learning algorithms need to be developed, resulting in new challenges with regard to model construction, computational tractability and model assessment.

As relations between objects can be observed in many different forms, this general problem setting provides links to several subfields of machine learning, like statistical relational learning [@Deraedt2009], graph mining [@Vert2005], metric learning [@Xing2002] and preference learning [@Hullermeier2010a]. More specifically, from a graph-theoretic perspective, learning a relation can be formulated as learning edges in a graph where the nodes represent information of the data objects; from a metric learning perspective, the relation that we aim to learn should satisfy some well-defined properties like positive definiteness, transitivity or the triangle inequality; and from a preference learning perspective, the relation expresses a (degree of) preference in a pairwise comparison of data objects.

The topic of learning relations between objects is also closely related to recent developments in fuzzy set theory. This article will elaborate on these connections via two important contributions: (1) the extension of the typical setting of learning crisp relations to real-valued and ordinal-valued relations and (2) the inclusion of domain knowledge about relations into the inference process by explicit modeling of mathematical properties of these relations. For algorithmic simplicity, 