Transitive closure

https://en.wikipedia.org/wiki/Transitive_closure

Transitive closure of a binary relation R on a set X is the smallest relation on X that contains R and is transitive.

For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs. For infinite sets, it is the unique minimal transitive superset of R.

For example, if X is a set of airports and xRy means "there is a direct flight from airport x to airport y" (for x and y in X), then the transitive closure of R on X is the relation R+ such that x R+ y means "it is possible to fly from x to y in one or more flights". Informally, the transitive closure gives you the set of all places you can get to from any starting place.

More formally, the transitive closure of a binary relation R on a set X is the transitive relation R+ on set X such that R+ contains R and R+ is minimal Lidl & Pilz (1998, p. 337). If the binary relation itself is transitive, then the transitive closure is that same binary relation; otherwise, the transitive closure is a different relation.

In a closely related area of graph theory, transitive reduction is analogous to adducing a minimal R from the given closure R+.