After defining semigroups, we now study a powerful criterion that turns a semigroup into a group. In a group, the equations $a\circ x=b$ and $y\circ a=b$ have unique solutions for all $a,b\in G$. The theorem in this lesson proves a converse: if a semigroup already has this unique solvability property, then it must contain an identity element and inverses, and hence it is a group. This is an important structural result because it shows that the ability to solve equations can replace the explicit assumption of identity and inverse elements.

Let $(S,\circ)$ be a semigroup. The equations $$\begin{equation*} a\circ x=b \end{equation*}$$ and $$\begin{equation*} y\circ a=b \end{equation*}$$ are said to be $\textbf{uniquely solvable}$ in $S$ for all $a,b\in S$ if for every choice of $a,b\in S$, each equation has exactly one solution in $S$.