Equationally definable principal congruences

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A (quasi)variety K of algebraic structures has equationally definable principal (relative) congruences (EDP(R)C) if there is a finite conjunction of atomic formulas φ(u,v,x,y) such that for all algebraic structures A ∈ K we have (x,y) ∈ Cg_K(u,v) ⇔ A |= φ(u,v,x,y). Here θ = Cg_K(u,v) denotes the smallest (relative) congruence that identifies the elements u,v, where "relative" means that A/θ ∈ K. Note that when the structures are algebras then the atomic formulas are simply equations.

Properties that imply EDP(R)C

[Discriminator variety]?

Properties implied by EDP(R)C

Relative congruence extension property

Relatively congruence distributive

Definable principal (relative) congruences

References

[W. J. Blok, D. Pigozzi, On the structure of varieties with equationally definable principal congruences. I, Algebra Universalis 15 (1982) 195--227 MRreview]

[W. J. Blok, P. Köhler, D. Pigozzi, On the structure of varieties with equationally definable principal congruences. II, Algebra Universalis 18 (1984) 334--379 MRreview]

[W. J. Blok, D. Pigozzi, On the structure of varieties with equationally definable principal congruences. III, Algebra Universalis 32 (1994) 545--608 MRreview]

[W. J. Blok, Don Pigozzi, On the structure of varieties with equationally definable principal congruences. IV, Algebra Universalis 31 (1994) 1--35 MRreview]