In abstract algebra, a derivative algebra is an algebraic structure of the signature
<A, ·, +, ', 0, 1, D> where
<A, ·, +, ', 0, 1> is a Boolean algebra and D is a unary operator, the derivative operator, satisfying the identities:
0D = 0 xDD ≤ x + xD (x + y)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.
References Esakia, L., Intuitionistic logic and modality via topology, Annals of Pure and Applied Logic, 127 (2004) 155-170 McKinsey, J.C.C. and Tarski, A., The Algebra of Topology, Annals of Mathematics, 45 (1944) 141-191