Example of a left compatible relation on a semigroup that is not right compatible. Definition : Let $S$ be a semigroup. let $R$ be a relation on $S$.


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*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)
