Ideals in universal algebras.

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

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##### References:

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