# 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:

- 0
^{D}= 0 *x*^{DD}≤*x*+*x*^{D}- (
*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

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