Dimensionality of ordinal structures Original Research Article

(S, (⩽n)nϵN) is called an ordinal structure if S is a set and (⩽n)nϵN a family of quasi-orders on S. Since ordinal structures are used as models for ordinal data, we assign to each ordinal structure a canonical conceptual structure, a so-called concept lattice. The ordinal dimension of an ordinal structure S ≔ (S, (⩽n)nϵN) is the smallest number of quasi-orders on S which determines the same conceptual structure as the concept lattice of S. It turns out that the ordinal dimension of S equals the chromatic number of a certain hypergraph. We show how to compute this hypergraph and analyse how the ordinal dimension behaves under several constructions. Furthermore, we discuss the linear case where all quasi-orders on S are assumed to be linear and the convex-ordinal case where, with every ⩽i in the family (⩽n)nϵN, its dual ⩾i is also in (⩽n)nϵN.