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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*) ∈ 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*.

### Properties that imply DP(R)C

Equationally definable principal (relative) congruences

### Properties implied by DP(R)C