Green's Relations and Their Congruences

[brydustin]

I don't understand how a left (right) Green relation is a right (left) congruence.

xLy <=> Sx = Sy (Green's Left Relation): where we join 1 to S if it doesn't have identity.
Left Congruence: aPb ==> caPcb for some c in the semigroup S.

Take this example table:

* a|b|c
a|a|b|c
b|b|a|c
c|c|b|c

aS = {a,b,c} = bS, so aRb (Green's Right Relation).
But IF it were a left congruence then any element,x, of the Semigroup S, would satisfy the following equation:

aRb ==> xaRxb; then let x=c. Notice ca=c, cb=b, then:

caS = cS = {b,c} </> {a,b,c} = bS = cbS. Therefore there is not a Right Green Relation
(cRb does not exists because cS <> bS, therefore the Right Green relation isn't a left congruence).

Clearly, there is a flaw in my logic, but I don't know where.

Thanks

 

[brydustin]

aLb iff Sa = Sb. Also acLbc iff Sac = Sbc. But if Sa=Sb then (Sa)c=(Sb)c trivially so. But this shows its a right congruence, by definition... Funny, no one answered this post, but me...