From wikipedia, for an arbitrary binary relation $R\subseteq S×S,$ and an arbitrary property $P$, the $P$ closure of $R$ is defined as:
The least relation $Q\subseteq S×S$ that contains $R$ (i.e. $R\subseteq Q$) and for which property $P$ holds (i.e. $P(Q)$ is true).
For instance, the symmetric closure is the least symmetric relation containing $R$, the reflexive transitive closure $R$* is the smallest preorder containing $R$, and the reflexive transitive symmetric closure $R\equiv$ is the smallest equivalence relation containing $R$.
For arbitrary $P$ and $R$, the $P$ closure of $R$ need not exist. In the above examples, these exist because reflexivity, transitivity and symmetry are closed under arbitrary intersections. In such cases, the $P$ closure can be directly defined as the intersection of all sets with property $P$ containing $R$.
- I was wondering what are some examples of properties $P$ that are not closed under arbitrary intersections?
- Is property $P$ being closed under arbitrary intersections the sufficient and necessary condition for existence of the $P$ closure of an arbitrary relation $R$?
Thanks and regards!