semigroup in which Green's relations are different

Question

I am beginning to study the Green relations and I would like to know if there is a semigroup in which the 5 Green relations are different. I would like to find an example where this occurs, I have tried some easy semigroups and have had no luck. Thank you!

Answer 1

It is easy to find examples of semigroups in which the four Green's relations $\cal R$, $\cal L$, $\cal H$ and $\cal D$ are different. The smallest example is $S = \{1, 2\} \times \{1, 2\}$ under the multiplication defined by $(r,s)(t,u) = (r, u)$.

However, since ${\cal D} = {\cal J}$ in every finite semigroup, you need an infinite semigroup to answer your question. Let $T$ be the infinite semigroup of matrices of the form $$ \begin{pmatrix} a&0\\ b&1 \end{pmatrix} $$ where $a$ and $b$ are strictly positive rational numbers, equipped with the usual multiplication of matrices. Then the four relations $\cal R$, $\cal L$, $\cal H$ and $\cal D$ coincide with the equality, but $T$ has a single $\cal J$-class. It follows that in the semigroup $S \times T$, the five Green's relations are distinct.

Another example is given in [1, Exercise 3.11]. Let $A = \{a,b,c\}$ and $M$ be the quotient of the free monoid $A^*$ by the congruence generated by the relation $abc = 1$. Then the five Green relations are distinct in $M$.

[1] A.J. Cain, Nine Chapters on the Semigroup Art