## Equationally definable principal congruences

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.

#### Properties that imply EDP(R)C

#### Properties implied by EDP(R)C

Relative congruence extension property

Relatively congruence distributive

#### References

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