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. 0D = 0
2. xDD ≤ x \+ xD
3. (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

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