# Differences

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+ | =====Congruence extension property===== | ||

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+ | 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 $\psi\cap(B\times B)=\theta$. | ||

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+ | 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 | ||

+ | $\psi\cap(B\times B)=\theta$. | ||

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+ | Note that if $\mathcal{K}$ is a variety and $B\in\mathcal{K}$ then every congruence of $\mathbf{B}$ is a $\mathcal{K}$-congruence. | ||

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+ | 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 $\mathbf{A}$. | ||

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+ | 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. | ||

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+ | === Properties that imply the (relative) congruence extension property === | ||

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+ | [[Equationally def. pr. cong.|Equationally definable principal (relative) congruences]] | ||

+ | |||

+ | === Properties implied by the (relative) congruence extension property === | ||

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