Morphological Representation Theory in the Fourier Inf-Semilattice: Universal Decomposition of Frequency-Domain Deep Learning Operators

Abstract

We develop a morphological representation theory for operators acting in the frequency domain of $L^2({\mathbb{R}}^n)$. Equipping the space with the \emph{Fourier inf-semilattice} order ${\le }_{\mathcal{F}}$ (spectral modulus inequality with phase equality), convolution becomes a morphological erosion and the adjoint dilation is the Wiener inverse filter. A key observation is that spatial translation-invariance is vacuous in the spectral order, replaced structurally by \emph{positive homogeneity} of the operator on spectral moduli. Under this hypothesis we prove the \emph{Morphological Representation Theorem} for the Fourier modulus lattice $\hat{L}$: any increasing, USC, positively homogeneous $\Psi :\hat{L}\to \hat{L}$ decomposes \emph{exactly} as a supremum of max-times erosions $\hat{f}/\psi$, indexed by a minimal morphological basis. Three structural corollaries follow: the spectral modulus map $|\cdot |:L^2({\mathbb{R}}^n)\to \hat{L}$ is an idempotent cross-lattice projection (the morphological analogue of ReLU) that makes depth non-trivial; wavelet scattering coefficients form the canonical dictionary for the basis via Littlewood--Paley density; and spatial pooling is a Fourier erosion, so that pool-then-unpool is the ideal band-pass opening, U-Net skip connections are Fourier top-hat transforms, and strided pooling preserves the morphological structure if and only if the Nyquist condition holds. The framework carries a natural ${\mathbb{C}}^{\ast }$-group morphology structure, positioning mathematical morphology as the constructive operator-theoretic complement of spectral bias, scattering, and tropical geometry theories.