Closure (mathematics) - P Closures of Binary Relations

P Closures of Binary Relations

The notion of a closure can be generalized for an arbitrary binary relation R ⊆ S×S, and an arbitrary property P in the following way: the P closure of R is the least relation Q ⊆ S×S that contains R (i.e. R ⊆ Q) and for which property P holds (i.e. P(Q) is true). For instance, one can define the symmetric closure as the least symmetric relation containing R. This generalization is often encountered in the theory of rewriting systems, where one often uses more "wordy" notions such as the reflexive transitive closure R*—the smallest preorder containing R, or the reflexive transitive symmetric closure R≡—the smallest equivalence relation containing R, and therefore also known as the equivalence closure. 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.