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intransitivity in mathematics
Property of mathematical relations
In mathematics, intransitivity (sometimes called nontransitivity) is a property of binary relations that are not transitive relations. This may include any relation that is not transitive, or the stronger property of antitransitivity, which describes a relation that is never transitive.

An antitransitive relation is always irreflexive. An antitransitive relation on a set of ≥4 elements is never connex. On a 3 element set, the depicted cycle has both properties. An irreflexive and left (or right ) unique relation is always anti transitive. An example of the former is the mother relation. If A is the mother of B, and B the mother of C, then A cannot be the mother of C.
If a relation R is antitransitive, so is each subset of R.

Intransitivity can occur under majority rule, in probabilistic outcomes of game theory, and in the Condorcet voting method in which ranking several candidates can produce a loop of preference when the weights are compared (see voting paradox). Intransitive dice demonstrate that the relation " X rolls a higher number than die Y more than half the time" need not be transitive. In psychology, intransitivity often occurs in a person's system of values (or preferences, or tastes), potentially leading to unresolvable conflicts. Analogously, in economics intransitivity can occur in a consumer's preferences. This may lead to consumer behaviour that does not conform to perfect economic rationality. Economists and philosophers have questioned whether violations of transitivity must necessarily lead to 'irrational behaviour' (see Anand (1993)).