By defining a closure operator on effective equivalence
relations in a regular category $\matcal C$, it is possible to establish a bijective correspondence between these closure operators and the regular epireflective subcategories $\matcal L$ of $\matcal C$. When $\matcal C$ is an exact Goursat
category this correspondence restricts to a bijection between the
Birkhoff closure operators on effective equivalence relations and
the Birkhoff subcategories of $\matcal C$. In this case it is possible to provide an explicit description of the closure, and to characterise the
congruence distributive Goursat categories.
