Definable principle congruences

A (quasi)variety $\mathcal{K}$ of algebraic structures has first-order definable principal (relative) congruences (DP(R)C) if there is a first-order formula $\phi(u,v,x,y)$ such that for all $\mathbf{A}\in\mathcal{K}$ 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 $\mathbf{A}//\theta\in\mathcal{K}$.

Properties that imply DP(R)C

Properties implied by DP(R)C