Boolean Algebra

A Boolean Algebra is a sextuple \((A, \land, \lor, \neg, \top, \bot)\) satisfying:

1. \((A, \land, \top)\) is a commutative Monoid.
2. \((A, \lor, \bot)\) is a commutative Monoid.
3. (Absorption) For all \(a, b, c \in A\), we have \(a \land (a \lor b) = a \lor (a \land b) = a\).

4. (Distributivity) For \(a, b, c \in A\), we have:

   \begin{aligned} a\land(b\lor c) &= (a\land b)\lor(a\land c) \\ a\lor(b\land c) &= (a\lor b)\land(a\lor c) \end{aligned}
5. (Annihilation) For all \(a \in A\), we have \(a \land \bot = \bot\), and \(a \lor \top = \top\).
6. (Complementation) For all \(a\in A\), we have \(a\land\lnot a=\bot\) and \(a\lor\lnot a=\top\).