In a congruence modular subtractive variety there are both the commutator of ideals and the commutator of congruences. We prove that, if I-delta is the smallest congruence having an ideal I as a congruence class, then [I, I] = 0/[I-delta, J(delta)]. The general identity [0/alpha, 0/beta] = 0/[alpha, beta] for alpha, beta congruences, does not always hold; we give several conditions equivalent to this identity and sufficient conditions for it to hold. In the meantime, we get some other characterizations of the commutator of ideals. We also deal with the equational definability of principal commutators in a subtractive variety and with the extension property of the commutator from ideals of a subalgebra to the commutator of ideals of the whole algebra.

Ursini, A. (2000). On subtractive varieties V: congruence modularity and the commutators. ALGEBRA UNIVERSALIS, 43(1), 52-78 [10.1007/s000120050145].

### On subtractive varieties V: congruence modularity and the commutators

#### Abstract

In a congruence modular subtractive variety there are both the commutator of ideals and the commutator of congruences. We prove that, if I-delta is the smallest congruence having an ideal I as a congruence class, then [I, I] = 0/[I-delta, J(delta)]. The general identity [0/alpha, 0/beta] = 0/[alpha, beta] for alpha, beta congruences, does not always hold; we give several conditions equivalent to this identity and sufficient conditions for it to hold. In the meantime, we get some other characterizations of the commutator of ideals. We also deal with the equational definability of principal commutators in a subtractive variety and with the extension property of the commutator from ideals of a subalgebra to the commutator of ideals of the whole algebra.
##### Scheda breve Scheda completa Scheda completa (DC) Ursini, A. (2000). On subtractive varieties V: congruence modularity and the commutators. ALGEBRA UNIVERSALIS, 43(1), 52-78 [10.1007/s000120050145].
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11365/7153`
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