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.

Discriminator variety

Relative congruence extension property

Relatively congruence distributive

Definable principal (relative) congruences

W. J. Blok and D. Pigozzi, \emph{On the structure of varieties with equationally definable principal congruences. I, II, III, IV}, Algebra Universalis, \textbf{15}, 1982, 195-227 MRreview, \textbf{18}, 1984, 334-379 MRreview, \textbf{32}, 1994, 545-608 MRreview, \textbf{31}, 1994, 1-35 MRreview