Definition:P-Closure of Relation

Definition

Let $\RR = \struct {S, T, R}$ be a relation on the sets $S$ and $T$:

$R \subseteq S \times T$

Let $P$ be a property of relations.

Let $\RR' = \struct {S, T, R'}$ be the relation on $S \times T$ such that:

$R \subseteq R'$

$\RR'$ is the smallest relation on $S \times T$ with respect to the subset ordering on $S \times T$

$R'$ has the property $P$.

Then $\RR'$ is known as the $P$-closure of $\RR$.