Intransitivity
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.
A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C. Some authors call a relation intransitive if it is not transitive, i.e. (if the relation in question is named )
This statement is equivalent to
For instance, in the food chain, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass. Thus, the feed on relation among life forms is intransitive, in this sense.
Another example that does not involve preference loops arises in freemasonry: in some instances lodge A recognizes lodge B, and lodge B recognizes lodge C, but lodge A does not recognize lodge C. Thus the recognition relation among Masonic lodges is intransitive.
Often the term intransitive is used to refer to the stronger property of antitransitivity.
We just saw that the feed on relation is not transitive, but it still contains some transitivity: for instance: humans feed on rabbits, rabbits feed on carrots, and humans also feed on carrots.
A relation is antitransitive if this never occurs at all, i.e.,
Many authors use the term intransitivity to mean antitransitivity.
An example of an antitransitive relation: the defeated relation in knockout tournaments. If player A defeated player B and player B defeated player C, A can have never played C, and therefore, A has not defeated C.
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