equationally_def._pr._cong

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=====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 ===

[[Discriminator variety]]

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

Relative [[congruence extension property]]

Relatively [[congruence distributive]]

[[Definable principal congruences|Definable principal (relative) congruences]]

=== 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 [[http://www.ams.org/mathscinet-getitem?mr=85g:08005|MRreview]], \textbf{18}, 1984, 334-379 [[http://www.ams.org/mathscinet-getitem?mr=86e:08005|MRreview]],
\textbf{32}, 1994, 545-608 [[http://www.ams.org/mathscinet-getitem?mr=96b:08007|MRreview]], \textbf{31}, 1994, 
1-35 [[http://www.ams.org/mathscinet-getitem?mr=95c:03144|MRreview]]

Trace: • commutative_involutive_fl-algebras • sqrt-quasi-mv-algebras • involutive_fl-algebras • commutative_idempotent_integral_involutive_fl-algebras • cyclic_involutive_fl-algebras • de_morgan_algebras • post_algebras • pure_discriminator_algebras • integral_involutive_fl-algebras • goedel_algebras