# Congruence extension property

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An algebraic structure A has the congruence extension property (CEP) if for any
algebraic substructure B ≤ A and
any congruence relation θ on B there exists a congruence relation ψ on A
such that ψ∩(B×B) = θ.

A class of algebraic structures has the congruence extension property if each of its members has the congruence extension
property.

For a class K of algebraic structures, a congruence θ on an algebra B is a K-congruence
if B/θ ∈ K. If B is a subalgebra of A, we say that a K-congruence
θ of B can be extended to A if there is a K-congruence ψ on A such that
ψ∩(B×B) = θ.

Note that if K is a variety and B ∈ K then every congruence of B is a K-congruence.

A class K of algebraic structures has the (principal) relative congruence extension property ((P)RCEP) if for every algebra
A ∈ K any (principal) K-congruence
of any subalgebra of A can be extended to A.

[ W. J. Blok,
D. Pigozzi,
On the congruence extension property,
Algebra Universalis
38
(1997)
391--394
MRreview ]
shows that for a quasivarieties K, PRCEP implies RCEP.

### Properties that imply the (relative) congruence extension property

[Equationally definable principal congruencesquationally definable principal (relative) congruences]?

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

An algebraic structure A has the congruence extension property (CEP) if for any
algebraic substructure B ≤ A and
any congruence relation θ on B there exists a congruence relation ψ on A
such that ψ∩(B×B) = θ.

A class of algebraic structures has the congruence extension property if each of its members has the congruence extension
property.

For a class K of algebraic structures, a congruence θ on an algebra B is a K-congruence
if B/θ ∈ K. If B is a subalgebra of A, we say that a K-congruence
θ of B can be extended to A if there is a K-congruence ψ on A such that
ψ∩(B×B) = θ.

Note that if K is a variety and B ∈ K then every congruence of B is a K-congruence.

A class K of algebraic structures has the (principal) relative congruence extension property ((P)RCEP) if for every algebra
A ∈ K any (principal) K-congruence
of any subalgebra of A can be extended to A.

[ W. J. Blok,
D. Pigozzi,
On the congruence extension property,
Algebra Universalis
38
(1997)
391--394
MRreview ]
shows that for a quasivarieties 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

An algebraic structure A has the congruence extension property (CEP) if for any algebraic substructure B ≤ A and any congruence relation θ on B there exists a congruence relation ψ on A such that ψ∩(B×B) = θ.

A class of algebraic structures has the congruence extension property if each of its members has the congruence extension property.

For a class K of algebraic structures, a congruence θ on an algebra B is a K-congruence if B/θ ∈ K. If B is a subalgebra of A, we say that a K-congruence θ of B can be extended to A if there is a K-congruence ψ on A such that ψ∩(B×B) = θ.

Note that if K is a variety and B ∈ K then every congruence of B is a K-congruence.

A class K of algebraic structures has the (principal) relative congruence extension property ((P)RCEP) if for every algebra A ∈ K any (principal) K-congruence of any subalgebra of A can be extended to A.

[W. J. Blok, D. Pigozzi, On the congruence extension property, Algebra Universalis 38 (1997) 391--394 MRreview] shows that for a quasivarieties K, PRCEP implies RCEP.

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

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