# Definable principal congruences

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Difference (from prior author revision) (major diff, minor diff)

Changed: 1,2c1,3
 An algebraic structure A has (first-order) definable principal congruences if there is a first-order formula φ(u,v,x,y) such that (u,v) ∈ cg(x,y)  ⇔   A  |=  φ(u,v,x,y).
 A (quasi)variety K of algebraic structures has first-order definable principal (relative) congruences (DP(R)C) if there is a first-order formula φ(u,v,x,y) such that for all A ∈ K we have (x,y) ∈ CgK(u,v)  ⇔   A  |=  φ(u,v,x,y).

Changed: 4,6c5,13
 A class of algebraic structures has (first-order) definable principal congruences if there is a first-order formula φ(u,v,x,y) that defines prinicipal congruences in each of the members of the class.
Here
θ = CgK(u,v) denotes the smallest (relative) congruence that identifies the elements
u,v, where "relative" means that A/θ ∈ K.

### Properties that imply DP(R)C

Equationally definable principal (relative) congruences

### Properties implied by DP(R)C

A (quasi)variety K of algebraic structures has first-order definable principal (relative) congruences (DP(R)C) if there is a first-order formula φ(u,v,x,y) such that for all A ∈ K we have (x,y) ∈ CgK(u,v)  ⇔   A  |=  φ(u,v,x,y).

Here θ = CgK(u,v) denotes the smallest (relative) congruence that identifies the elements u,v, where "relative" means that A/θ ∈ K.

### Properties implied by DP(R)C

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