Green’s equivalences

Let $S$ be a semigroup. Green’s equivalences are five equivalences (http://planetmath.org/EquivalenceRelation) on $S$: $ℒ,ℛ,\mathscr{H},𝒟,𝒥$

For all $x,y\in S$,
$xℒy$ if $S^{1}x=S^{1}y$, i.e. $sx=y,ty=x$ for some $s,t\in S^1$
$xℛy$ if $x{S}^1=y{S}^1$, i.e. $xs=y,yt=x$ for some $s,t\in S^1$

$x𝒥y$ if $S^{1}x{S}^1=S^{1}y{S}^1$, i.e. $sxt=y,uyv=x$ for some $s,t,u,v\in S^1$

$x\mathscr{H}y$ if $xℒy$ and $xℛy$, i.e. $\mathscr{H}=ℒ\cap ℛ$
$x𝒟y$ if $\exists z\in S$ such that $xℒz$ and $zℛy$, i.e. $𝒟=ℒ\circ ℛ$

It is clear that $\mathscr{H}\subseteq ℒ,\mathscr{H}\subseteq ℛ,ℒ\subseteq 𝒟,ℛ\subseteq 𝒟,𝒟\subseteq 𝒥$

These play a fundamental role in understanding the of semigroups.