Idempotent relation

In mathematics, an idempotent binary relation R ⊆ X × X is one for which R ∘ R = R.[1][2] 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 ℚ is idempotent. In contrast, < on ℤ is not, since (<) ∘ (<) ⊇ (<) does not hold: e.g. 1 < 2, but 1 < x < 2 is false for every x ∈ ℤ.

Idempotent relations have been used as an example to illustrate the application of Mechanized Formalisation of mathematics using the interactive theorem prover Isabelle/HOL. Besides checking the mathematical properties of finite idempotent relations, an algorithm for counting the number of idempotent relations has been derived in Isabelle/HOL. [3][4]

References

  1. ↑ Florian Kammüller, J. W. Sanders (2004). Idempotent Relation in Isabelle/HOL (PDF) (Technical report). TU Berlin. p. 27. 2004-04. Here:p.3
  2. ↑ Florian Kammüller (2011). "Mechanical Analysis of Finite Idempotent Relations". Fundamenta Informaticae. 107. pp. 43–65. doi:10.3233/FI-2011-392.
  3. ↑ Florian Kammüller (2006). "Number of idempotent relations on n labeled elements". The On-Line Ecyclopedea of Integer Sequences (A12137).
  4. ↑ Florian Kammüller (2008). Counting Idempotent Relations (PDF) (Technical report). TU Berlin. p. 27. 2008-15.
  • Berstel, Jean; Perrin, Dominique; Reutenauer, Christophe (2010). Codes and automata. Encyclopedia of Mathematics and its Applications. 129. Cambridge: Cambridge University Press. p. 330. ISBN 978-0-521-88831-8. Zbl 1187.94001.


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