Equationally definable principal congruences

A (quasi)variety $\mathcal{K}$ of algebraic structures has 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.

Properties that imply EDP(R)C

Properties implied by EDP(R)C

References

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