This paper presents a theoretical framework that integrates mathematical morphology with deep learning, focusing on the construction of neural network layers that inherently converge to fixed points through iterative application. Drawing from the principles of idempotence and convergence in complete lattices, we propose a class of nonlinear operators that can be embedded into deep architectures to enhance stability and reduce parameter complexity. The framework is extended to group-equivariant settings on homogeneous spaces, enabling the design of layers that respect symmetries in the data. We formalize the construction of equivariant fixed-point layers using group convolutions and max-plus algebra operators, and we characterize their convergence properties. This work lays the mathematical foundation for future implementations of fixed-point layers in deep learning, particularly in contexts where equivariance and stability are desirable.