A closure operator on a partially ordered set is an increasing, extensive, idempotent endofunction. Of particular interest to us are closure operators on the boolean algebra which can in fact be described by their set of fixed points which form an upper bounded meet subsemilattice of the boolean algebra.
A preorder can be produced from a closure operator on a boolean algebra by the inclusion order on the closure of singletons. Set systems that are entirely determined by their closure preorder are equivalent to Alexandrov topologies.
Besides the reachability closure on any binary relation we have the closure of upper sets, lower sets, convex sets, rank complete suborders, join subsemilattices, meet subsemilattices, sublattices, and complemented sublattices among others.
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