A Kleene algebra is a de Morgan algebra$D$ satisfying $x \wedge \neg x \le y \vee \neg y$ for all $x,y\in D$. Since the order is definable in terms of the lattice operators, this can be stated as the equation

$x \wedge \neg x \wedge (y \vee \neg y) = x \wedge \neg x.$

Examples

Any Boolean algebra is a Kleene algebra, with $\neg$ the logical negation.

The unit interval $[0,1]$ is a Kleene algebra, with $\neg x = (1-x)$.