Definable principal congruences

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) ∈ Cg_K(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 = Cg_K(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) ∈ Cg_K(u,v) ⇔ A |= φ(u,v,x,y).

Here θ = Cg_K(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