I would like to ask you for help with proving the following theorem from our textbook:
Any semigroup is isomorphic to a subsemigroup of ($A^A, \cdot$) for a suitable set A.
The theorem is then proved somewhat briefly for me, claiming that:
Let (S, $\cdot$) be a semigroup. For a $\in$ G, let $\rho_a: S \to S$ be a function defined as $\rho_a(x) = a \cdot x$ for any x $\in$ S. Then $\rho$ is an injective function from (S, $\cdot$) to ($S^S, \circ$). To prove that $\rho$ is an isomorphism: (...)
In a recommended literature borrowed from our university's library, I could only find a very similar theorem saying thet for every
group G, $\rho: G \to Sym(G)$ is an injective homomorphism. This is proved by saying that $\rho_a: G \to G$ is a bijective function as implied by the law of reduction ($a \cdot b = a \cdot c \implies b = c$) proved for groups using the inverse of a which doesn't have to exist in a semigroup.
I am, however, very confused by the relation of $\rho_a$ as a function of x $\in$ G and $\rho(a) = \rho_a(x), a \in G$. Could please anyone explain thoroughly how do they relate to each other forming $\rho: G \to Sym(G)$ from $\rho_a: G \to G$ and prove that the original theorem from our textbook concerning semigroups was right or wrong?