Congruence extension property

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 $\mathbf{A}$.

[W. J. Blok and D. Pigozzi, On the congruence extension property, Algebra Universalis, 38, 1997, 391--394 MRreview] 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