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
This algebra-related article is a stub. You can help Wikipedia by expanding it. |
This page is based on the copyrighted Wikipedia article "Derivative algebra (abstract algebra)"; it is used under the Creative Commons
Attribution-ShareAlike 3.0 Unported License (CC-BY-SA). You may
redistribute it, verbatim or modified, providing that you comply with
the terms of the CC-BY-SA