# Does the inverse function theorem have an analogue for Boolean algebra?

This is a more focused version of a question (unresolved) about finding the inverse of a system of Boolean equations. For a vector space over the reals, the inverse function theorem says that the inverse of the Jacobian matrix gives us the Jacobian of the inverse. Can we do likewise for a module endomorphism $$\mathbf{f} : \mathbb{B}^n \to \mathbb{B}^n$$?

In the Boolean differential calculus, the derivative for a scalar is defined as $$\frac{df}{dx} = f\rvert_{x=0} + f\rvert_{x=1}$$ where $$(+)$$ indicates exclusive OR. Up to chain and product rules, this works like "normal" differentiation. There is a short paper [1] (h/t this post) assessing the criteria for a Boolean matrix to be invertible, the main one being orthogonality; do its results extend to a matrix of Boolean polynomials, namely the Jacobian $$J_{\mathbf{f}}$$ using Boolean partial derivatives?

Conjecture (false): If $$J_{\mathbf{f}} \cdot J_{\mathbf{f}}^{T} = I$$, then $$\mathbf{f}^{-1}$$ exists, and I can find it by iterated antidifferentiation on the rows of $$J_{\mathbf{f}}^{T}$$.

(False because a Boolean function is always polylinear, and $$\forall x\ \frac{d^2 f}{dx^2}=0$$. This can easily turn out not the case for $$J_{\mathbf{f}}^{T}$$.)

What next?

[1] Rutherford, D. E. (1963). Inverses of Boolean matrices. Proceedings of the Glasgow Mathematical Association, 6(1), 49–53. doi: 10.1017/s2040618500034705