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## Main Question or Discussion Point

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

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