Varieties whose skeletons are lattices.

*(English. Russian original)*Zbl 0787.08007
Algebra Logic 31, No. 1, 48-53 (1992); translation from Algebra Logika 31, No. 1, 74-82 (1992).

In Algebra Logika 24, No. 5, 588-607 (1985; Zbl 0619.08005), we introduced the notions of epimorphism and embeddability skeletons of a variety of algebras and studied them in more detail for the case of congruence distributive (c.d.) varieties in a series of subsequent publications. In Algebra Logika 27, No. 3, 316-326 (1988; Zbl 0666.08004), we noted that the skeletons in question take the position between “thin” congruence lattices of free algebras, lattices of subalgebras of universal algebras (their skeletons are, respectively, antiisotonical and isotonical images), and “rough” lattices of subvarieties (these lattices are isotonical images of skeletons).

In connection with the above we raise the following question: Is it always the case, and if not then in what cases, are the epimorphism and embeddability skeletons of a variety of algebras lattices in their own right? In the present note we consider this problem for the case of c.d. varieties.

In connection with the above we raise the following question: Is it always the case, and if not then in what cases, are the epimorphism and embeddability skeletons of a variety of algebras lattices in their own right? In the present note we consider this problem for the case of c.d. varieties.

##### MSC:

08B10 | Congruence modularity, congruence distributivity |

08A35 | Automorphisms and endomorphisms of algebraic structures |

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\textit{A. G. Pinus}, Algebra Logic 31, No. 1, 48--53 (1992; Zbl 0787.08007); translation from Algebra Logika 31, No. 1, 74--82 (1992)

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##### References:

[1] | A. G. Pinus, ”Relations of epimorphism and embeddability on congruence-distributive varieties,” Algebra Logika,24, No. 5, 588–607 (1985). · Zbl 0619.08005 |

[2] | A. G. Pinus, ”Congruence-distributive varieties of algebras,” Itogi Nauki i Tekhniki (Algebra, Topology, and Geometry),26, 45–83 (1988). · Zbl 0679.08004 |

[3] | A. G. Pinus, ”Boolean constructions in universal algebras,” Usp. Mat. Nauk,47 (1992). · Zbl 0792.08002 |

[4] | A. G. Pinus, ”Coverings in epimorphism skeletons of varieties of algebras,” Algebra Logika,27, No. 3, 316–326 (1988). · Zbl 0666.08004 |

[5] | A. G. Pinus, Congruence-Modular Varieties of Algebras [in Russian], Irkutsk (1986). |

[6] | S. Shelah, ”Constructions of many complicated uncountable structures and Boolean algebras,” Israel J. Math.,45, No. 2–3, 100–146 (1983). · Zbl 0552.03018 |

[7] | A. G. Pinus, ”Elementary theory of embeddability skeletons for discriminator varieties,” Sib. Mat. Zh.,32, No. 5, 126–131 (1991). · Zbl 0756.08003 |

[8] | R. Bonnet, ”Very strongly rigid Boolean algebras, continuum, discrete set condition, countable antichain condition (I),” Alg. Univ.,11, No 3, 341–364 (1980). · Zbl 0467.06008 |

[9] | S. Burris and H. Werner, ”Sheaf constructions and their elementary properties,” Trans. Am. Math. Soc.,248, No. 2, 269–309 (1979). · Zbl 0411.03022 |

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