On subtractive varieties. III: From ideals to congruences. (English) Zbl 0906.08005

Agliano, P.; Ursini, A.

Algebra Univers. 37, No. 3, 296-333 (1997).

[For Parts I and II see ibid. 31, No. 2, 204-222 (1994; Zbl 0799.08010) and ibid. 36, No. 2, 222-259 (1996; Zbl 0902.08010), respectively.]
If \(\mathcal V\) is a subtractive (i.e. permutable at 0) variety, then every ideal \(I\) of \(A\in \mathcal V\) is a 0-class of some \(\theta \in\text{Con }A\). In fact, the set \(\text{CON}(I)\) of all \(\theta \in\text{Con }A\) having \(I\) as a 0-class forms an interval of Con\(A\). The authors study how to recover the congruence structure from the ideal structure of these varieties, especially the last and the greatest element of CON\((I)\). The paper contains a number of technical results describing the properties of congruence-ideal connections using the bounds of \(\text{CON}(I)\) and it is completed by five important examples (left-complementary monoids, BCK-algebras, varieties having a single 0-permutability term, pseudocomplemented semilattices, Hilbert algebras).

Reviewer: I.Chajda (Olomouc)

MSC:

08B05	Equational logic, Mal’tsev conditions
08A30	Subalgebras, congruence relations

Keywords:

ideal of algebra; congruence; subtractive variety; permutability at 0

Citations:

Zbl 0902.08010; Zbl 0799.08010