##
**Ideals in universal algebras.**
*(English)*
Zbl 0547.08001

The authors consider ideals in varieties with 0, which were introduced by A. Ursini [Boll. Un. Mat. Ital., IV. Ser. 6, 90-95 (1972; Zbl 0263.08006)] as a generalization of ideals in rings, normal subgroups etc. Ideals are defined by equations in classes of algebras with a constant 0 and for every congruence \(\theta\) [0]\(\theta\) is an ideal. Important is the case that also every ideal is [0]\(\theta\) for a unique congruence \(\theta\), in which case the lattices of ideals and congruences are isomorphic. These varieties are called ideal determined. Using a result of Fichtner on 0-regular varieties a Mal’cev condition for ideal determined varieties can be stated. Some characterizations on the congruence lattice follow. In the second part, in ideal determined varieties commutators on congruences (and ideals) are considered. Here a characterization of the commutator [I,J] of ideals I, J is given using terms.

Reviewer: G.Matthiesen

### MSC:

08A30 | Subalgebras, congruence relations |

08B05 | Equational logic, Mal’tsev conditions |

08B10 | Congruence modularity, congruence distributivity |

### Keywords:

ideals in varieties; lattices of ideals; 0-regular varieties; Mal’cev condition; ideal determined varieties; congruence lattice; commutators### Citations:

Zbl 0263.08006
PDF
BibTeX
XML
Cite

\textit{H. P. Gumm} and \textit{A. Ursini}, Algebra Univers. 19, 45--54 (1984; Zbl 0547.08001)

Full Text:
DOI

### References:

[1] | K. Fichtner,Fine Bermerkung ?ber Mannigfaltigkeiten universeller Algebren mit Idealen. Monatsh. d. Deutsch. Akad. d. Wiss. (Berlin)12 (1970), 21-25. · Zbl 0198.33601 |

[2] | R.Freese and R.McKenzie,The modular commutator and its identities. Preprint 1980. |

[3] | H. P. Gumm,An easy way to the commutator in modular varieties. Arch. Math.,34 (1980), 220-228. · Zbl 0438.08004 |

[4] | H. P.Gumm,Geometrical methods in congruence modular algebras. Habilitationsschrift, Darmstadt 1981. To appear in ?Memoirs of the AMS?. |

[5] | J.Hagemann,On regular and weakly regular congruences. Preprint Nr. 75, TH Darmstadt, 1973. · Zbl 0273.08001 |

[6] | J. Hagemann andCh. Herrmann,A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity. Arch. Math.32 (1979), 234-245. · Zbl 0419.08001 |

[7] | P. J. Higgins,Groups with multiple operators. Proc. London Math. Soc. (3)6 (1956), 366-416. · Zbl 0073.01704 |

[8] | B. J?nsson,On the representation of lattices. Math. Scand.1 (1953), 193-206. · Zbl 0053.21304 |

[9] | R. Magari,Su una classe equazionale di algebre. Ann. di Mat. pura e applic. (4)65 (1967), 277-311. · Zbl 0159.01902 |

[10] | A. I. Mal’cev,On the general theory of algebraic systems (Russian). Mat. Sb. (New Series)35 (77), (1954), 3-20. |

[11] | A. Mitschke,Implication algebras are 3-permutable and 3-distributive. Algebra Universalis1 (1971), 182-186. · Zbl 0242.08005 |

[12] | A. Ursini,Sulle variet? di algebre con una buona teoria degli ideali. Boll. U.M.I. (4)6 (1972), 90-95. · Zbl 0263.08006 |

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.