nLab
De Morgan algebra

This page is about distributive lattices equipped with a contravariant involution. For Heyting algebras satisfying de Morgan's law, which are also sometimes called “De Morgan algebras”, see De Morgan Heyting algebra.

De Morgan algebras

Definition

A De Morgan algebra is a distributive lattice DD equipped with a contravariant involution, i.e. a map ¬:D opD\neg : D^{op}\to D such that ¬¬A=A\neg\neg A = A. This implies that ¬\neg satisfies De Morgan’s laws: ¬(AB)=¬A¬B\neg(A\wedge B) = \neg A \vee \neg B and ¬(AB)=¬A¬B\neg (A\vee B) = \neg A \wedge \neg B.

Examples

  • Any Boolean algebra is a De Morgan algebra, with ¬\neg the logical negation.
  • Any star-autonomous poset that happens to be a distributive lattice is a De Morgan algebra, with ¬\neg the star-autonomous involution () *(-)^*.
  • The unit interval [0,1][0,1] is a De Morgan algebra, with ¬x=(1x)\neg x = (1-x).

Created on December 16, 2017 at 06:13:37. See the history of this page for a list of all contributions to it.