Extending congruence relations

Abstract:

If $\mathfrak {A}$ and $\mathfrak {B}$ are algebras, where $\mathfrak {A} \subseteq \mathfrak {B}$, and $\theta$ is a congruence relation on $\mathfrak {A}$, let ${\theta ^\mathfrak {B}}$ be the smallest congruence relation on $\mathfrak {B}$ containing $\theta$. $\mathfrak {A}$ is called a congruence subalgebra of $\mathfrak {B}$ if $\mathfrak {A} \subseteq \mathfrak {B}$ and, for every congruence relation $\theta$ on $\mathfrak {A},{\theta ^\mathfrak {B}} \cap |\mathfrak {A}{|^2} = \theta$. Elementary subalgebras are congruence subalgebras, and there are Directed Union and Loewenheim-Skolem Theorems for congruence subalgebras analogous to those for elementary subalgebras. Consequently we obtain full analogues of the Jónsson-Morley-Vaught results concerning homogeneous-universal algebras, where the notion of “subalgebra” is everywhere replaced by “congruence subalgebra".