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A (quasi)variety K of algebraic structures has first-order definable principal (relative) congruences (DP(R)C) if
there is a first-order formula φ(u,v,x,y) such that for all
A ∈ K we have (x,y) ∈ CgK(u,v) ⇔ A |= φ(u,v,x,y).
θ = CgK(u,v) denotes the smallest (relative) congruence that identifies the elements
u,v, where "relative" means that A/θ ∈ K.
Properties that imply DP(R)C
Equationally definable principal (relative) congruences
Properties implied by DP(R)C