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definable_principal_congruences [2010/08/20 20:21] (current)
jipsen created
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 +=====Definable principle congruences=====
 +A (quasi)variety $\mathcal{K}$ of algebraic structures has \emph{first-order definable principal (relative) congruences} (DP(R)C) if
 +there is a first-order formula $\phi(u,v,x,y)$ such that for all
 +$\mathbf{A}\in\mathcal{K}$ we have $\langle x,y\rangle\in\mbox{Cg}_{\mathcal{K}}(u,v)\iff \mathbf{A}\models \phi(u,v,x,y)$.
 +$\theta=\mbox{Cg}_{\mathcal{K}}(u,v)$ denotes the smallest (relative) congruence that identifies the elements
 +$u,v$, where "relative" means that $\mathbf{A}//\theta\in\mathcal{K}$.
 +=== Properties that imply DP(R)C ===
 +[[Equationally def. pr. cong.|Equationally definable principal (relative) congruences]]
 +=== Properties implied by DP(R)C ===