Green's relations

Remarks

All five are equivalence relations; for $\mathcal{D}$ this uses the fact that $\mathcal{L}$ and $\mathcal{R}$ commute as relations. Equivalently, $a \mathrel{\mathcal{L}} b$ if and only if $a = xb$ and $b = ya$ for some $x, y \in S^1$. Always $\mathcal{H} \subseteq \mathcal{L}, \mathcal{R} \subseteq \mathcal{D} \subseteq \mathcal{J}$, and $\mathcal{D} = \mathcal{J}$ when $S$ is finite.

Example: let $T_X$ be the monoid of all maps of a finite set $X$ to itself, with $(fg)(x) = f(g(x))$. If $f = gh$, then the image of $f$ is contained in the image of $g$. Conversely, if the images are equal, choose $h(x)$ to be a preimage of $f(x)$ under $g$; then $f = gh$. So $f \mathrel{\mathcal{R}} g$ if and only if $f$ and $g$ have the same image. Similarly, $f \mathrel{\mathcal{L}} g$ if and only if $f$ and $g$ have the same fibres, and $f \mathrel{\mathcal{D}} g$ if and only if their images have the same size.

Non-example: in the monoid $(\N, +)$, the left ideal generated by $a$ is $a + \N$, so $a \mathrel{\mathcal{L}} b$ only when $a = b$. Unlike in a group, the identity $0$ and the element $1$ are not $\mathcal{L}$-related.