congruence_extension_property

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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 ===

Trace: • commutative_idempotent_integral_involutive_fl-algebras

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