Mathematical Structures: Congruence extension property

# Congruence extension property

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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.