Concept lattices are well-known conceptual structures that organise interesting patterns — the concepts — extracted from data. In some applications, the size of the lattice can be a problem, as it is often too large to be efficiently computed and too complex to be browsed. In others, redundant information produces noise that makes understanding the data difficult. In classical FCA, those two problems can be attenuated by, respectively, computing a substructure of the lattice — such as the AOC-poset — and reducing the context. These solutions have not been studied in $d$-dimensional contexts for $d > 3$. In this paper, we generalise the notions of AOC-poset and reduction to $d$-lattices, the structures that are obtained from multidimensional data in the same way that concept lattices are obtained from binary relations.