## 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 MRreview shows that for a quasivarieties $\mathcal{K}$, PRCEP implies RCEP.

#### Properties implied by the (relative) congruence extension property

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