##
**Semidirect products of pseudovarieties from the universal algebraist’s point of view.**
*(English)*
Zbl 0687.20053

The following question is considered. If membership of two pseudovarieties of finite semigroups can be algorithmically determined, does an effective algorithm exist to determine membership of their semidirect product by a given finite semigroup? Of special interest is the case where one of the pseudovarieties consists of all finite semigroups in which every idempotent is a right zero. In tackling such problems the author illustrates the use of some results from C. J. Ash [J. Algebra 92, 104-115 (1985; Zbl 0548.08007)] who explored the relationship between varieties, pseudovarieties and generalized varieties. He begins by proving that a generalized variety is locally finite if and only if it is generated by the class of all its finite members. He also characterizes the class of all finite members of the semidirect product of two locally finite generalized varieties as the semidirect product of the two classes of finite algebras obtained from each of the generalized varieties. Using these and other results he obtains characterizations of various semidirect products of pseudovarieties of semigroups.

Reviewer: G.Clarke

### MSC:

20M07 | Varieties and pseudovarieties of semigroups |

08B25 | Products, amalgamated products, and other kinds of limits and colimits |

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

08C15 | Quasivarieties |

### Keywords:

pseudovarieties of finite semigroups; effective algorithm; membership; generalized varieties; semidirect products### Citations:

Zbl 0548.08007
PDFBibTeX
XMLCite

\textit{J. Almeida}, J. Pure Appl. Algebra 60, No. 2, 113--128 (1989; Zbl 0687.20053)

Full Text:
DOI

### References:

[1] | Almeida, J., Some order properties of the lattice of varieties of commutative semigroups, Canad. J. Math., 38, 19-47 (1986) · Zbl 0594.20053 |

[2] | Almeida, J., (Proc. First Meeting of Portuguese Algebraists, Lisboa, 1986. Proc. First Meeting of Portuguese Algebraists, Lisboa, 1986, Pseudovarieties of semigroups (in Portuguese) (1988), Universidade de Lisboa), 11-46 |

[3] | Almeida, J., The algebra of implicit operations, Algebra Universalis, 26, 16-32 (1989) · Zbl 0671.08003 |

[4] | J. Almeida, On pseudovarieties, varieties of languages, filters and congruences, pseudoidentities and related topics, Algebra Universalis, to appear.; J. Almeida, On pseudovarieties, varieties of languages, filters and congruences, pseudoidentities and related topics, Algebra Universalis, to appear. · Zbl 0715.08006 |

[5] | Ash, C. J., Pseudovarieties, generalized varieties and similarly described classes, J. Algebra, 92, 104-115 (1985) · Zbl 0548.08007 |

[6] | Banaschewski, B., The Birkhoff Theorem for varieties of finite algebras, Algebra Universalis, 17, 360-368 (1983) · Zbl 0534.08005 |

[7] | Brzozowski, J. A.; Simon, I., Characterizations of locally testable events, Discrete Math., 4, 243-271 (1973) · Zbl 0255.94032 |

[8] | Burris, S.; Sankappanavar, H. P., A Course in Universal Algebra (1981), Springer: Springer Berlin · Zbl 0478.08001 |

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

[10] | Eilenberg, S.; Schützenberger, M. P., On pseudovarieties, Adv. Math., 19, 413-418 (1976) · Zbl 0351.20035 |

[11] | Irastorza, C., Base non finie de variétés, (STACS 85, Saarbrücken, 1985, 182 (1985), Springer: Springer Berlin), 173-179, Lecture Notes in Computer Science · Zbl 0572.20041 |

[12] | Krohn, K.; Rhodes, J., Algebraic theory of machines, Trans. Amer. Math. Soc., 116, 450-464 (1965) · Zbl 0148.01002 |

[13] | Perkins, P., Bases for equational theories of semigroups, J. Algebra, 11, 298-314 (1968) · Zbl 0186.03401 |

[14] | Reiterman, J., The Birkhoff theorem for finite algebras, Algebra Universalis, 14, 1-10 (1982) · Zbl 0484.08007 |

[15] | Rhodes, J., Infinite iteration of matrix semigroups, II, J. Algebra, 100, 25-137 (1986) · Zbl 0626.20050 |

[16] | Simon, I., Piecewise testable events, (Lecture Notes in Computer Science, 33 (1975), Springer: Springer Berlin), 214-222 · Zbl 0316.68034 |

[17] | Straubing, H., Finite semigroup varieties of the form \(V\) ∗ \(D\), J. Pure Appl. Algebra, 36, 53-94 (1985) · Zbl 0561.20042 |

[18] | Thérien, D.; Weiss, A., Graph congruences and wreath products, J. Pure Appl. Algebra, 36, 205-215 (1985) · Zbl 0559.20042 |

[19] | B. Tilson, Chapter XII Eilenberg [9].; B. Tilson, Chapter XII Eilenberg [9]. |

[20] | Tilson, B., Categories as algebra: An essential ingredient in the theory of monoids, J. Pure Appl. Algebra, 48, 83-198 (1987) · Zbl 0627.20031 |

[21] | Trahtman, A. N., The varieties of \(n\)-testable semigroups, Semigroup Forum, 27, 309-318 (1983) · Zbl 0525.20043 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.