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**Wreath products of varieties, and semi-Archimedean semigroup varieties.**
*(English.
Russian original)*
Zbl 0899.20032

Trans. Mosc. Math. Soc. 1996, 203-222 (1996); translation from Tr. Mosk. Mat. O.-va 57, 218-238 (1996).

The paper may be subdivided into 3 more or less independent parts. The first part is mainly devoted to a comparison between various possible definitions of the wreath product of semigroup varieties. This part is almost identical with Sections 1-5 of the author’s paper [Fundam. Prikl. Mat. 2, No. 1, 233-249 (1996)].

By a semi-Archimedean semigroup the author means a semilattice of Archimedean semigroups. The second part of the paper under review is focused on the question when the monoidal wreath product \({\mathbf U}\circ{\mathbf V}\) of two semigroup varieties consists entirely of periodic semi-Archimedean semigroups. An answer is given by Theorem 4.4: both \(\mathbf U\) and \(\mathbf V\) should consist of periodic semi-Archimedean semigroups, and, in addition, either \(\mathbf U\) consists of Archimedean semigroups or \(\mathbf V\) consists of semilattices of left nilsemigroups.

The main result of the third part is the following: for every \(n\geq 2\), the monoidal wreath product \(\mathbf{Sl}\circ{\mathbf A}_n\) of the variety \(\mathbf{Sl}\) of semilattices with the variety \({\mathbf A}_n\) of abelian groups of exponent \(n\) is inherently non-finitely based (Theorem 7.3). This extends a similar result for \(n=2\) found by C. Irastorza [Lect. Notes Comput. Sci. 182, 180-186 (1985; Zbl 0572.20041)].

Reviewer’s remark. The crucial point in the proofs of both the cited results is the observation that the six-element Brandt monoid divides a semidirect product of a semilattice with any non-trivial group (Lemma 6.1 in the paper under review). The author appears to believe that this lemma is new although in fact it is well known and may be found, e.g., in J. Almeida’s book [Finite semigroups and universal algebra, World Scientific (1994; Zbl 0844.20039)], see Proposition 10.10.12 in the book.

By a semi-Archimedean semigroup the author means a semilattice of Archimedean semigroups. The second part of the paper under review is focused on the question when the monoidal wreath product \({\mathbf U}\circ{\mathbf V}\) of two semigroup varieties consists entirely of periodic semi-Archimedean semigroups. An answer is given by Theorem 4.4: both \(\mathbf U\) and \(\mathbf V\) should consist of periodic semi-Archimedean semigroups, and, in addition, either \(\mathbf U\) consists of Archimedean semigroups or \(\mathbf V\) consists of semilattices of left nilsemigroups.

The main result of the third part is the following: for every \(n\geq 2\), the monoidal wreath product \(\mathbf{Sl}\circ{\mathbf A}_n\) of the variety \(\mathbf{Sl}\) of semilattices with the variety \({\mathbf A}_n\) of abelian groups of exponent \(n\) is inherently non-finitely based (Theorem 7.3). This extends a similar result for \(n=2\) found by C. Irastorza [Lect. Notes Comput. Sci. 182, 180-186 (1985; Zbl 0572.20041)].

Reviewer’s remark. The crucial point in the proofs of both the cited results is the observation that the six-element Brandt monoid divides a semidirect product of a semilattice with any non-trivial group (Lemma 6.1 in the paper under review). The author appears to believe that this lemma is new although in fact it is well known and may be found, e.g., in J. Almeida’s book [Finite semigroups and universal algebra, World Scientific (1994; Zbl 0844.20039)], see Proposition 10.10.12 in the book.

Reviewer: M.V.Volkov (Ekaterinburg)

### MSC:

20M07 | Varieties and pseudovarieties of semigroups |

20M10 | General structure theory for semigroups |