Derivative algebra (abstract algebra)

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 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