Green’s equivalences

Let S be a semigroup. Green’s equivalences are five equivalences ( on S: ,,,𝒟,𝒥

For all x,yS,
xy if S1x=S1y, i.e. sx=y,ty=x for some s,tS1
xy if xS1=yS1, i.e. xs=y,yt=x for some s,tS1

x𝒥y if S1xS1=S1yS1, i.e. sxt=y,uyv=x for some s,t,u,vS1

xy if xy and xy, i.e. =
x𝒟y if zS such that xz and zy, i.e. 𝒟=

It is clear that ,,𝒟,𝒟,𝒟𝒥

These play a fundamental role in understanding the of semigroups.

Title Green’s equivalences
Canonical name GreensEquivalences
Date of creation 2013-03-22 14:23:12
Last modified on 2013-03-22 14:23:12
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Definition
Classification msc 20Mxx
Synonym Green’s relations