Quasi-identities of finite semigroups and symbolic dynamics (Q1905790)

An algebra is called inherently non-finitely \((Q-)\) based if it is not contained in any locally finite and finitely based (quasi-)variety. \textit{M. V. Sapir} first proved that there exist finite inherently non-finitely based semigroups [Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 2, 319-340 (1987; Zbl 0646.20047)]\ and then effectively described them all [Mat. Sb., Nov. Ser. 133, 154-166 (1987; Zbl 0634.20027)]. In the paper under review the authors show that, in strong contrast with the variety case, no finite semigroup is inherently non-finitely \(Q\)-based. The proof makes essential use of the connection between Burnside type problems for semigroups and symbolic dynamics established in the first of the cited papers by M. V. Sapir.