information:filtering, stylistic:style_transfer
Related properties

A relation R is called intransitive if it is not transitive, that is, if xRy and yRz, but not xRz, for some x, y, z.In contrast, a relation R is called antitransitive if xRy and yRz always implies that xRz does not hold.For example, the relation defined by xRy if xy is an even number is intransitive, but not antitransitive. The relation defined by xRy if x is even and y is odd is both transitive and antitransitive. The relation defined by xRy if x is the successor number of y is both intransitive and antitransitive. Unexpected examples of intransitivity arise in situations such as political questions or group preferences.

Generalized to stochastic versions (stochastic transitivity), the study of transitivity finds applications of in decision theory, psychometrics and utility models.

A quasitransitive relation is another generalization; it is required to be transitive only on its non-symmetric part. Such relations are used in social choice theory or microeconomics.

Proposition: If R is a univalent, then R;R is transitive.

Corollary: If R is univalent, then R;R is an equivalence relation on the domain of R.