A (quasi)variety $\mathcal{K}$ of algebraic structures has \emph{equationally definable principal (relative) congruences} (EDP(R)C) if there is a finite conjunction of atomic formulas $\phi(u,v,x,y)$ such that for all algebraic structures $\mathbf{A}\in\mathcal{K}$ we have $\langle x,y\rangle\in\mbox{Cg}_{\mathcal{K}}(u,v)\iff \mathbf{A}\models \phi(u,v,x,y)$. Here $\theta=\mbox{Cg}_{\mathcal{K}}(u,v)$ denotes the smallest (relative) congruence that identifies the elements $u,v$, where “relative” means that $\mathbf{A}//\theta\in\mathcal{K}$. Note that when the structures are algebras then the atomic formulas are simply equations.
Relative congruence extension property
Relatively congruence distributive