Regular semigroup

Remarks

If $axa = a$, then $y = xax$ satisfies $aya = a$ and $yay = y$; such a $y$ is called an inverse of $a$. So $S$ is regular if and only if every element has at least one inverse in this sense. If $axa = a$, then $ax$ and $xa$ are idempotents. A semigroup is regular if and only if every $\mathcal{R}$-class, or equivalently every $\mathcal{L}$-class, of Green's relations contains an idempotent.

Example: the monoid $T_X$ of all maps $X \to X$ under composition, $(fg)(x) = f(g(x))$. Given $f$, choose $g$ such that $g(y)$ is a preimage of $y$ under $f$ for each $y$ in the image of $f$, and $g(y)$ is arbitrary otherwise. Then $f(g(f(x))) = f(x)$ for all $x$, so $fgf = f$.

Non-example: the positive integers under addition. For $a = 1$, the equation $1 + x + 1 = 1$ has no positive solution $x$, so $1$ is not regular.