A semigroup presentation $ \langle A | R\rangle$ is called a tree-like if every relation has the form $ ab=c$ , $ a,b,c$ are in $ A$ and if two relations $ ab=c, a’b’=c’$ belong to $ R$ , then $ c=c’$ if and only if $ a=a’$ and $ b=b’$ . Example: $ P=\langle x | xx=x\rangle$ . These presentations correspond to certain closed subgroups of the R. Thompson group $ F$ (if you are interested, see this paper,but the question is purely semigroup theoretic). For example, the presentation $ P$ corresponds to $ F$ itself.

Question. Is it decidable that a semigroup given by a tree-like presentation contains an idempotent?

The answer is yes if and only if it is decidable if a closed subgroup of $ F$ contains a copy of $ F$ (by this paper).