." Surely if $x/\rho$ is an idempotent, clearly $x/\rho$ $\in$ $x/\rho$? I dont really get what this is asking or why the finiteness condition is needed... Any help would be appreciated.
It is simpler to state your question in terms of homomorphisms. Let $\rho: S \to T$ be a semigroup homomorphism. If $e$ is idempotent in $S$, then $\rho(e)$ is idempotent in $T$. Furthermore, the set $$ \{s \in S \mid \rho(s) = \rho(e) \} $$ is equal to $\rho^{-1}(\rho(e))$ and hence is a subsemigroup of $S$. Finally if $\rho(x)$ is equal to an idempotent $f$ of $T$, then $$ R = \{s \in S \mid \rho(s) = \rho(x) \} = \{s \in S \mid \rho(s) = f\} = \rho^{-1}(f) $$ is a nonempty subsemigroup of $S$. If $S$ is finite, then so is $R$ and thus $R$ contains an idempotent.