##
**On iterated semidirect products of finite semilattices.**
*(English)*
Zbl 0743.20056

In [Semigroup Forum 28, 73-81 (1984; Zbl 0527.20046)], J.-E. Pin proved that the pseudovariety of monoids generated by all semidirected products of two semilattice monoids is defined by the identities \(xuyvxy=xuyvyx\) and \(xux=xux^ 2\), and later posed the question as to whether, for any \(n\), the pseudovariety of monoids generated by all semidirect products of \(n\) semilattice monoids is also finitely based. In the current paper, the author answers this question in the negative, as a consequence of the more general consideration of the pseudovarieties \({\mathcal S}\ell^ n\) of semigroups, rather than monoids, generated by semidirect products of \(n\) semilattices. He finds a fairly simple basis of identities for \({\mathcal S}\ell^ n\) and shows that, for \(n>2\), no finite number of these identities will suffice. An explicit connection is set up between the monoid and semigroup pseudovarieties of these types. Various other interesting properties of these pseudovarieties are investigated.

Reviewer: P.R.Jones (Milwaukee)

### MSC:

20M07 | Varieties and pseudovarieties of semigroups |

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

### Keywords:

pseudovarieties of monoids; semidirect products; semilattice monoids; bases of identities; semigroup pseudovarieties### Citations:

Zbl 0527.20046
Full Text:
DOI

### References:

[1] | Almeida, J, Pseudovarieties of semigroups, (), 11-46, [In Portuguese] |

[2] | Almeida, J, Semidirect products of pseudovarieties from the universal Algebraist’s point of view, J. pure appl. algebra, 60, 113-128, (1989) · Zbl 0687.20053 |

[3] | Almeida, J, Equations for pseudovarieties, (), 148-164 |

[4] | Burris, S; Sankappanavar, H.P, A course in universal algebra, (1981), Springer-Verlag New York · Zbl 0478.08001 |

[5] | Eilenberg, S, () |

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

[7] | Pin, J.-E, Hiérarchies de concaténation, RAIRO inform. théor., 18, 23-46, (1984) · Zbl 0559.68062 |

[8] | Pin, J.-E, On the semidirect product of two finite semilattices, (), 73-81 · Zbl 0527.20046 |

[9] | Pin, J.-P, Varieties of formal languages, (1986), Plenum London |

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

[11] | Sapir, M.V, On the finite basis property for pseudovarieties of finite semigroups, C. R. acad. sci. Paris, 306, 795-797, (1988), série I · Zbl 0658.20033 |

[12] | Stiffler, P, Extension of the fundamental theorem of finite semigroups, Adv. in math., 11, 159-209, (1973) |

[13] | Straubing, H, Finite semigroup varieties fo the form V ∗ D, J. pure apl. algebra, 36, 53-94, (1985) · Zbl 0561.20042 |

[14] | Tilson, B, Categories as algebra: an essential ingredient in the theory of monoids, J. pure appl. algebra, 48, 83-198, (1987) · Zbl 0627.20031 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.