In abstract algebra, a derivative algebra is an algebraic structure of the signature

:<A, ·, +, ', 0, 1, <sup>D</sup>>

where

:<A, ·, +, ', 0, 1>

is a Boolean algebra and <sup>D</sup> is a unary operator, the derivative operator, satisfying the identities:

  1. 0<sup>D</sup> = 0
  2. x<sup>DD</sup> ≤ x + x<sup>D</sup>
  3. (x + y)<sup>D</sup> = x<sup>D</sup> + y<sup>D</sup>.

x<sup>D</sup> 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&nbsp;+ (p∧□p&nbsp;→&nbsp;□□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