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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

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]