Computational mathematical-morphology has been developed to provide a directly computable alternative to classical gray-scale morphology that is range preserving and compatible with the design of statistically optimal filters based on morphological representation. It serves as an image algebra because of the expressive capability of its image-operator representations. Because representations are based on binary comparators used in conjunction with AND and OR operations, it provides a low-level, efficient computational environment that is a direct extension of the finite Boolean operational environment. The paper focuses on development of the comparator-based representation, providing the relevant representation theory for lattice operators and lattice-vector operators. Computational lattice-operator theory represents a comparator-based alternative to the classical morphological lattice theory, one that is directly implementable in logic. For totally ordered valuation spaces, the lattice theory reduces to a simplified form appropriate to straightforward optimization. The present paper treats architectural considerations and a second part considers the implications for latticevalued image operators.