# 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

## Definition

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

## 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 $[0,1]$ is a De Morgan algebra, with $\neg x = (1-x)$.

Last revised on June 14, 2021 at 19:27:24. See the history of this page for a list of all contributions to it.