Boolean–Lie algebras and the Leibniz rule

Using internal negations acting on Boolean functions, the notion of Boolean–Lie algebra is introduced. The underlying Lie product is the Boolean analogue of the Poisson bracket. The structure of a Boolean–Lie algebra is determined; it turns out to be solvable, but not nilpotent. We prove that the adjoint representation of an element of the Boolean–Lie algebra acts as a derivative operator on the space of Boolean functions. The adjoint representation is related to the previously known concept of the sensitivity function. Using the notion of adjoint representation we give the definition of a temporal derivative applicable to iterative dynamics of Boolean mappings.

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