Inverse semigroups and varieties of finite semigroups.

*(English)*Zbl 0625.20045This third paper of the series of three [see the preceding reviews] is devoted to the study of varieties of finite monoids - classes of (finite) monoids closed under the taking of submonoids, quotients and finite direct products, and in particular the variety Inv generated by finite inverse monoids. The main conjecture, which is shown to be equivalent to a finite version of a conjecture form their first paper, is that Inv equals the variety of all finite monoids whose idempotents from a semilattice. The authors note at the end of the paper that this has been proved to be true by C. J. Ash [J. Aust. Math. Soc., Ser. A 43, 81- 90 (1987)]. Ash has also published an expository paper on the proof of this result [in Semigroups and their applications, Proc. Int. Conf., Chico/Calif. 1986, 13-23 (1987; Zbl 0623.20048)] in which there is also a paper by J.-C. Birget, S. Margolis and J. Rhodes [ibid. 25-35 (1987; Zbl 0622.20052)] in which Ash’s result is extended to prove that the variety of finite monoids generated by the orthodox monoids is the variety of finite monoids whose idempotents form a band.

In this paper the authors show that Inv can be generated by four other classes of monoids including the class of all semidirect products of a semilattice by a group. A three-way characterization of the variety of languages corresponding to Inv includes their realization as those languages of the form L or KaL where K and L are group languages and \(a\in A\), the base alphabet. In addition, the variety of all monoids on Inv whose groups are in a given variety of groups can be described by its finite biprefix codes.

In this paper the authors show that Inv can be generated by four other classes of monoids including the class of all semidirect products of a semilattice by a group. A three-way characterization of the variety of languages corresponding to Inv includes their realization as those languages of the form L or KaL where K and L are group languages and \(a\in A\), the base alphabet. In addition, the variety of all monoids on Inv whose groups are in a given variety of groups can be described by its finite biprefix codes.

Reviewer: P.M.Higgins

##### MSC:

20M10 | General structure theory for semigroups |

20M07 | Varieties and pseudovarieties of semigroups |

20M05 | Free semigroups, generators and relations, word problems |

20M35 | Semigroups in automata theory, linguistics, etc. |

20E10 | Quasivarieties and varieties of groups |

##### Keywords:

varieties of finite monoids; finite inverse monoids; idempotents; orthodox monoids; semidirect products; variety of languages; group languages; variety of groups; finite biprefix codes
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\textit{S. W. Margolis} and \textit{J. E. Pin}, J. Algebra 110, 306--323 (1987; Zbl 0625.20045)

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