Derivative algebra (abstract algebra)

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (May 2024) (Learn how and when to remove this message)

|

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

[edit]

* 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

This algebra-related article is a stub. You can help Wikipedia by expanding it.

|

* v
* t
* e

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template