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## 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

Relative congruence extension property

Relatively congruence distributive

#### 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]

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