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

[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

### Citations:

Zbl 0902.08010; Zbl 0799.08010
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