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**Solid varieties of semigroups.**
*(English)*
Zbl 0797.20045

Let \(\tau = (n_ i)_{i \in I}\) be a type and \((f_ i)_{i \in I}\) operation symbols such that the arity of \(f_ i\) is \(n_ i\). Let \(W_ \tau(X)\) be the set of all terms of type \(\tau\) in an alphabet \(X\). A map \(\sigma: (f_ i)_{i \in I} \to W_{\tau}(X)\) is a hypersubstitution. If \(t \approx t'\) is an equation, then \(\Xi[t \approx t']\) denotes the set of all equations obtained from \(t \approx t'\) by hypersubstitution. The equation \(t \approx t'\) is a type \(\tau\) hyperidentity of an algebra \(A\) if \(A\) satisfies all the equations in \(\Xi[t \approx t']\). Similarly, an equation is a hyperidentity of a variety \(V\) if it is a hyperidentity of every algebra in \(V\). A variety \(V\) of type \(\tau\) is solid if every identity of \(V\) is also a type \(\tau\) hyperidentity in \(V\).

It was shown by K. Denecke, D. Lau, R. Pöschel and D. Schwiegert [Contrib. Gen. Algebra 7, 97-118 (1991; Zbl 0759.08005)] that a variety is solid if and only if it is a hypervariety. This paper gives other results of this nature for algebras in general, and then turns to a study of solid varieties of semigroups. Many new examples of such varieties are given, and a picture of the bottom of the lattice of such varieties is produced.

It was shown by K. Denecke, D. Lau, R. Pöschel and D. Schwiegert [Contrib. Gen. Algebra 7, 97-118 (1991; Zbl 0759.08005)] that a variety is solid if and only if it is a hypervariety. This paper gives other results of this nature for algebras in general, and then turns to a study of solid varieties of semigroups. Many new examples of such varieties are given, and a picture of the bottom of the lattice of such varieties is produced.

Reviewer: S.Oates-Williams (St.Lucia)

### MSC:

20M07 | Varieties and pseudovarieties of semigroups |

08B15 | Lattices of varieties |

08B05 | Equational logic, Mal’tsev conditions |

### Keywords:

lattice of varieties; type; operation symbols; hypersubstitution; hyperidentity; hypervariety; solid varieties of semigroups### Citations:

Zbl 0759.08005
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\textit{K. Denecke} and \textit{S. L. Wismath}, Semigroup Forum 48, No. 2, 219--234 (1994; Zbl 0797.20045)

### References:

[1] | [DK93] Denecke, K. and J. Koppitz,Hyperassociative semigroups, preprint, 1993. |

[2] | [DL91] Denecke, K., D. Lau, R. Pöschel and D. Schweigert,Hyperidentities, hyperequational classes and clone congruences, Contributions to General Algebra7(1991), 97–118. · Zbl 0759.08005 |

[3] | [GS90] Graczynka, E. and D. Schweigert,Hyperidentities of a given type, Algebra Universalis27 (1990), 305–318. · Zbl 0715.08002 |

[4] | [Kop93] Koppitz, J.,On equational descriptions of solid semigroup varieties, preprint, 1993. |

[5] | [Per69] Perkins, P.,Bases for equational theories of semigroups, J. Algebra11 (1969), 298–314. · Zbl 0186.03401 |

[6] | [Sch92] Schweigert, D.,Hyperidentities, Universitat Kaiserslautern Preprint No. 220, 1992. · Zbl 0795.08003 |

[7] | [Tay81] Taylor, W.,Hyperidentities and hypervarieties, Aequationes Mathematicae23(1981), 30–49. · Zbl 0491.08009 |

[8] | [Wis90] Wismath, S.,Hyperidentities for some varieties of semigroups, Algebra Universalis27(1990), 111–127. · Zbl 0692.08008 |

[9] | [Wis91] Wismath, S.,Hyperidentities for some varieties of commutative semigroups, Algebra Universalis28(1991), 245–273. · Zbl 0741.20043 |

[10] | [Wis93] Wismath, S.,Hyperidentily Bases for rectangular bands and other semigroup varieties, to appear in Journal of the Australian Math. Society (A). |

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