Let $R\subseteq X\times X$ denote a relation on the set $X$.
There are various names for properties that $R$ can satisfy: $R$ is
- reflexive if $\forall x.xRx$,
- symmetric if $\forall x,y.(xRy\implies yRx)$,
- transitive if $\forall x,y,z.(xRy \wedge yRz\implies xRz)$,
- an equivalence if it is all of the above.
For each of these properties, given an arbitrary relation $R$, there is always a well-defined 'smallest' reflexive closure/symmetric closure/transitive closure/equivalence generated by $R$. It can be defined by taking the intersection of the family of relations that 1. contain $R$, and 2. satisfy the relevant property.
This seems to work because for each of the properties, 1. taking intersections of relations with the property preserves the property, and 2. the family is nonempty, since the maximal relation $X\times X$ has the property.
Beyond the 4 properties of basic relations listed above, something similar holds for more general properties on sets with more structure, e.g. we can also generate the 'smallest monoidal congruence'.
My question is -- what is the correct general notion that generalizes these properties? Is there some notion from order theory, or some correct setting, that unifies all of these properties (and more)? Binary relations might not even be the correct most general setting to consider here -- something analogous holds for topologies and sigma algebras, for instance, where we can 'generate the smallest such instance by intersecting a bunch of sets'.