I've begun reading Steinberg's book on representations of finite monoids. In the very first chapter, he introduces congruences on a monoid as an equivalence relation $\equiv$ such that $x \equiv y$ implies $zxw \equiv zyw$ for all $x, y, z, w$ in the monoid. From this arises the quotient monoid $M / \equiv$ with the obvious multiplication.
The associative and identity axioms of this construction are pretty automatic, so that only well-definedness has to be shown. And I believe the proof carries over to semigroups, provided one replaces the condition by $x \equiv y$ implies $zx \equiv zy$ and $xw \equiv yw$ for all $x, y, z, w$ in the monoid.
These conditions both generalise to universal algebra, by taking an equivalence relation that is also a subalgebra of $A \times A$, where $A$ is an algebra with certain operations.
My question is: Does this generalise to model theory?