Derivative algebra (abstract algebra)


In?? abstract algebra, a derivative algebra is an algebraic structure of the signature


where
is a Boolean algebra and D is a unary operator, the derivative operator, satisfying the identities:

  1. 0D = 0
  2. xDD ≤ x + xD
  3. D = xD + yD.
xD is called the derivative of x. Derivative algebras provide an algebraic abstraction of the derived set operator in topology. They also play the same role for the modal logic wK4 = K + p∧?p → ??p that Boolean algebras play for ordinary propositional logic.