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**Further properties of lattices of equational theories.**
*(English)*
Zbl 0756.08006

\(\Sigma\) is an equational theory iff \(\Sigma\) is closed under deduction. \(L(\Sigma)\) stands for the lattice of all equational theories contained in \(\Sigma\). \(L(\Sigma)\) is algebraic, fails every nontrivial lattice- identity but satisfies the (restrictive) Zipper Condition, as shown by the author [Algebra Univers. 23, 61-69 (1986; Zbl 0604.08003)].

The present paper considers several even more restrictive conditions satisfied by every \(L(\Sigma)\), due to the author, Erné and Tardos. In the colorful nomenclature introduced by the author, he proves that Velcro \(\Rightarrow\) finitary Velcro \(\Rightarrow\) F88 \(\Rightarrow\) Erné- Tardos \(\Rightarrow\) Zipper — see the paper for the exact definitions. Erné and Tardos showed that the two last implications are not reversible. A central topic of this paper is the construction of two infinite lattices witnessing that also the two first implications cannot be reversed. The main tool used is that of colored graphs as introduced by Pudlák.

The other main result is that for algebraic lattices of finite length, all the conditions listed are equivalent, supporting the conjecture that among finite lattices, the Zipper Condition indeed characterizes those isomorphic to some lattice of equational theories.

The present paper considers several even more restrictive conditions satisfied by every \(L(\Sigma)\), due to the author, Erné and Tardos. In the colorful nomenclature introduced by the author, he proves that Velcro \(\Rightarrow\) finitary Velcro \(\Rightarrow\) F88 \(\Rightarrow\) Erné- Tardos \(\Rightarrow\) Zipper — see the paper for the exact definitions. Erné and Tardos showed that the two last implications are not reversible. A central topic of this paper is the construction of two infinite lattices witnessing that also the two first implications cannot be reversed. The main tool used is that of colored graphs as introduced by Pudlák.

The other main result is that for algebraic lattices of finite length, all the conditions listed are equivalent, supporting the conjecture that among finite lattices, the Zipper Condition indeed characterizes those isomorphic to some lattice of equational theories.

Reviewer: J.Schmid (Bern)

### MSC:

08B15 | Lattices of varieties |

08B05 | Equational logic, Mal’tsev conditions |

06B15 | Representation theory of lattices |

05C15 | Coloring of graphs and hypergraphs |

06B05 | Structure theory of lattices |

### Keywords:

lattices of equational theories; Zipper condition; finite lattices; colored graphs; algebraic lattices### Citations:

Zbl 0604.08003
Full Text:
DOI

### References:

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