# Green’s equivalences

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

For all $x,y\in S$,
$x\mathcal{L}y$ if $S^{1}x=S^{1}y$, i.e. $sx=y,ty=x$ for some $s,t\in S^{1}$
$x\mathcal{R}y$ if $xS^{1}=yS^{1}$, i.e. $xs=y,yt=x$ for some $s,t\in S^{1}$

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

$x\mathcal{H}y$ if $x\mathcal{L}y$ and $x\mathcal{R}y$, i.e. $\mathcal{H=L\cap R}$
$x\mathcal{D}y$ if $\exists z\in S$ such that $x\mathcal{L}z$ and $z\mathcal{R}y$, i.e. $\mathcal{D=L\circ R}$

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

These play a fundamental role in understanding the of semigroups.

Title Green’s equivalences GreensEquivalences 2013-03-22 14:23:12 2013-03-22 14:23:12 mathcam (2727) mathcam (2727) 9 mathcam (2727) Definition msc 20Mxx Green’s relations