Derivative algebra (abstract algebra)

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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. xDDx + 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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