Beziehungsgleich

Beziehungsgleich is a term used in some German-language discussions of relational logic, linguistics, and information theory to denote a specific property of two relationships. It signifies that the two relationships are equal or equivalent in how they relate entities within a given domain. Because it is not part of standard terminology, its precise sense can vary between sources.

Etymology and scope. The word combines Beziehung (relation) with Gleich (equal). In different contexts, Beziehungsgleich may refer to strict equality of two relations (the two have the same extension for all argument pairs) or to a looser notion of equivalence under some transformation or renaming of elements.

Formal meaning. In a strict reading, two binary relations R and S on a domain D are Beziehungsgleich if for all x, y in D, R(x, y) holds exactly when S(x, y) holds. This is the usual notion of equality of relations, i.e., R and S have identical truth values across all inputs. In a broader reading, Beziehungsgleich can describe isomorphism of relational structures, where R and S correspond under a bijection of the domain.

Usage and contexts. The term appears in discussions about model theory, database schema comparison, and semantic representations in linguistics. It is typically used to express that two relational descriptions encode the same relational pattern, either directly (identical extension) or up to a structural renaming.

Example. If the domain is {a, b} and R = {(a, b)} while S = {(a, b)}, then R and S are Beziehungsgleich. If a bijection maps a to b and b to a and aligns R with S accordingly, some authors would also describe them as Beziehungsgleich in a looser sense.

See also. Relation, Gleichheit, Äquivalenzrelation, Isomorphismus, Relationale Modelle.