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congruence_extension_property [2010/08/20 20:19] (current)
jipsen created
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+=====Congruence extension property=====
+
+An algebraic structure $\mathbf{A}$ has the \emph{congruence extension property} (CEP) if for any
+algebraic substructure $\mathbf{B}\le\mathbf{A}$ and
+any congruence relation $\theta$ on $\mathbf{B}$ there exists a congruence relation $\psi$ on $\mathbf{A}$
+such that $\psi\cap(B\times B)=\theta$.
+
+A class of algebraic structures has the \emph{congruence extension property} if each of its members has the congruence extension
+property.
+
+For a class $\mathcal{K}$ of algebraic structures, a congruence $\theta$ on an algebra $\mathbf{B}$ is a $\mathcal{K}$-congruence
+if $\mathbf{B}//\theta\in\mathcal{K}$. If $\mathbf{B}$ is a subalgebra of $\mathbf{A}$, we say that a $\mathcal{K}$-congruence
+$\theta$ of $\mathbf{B}$ can be extended to $\mathbf{A}$ if there is a $\mathcal{K}$-congruence $\psi$ on $\mathbf{A}$ such that
+$\psi\cap(B\times B)=\theta$.
+
+Note that if $\mathcal{K}$ is a variety and $B\in\mathcal{K}$ then every congruence of $\mathbf{B}$ is a $\mathcal{K}$-congruence.
+
+A class $\mathcal{K}$ of algebraic structures has the \emph{(principal) relative congruence extension property} ((P)RCEP) if for every algebra
+$\mathbf{A}\in\mathcal{K}$ any (principal) $\mathcal{K}$-congruence
+of any subalgebra of $\mathbf{A}$ can be extended to $\mathbf{A}$.
+
+W. J. Blok and D. Pigozzi, \emph{On the congruence extension property}, Algebra Universalis, \textbf{38}, 1997,
+391--394 [[http://www.ams.org/mathscinet-getitem?mr=99m:08007|MRreview]] shows that for a quasivarieties $\mathcal{K}$, PRCEP implies RCEP.
+
+=== Properties that imply the (relative) congruence extension property ===
+
+[[Equationally def. pr. cong.|Equationally definable principal (relative) congruences]]
+
+=== Properties implied by the (relative) congruence extension property ===
+