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**Semidirectly closed pseudovarieties of locally trivial semigroups.**
*(English)*
Zbl 0711.20032

Semidirect products of pseudovarieties of semigroups have previously been investigated by the author [J. Pure Appl. Algebra 60, No.2, 113-128 (1989; Zbl 0687.20053)]. In the present paper he considers the question of which pseudovarieties of semigroups are semidirectly closed. In the process, he determines all semidirectly closed varieties of semigroups and all semidirectly closed generalized varieties of nilpotent extensions of rectangular bands. Using the work of C. J. Ash [J. Algebra 92, 104-115 (1985; Zbl 0548.08007)] on the relationship between pseudovarieties and generalized varieties, he shows that the generalized variety generated by a semidirectly closed pseudovariety is also semidirectly closed. Finally he describes all semidirectly closed pseudovarieties of locally trivial semigroups, and shows that they form a complete distributive lattice.

Reviewer: G.Clarke

### MSC:

20M07 | Varieties and pseudovarieties of semigroups |

08B15 | Lattices of varieties |

08C15 | Quasivarieties |

08B26 | Subdirect products and subdirect irreducibility |

### Keywords:

Semidirect products; pseudovarieties of semigroups; semidirectly closed varieties of semigroups; generalized varieties; nilpotent extensions of rectangular bands; locally trivial semigroups; complete distributive lattice### References:

[1] | Almeda, J.,Minimal non-permutative pseudovarieties of semigroups I, Pacific J. Math.121 (1986) 257–270. · Zbl 0582.20040 |

[2] | –,Minimal non-permutative pseudovarieties of semigroups III, Algebra Universalis21 (1985) 256–279. · doi:10.1007/BF01188061 |

[3] | –,Power pseudovarieties of semigroups I, Semigroup Forum33 (1986) 357–373. · Zbl 0588.20033 · doi:10.1007/BF02573209 |

[4] | –,On power varieties of semigroups, J. Algebra120 (1989) 1–17. · Zbl 0679.20052 · doi:10.1016/0021-8693(89)90183-X |

[5] | Almeida, J.,Semidirect products of pseudovarieties from the universal algebraist’s point of view, J. Pure and Applied Algebra, to appear. · Zbl 0687.20053 |

[6] | Ash, C. J.,Pseudovarieties, generalized varieties and similarly described classes, J. Algebra92 (1985) 104–115. · Zbl 0548.08007 · doi:10.1016/0021-8693(85)90147-4 |

[7] | Burris and Sankappanavar,A Course in Universal Algebra, Springer-Verlag, New York, 1981. · Zbl 0478.08001 |

[8] | Eilenberg, S.,Automata, Languages and Machines, Vol.B, Academic Press, New York, 1976. · Zbl 0359.94067 |

[9] | Pin, J.-E.,Varieties of Formal Languages, Plenum, London, 1986. · Zbl 0632.68069 |

[10] | Tilson, B.,Categories as Algebra:an essential ingredient in the theory of monoids, J. Pure and Applied Algebra48 (1987) 83–198. · Zbl 0627.20031 · doi:10.1016/0022-4049(87)90108-3 |

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