hyperrelations

Hyperrelation is a generalization of the mathematical concept of a relation that relates more than two elements. Formally, an n-ary hyperrelation R of arity n ≥ 2 is a subset R ⊆ D1 × D2 × ... × Dn where D1, ..., Dn are domains. When n = 2, R is a binary relation. In this sense, hyperrelations extend the familiar idea of pairwise relations to relationships among three or more components.

Hyperrelations can be visualized as hyperedges in a hypergraph, where each tuple (d1, d2, ..., dn) in R corresponds to a multi-way connection among the involved elements. They are used in database theory to represent multi-attribute relations in a single table, and the standard relational operations extend to them: projection reduces arity by dropping components, selection filters tuples, and join combines compatible relations on common attributes.

In constraint satisfaction problems, variables with finite domains are constrained by hyperrelations that encode allowable combinations of variable assignments. Solving a CSP involves finding an assignment that lies in the natural join of all constraint relations, effectively satisfying all hyperconstraints simultaneously.

Hyperrelations are not inherently functional or deterministic; a single element from one domain can participate in many tuples. They may be symmetric or asymmetric, depending on the predicate they model.

The term is sometimes used loosely to refer to any n-ary relation or to a family of relations of varying arity. In practice, hyperrelations provide a flexible framework for modeling multi-way dependencies in databases, CSPs, knowledge representation, and related fields.