nLab De Morgan algebra

De Morgan algebras

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


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.


  • 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).
  • The four-point algebra with two parallel lines: {0,a,b,1}\{0, a, b, 1\} with ¬a=a\neg a = a, ¬b=b\neg b = b, ab=1a \vee b = 1 and ab=0a \wedge b = 0 is a small example of a DeMorgan algebra which is not Boolean.

Last revised on October 9, 2022 at 23:42:44. See the history of this page for a list of all contributions to it.