Derivative algebra (abstract algebra) explained

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:

1. 0^D = 0
2. x^DD ≤ x + x^D
3. (x + y)^D = x^D + y^D.

x^D 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