Definition : Let $S$ be a semigroup. let $R$ be a relation on $S$.
Left compatibility: $R$ is left compatible if $$ (\forall a , s ,t \in S) \ \ (s,t) \in R \ \ \Rightarrow (as , at) \in R $$
Right compatibility: $R$ is is right compatible if $$ (\forall a , s ,t \in S) \ \ (s,t) \in R \ \ \Rightarrow (sa , ta) \in R $$
Give a counterexample of a relation which is left compatible but not right compatible. Similarly, a counter example of a relation which is right compatible but not left compatible.
Any help would be appreciated. Thank you.
The Green's relation $\mathcal{R}$ that you mentioned in this question is left compatible (but in general not right compatible). Dually the Green's relation $\mathcal{L}$ is right compatible (but in general not left compatible)