Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A (binary) relation on a set is transitive if in every chain of pairwise related elements, the first and last elements are also related:
which generalises from to any finite, positive number of elements.
In the language of the -poset Rel of sets and relations, a relation is transitive if it contains its composite with itself:
from which it follows that for any positive natural number . To include the case where , we must explicitly state that the relation is reflexive.
Transitive relations are often understood as orders.
Last revised on December 2, 2012 at 20:18:21. See the history of this page for a list of all contributions to it.