19.76 (2018)

A semigroup presentation is called tree-like if all relations have the form $a = bc$ where $a, b, c$ are letters and no two relations share the left-hand side or the right-hand side. Is it decidable whether the semigroup given by a finite tree-like presentation contains an idempotent?

This is equivalent to the question whether the closure of a finitely generated subgroup of R. Thompson’s group $F$ contains an isomorphic copy of $F$.