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.

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

Related concepts

• De Morgan duality
• De Morgan algebras are used in one form of cubical type theory