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Zbl 0547.08001
Gumm, H.Peter; Ursini, Aldo
Ideals in universal algebras. (English)
[J] Algebra Univers. 19, 45-54 (1984). ISSN 0002-5240; ISSN 1420-8911/e

The authors consider ideals in varieties with 0, which were introduced by {\it 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.
[G.Matthiesen]

MSC 2000:: *08A30 Subalgebras of general algebraic systems
08B05 Equational logic in varieties of algebras
08B10 Congruence modularity and generalizations in varieties of algebras

Keywords: ideals in varieties; lattices of ideals; 0-regular varieties; Mal'cev condition; ideal determined varieties; congruence lattice; commutators

Citations: Zbl 0263.08006

Cited in: Zbl 0922.08003 Zbl 0803.08001 Zbl 0780.08001

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