Kourovka Notebook Problem 19.76

Definitions and mathematical statement

Let $A$ be a finite alphabet, and let $A^+$ denote the semigroup of nonempty finite words over $A$, with concatenation as multiplication. A finite semigroup presentation \[ P=\langle A\mid a_i=b_i c_i\ (1\leq i\leq m)\rangle \] is tree-like if $a_i,b_i,c_i\in A$ and, for distinct indices $i,j$, \[ a_i\neq a_j\qquad\text{and}\qquad b_i c_i\neq b_j c_j \] as words. No additional distinctness condition on the letters within a relation is imposed. Let $\equiv_P$ be the least semigroup congruence on $A^+$ containing the defining relations, and write $S(P)=A^+/\!\equiv_P$. An element $e\in S(P)$ is an idempotent if $e^2=e$.

Does there exist an algorithm which, for every finite tree-like presentation $P$, halts and correctly decides whether \[ \exists e\in S(P)\quad e^2=e? \] Equivalently, the property to be decided is \[ \exists w\in A^+\quad ww\equiv_P w. \] The empty word is excluded, and no identity is adjoined.

The supplied source also asserts an equivalent decision question: given a finitely generated subgroup $H$ of R. Thompson's group $F$, decide whether its closure contains a subgroup isomorphic to $F$. Here $F$ is the group of increasing piecewise linear homeomorphisms of $[0,1]$ with finitely many dyadic rational breakpoints and slopes that are integral powers of $2$. The supplied record does not define the closure operation on subgroups of $F$. Consequently, this accompanying formulation and its asserted equivalence require clarification from the cited source; no particular meaning of closure is assumed here.

Short English statement

M. Sapir asks whether an algorithm can always decide if a semigroup given by a finite tree-like presentation has an idempotent. The source asserts an equivalent question about closures of finitely generated subgroups of Thompson's group $F$, but does not specify the closure operation. The registry labels the problem open; that status is supplied metadata, not an independently verified assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701624$, exact source code catalogue entry, title Kourovka Notebook Problem 19.76. The record attributes the problem to M. Sapir, proposed in 2018, and labels its status open.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 19.76, 19th issue (2018); the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. This bibliographic attribution is supplied evidence and has not been independently checked.