Algebra univers. 46 (2001) 97 – 103 0002–5240/01/020097 – 07 $ 1.50 + 0.20/0 © Birkh¨ auser Verlag, Basel, 2001 Decidability of finite quasivarieties generated by certain transformation semigroups M. V. Volkov Dedicated to the memory of Victor Aleksandrovich Gorbunov Introduction Recall that a semigroup pseudovariety is a class of finite semigroups closed under the operators H (taking homomorphic images), S (taking subsemigroups), and P fin (forming finitary direct products). The notion of a pseudovariety is rather natural per se but it has got a strong additional motivation coming from some recent developments in finite automata and formal languages [6, 11, 2]. It is clear from the very definition that the notion of a pseudovariety is closely related to the classical notion of a variety. However, unlike varieties, not every semigroup pseudovar- iety is defined by a set of identities, and any of syntactic characterizations of pseudovarieties (cf. [7, 12, 17] for various approaches) is inevitably more complicated than that of varieties. By a finite quasivariety of semigroups we understand a class of finite semigroups closed under the operators S and P fin . Such classes of finite semigroups were first considered by C. J. Ash [3] who called them pseudo-quasivarieties; the term “finite quasivariety”, though not perfect from the logical point of view, reflects the fact that, in a strong contrast to the varietal case, finite quasivarieties are indeed so to speak finite traces of “infinite” (that is, usual) quasivarieties. Thus, every finite quasivariety (in particular, every pseudovariety!) can be defined within the class of all finite semigroups by a suitable set of quasi-identities. These important properties of finite quasivarieties first observed by V. A. Gorbunov [9, Corollary 2.9] have given a strong impetus to studying the latter from the pseudovariety standpoint; see [15, 16] as examples of such studies. In the present note, we also “project onto finite quasivarieties” a problem which plays a distinguished role in the theory of semi- group pseudovarieties. Given a class C of finite semigroups, the least pseudovariety containing C is known to be equal to HSP fin (C ). Many famous open questions about formal languages and finite Presented by Professors Kira Adaricheva and Wieslaw Dziobiak. Received January 2, 2000; accepted in final form August 28, 2000. 2000 Mathematics Subject Classification: 08C15, 20M20. Key words: Pseudovariety, quasivariety, transformation semigroup, pseudovariety membership problem, quasivariety membership problem, algorithmic decidability. 97