An algebraic structure $\mathbf{A}$ has the
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
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
(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 $\mathbf{A}$.
<bibxml:entry id="MR99m:08007">
<bibxml:article>
<bibxml:author>W. J. Blok</bibxml:author>
<bibxml:author>D. Pigozzi</bibxml:author>
<bibxml:title>On the congruence extension property</bibxml:title>
<bibxml:journal>Algebra Universalis</bibxml:journal>
<bibxml:fjournal>Algebra Universalis</bibxml:fjournal>
<bibxml:volume>38</bibxml:volume>
<bibxml:year>1997</bibxml:year>
<bibxml:number>4</bibxml:number>
<bibxml:pages>391--394</bibxml:pages>
<bibxml:issn>0002-5240</bibxml:issn>
<bibxml:coden>AGUVA3</bibxml:coden>
<bibxml:mrclass>08A30 (08C15)</bibxml:mrclass>
<bibxml:mrnumber>99m:08007</bibxml:mrnumber>
<bibxml:mrreviewer>V. N. Sali\\u\\i</bibxml:mrreviewer>
</bibxml:article>
</bibxml:entry>
shows that for a quasivarieties $\mathcal{K}$, PRCEP implies RCEP.
Properties that imply the (relative) congruence extension property
Equationally definable principal (relative) congruences
Properties implied by the (relative) congruence extension property
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