Idempotent relation

From Wikipedia, the free encyclopedia
Jump to: navigation, search

In mathematics, an idempotent binary relation R ⊆ X × X is one for which R  R = R.[1] This notion generalizes that of an idempotent function to relations. Each idempotent relation is necessarily transitive, as the latter means R ∘ R ⊆ R.

For example, the relation < on is idempotent. In contrast, < on is not, since (<) ∘ (<) ⊉ (<), e.g. 1 < 2, but it is not true that 1 < x < 2 for every x ∈ ℤ.

References[edit]

  1. ^ Florian Kammüller, Jeffrey W. Saunders (2004). Idempotent Relation in Isabelle/HOL (Technical report). TU Berlin. p. 27. 2004-04.  Here:p.3