Transitivity

In mathematics, the word transitive admits at least four distinct meanings:

• A group G acts transitively on a set S if for any x, y ∈ S, there is some g ∈ G such that gx = y. See group action. A somewhat related meaning is explained at ergodic theory.
• A binary relation is transitive if whenever A is related to B and B is related to C, then A is related to C, for all A, B, and C in the domain of the relation. See transitive relation.
• A transitive set is a set A such that whenever x ∈ A, and y ∈ x, then y ∈ A. The smallest transitive set containing a set A is called the transitive closure of A.
• A discrete dynamical system f is topologically transitive if every open subset U' of the phase space intersects every other open subset V, when going along trajectory, i.e. there exists an integer n, for which $f^{n}(U)\cap V\neq \emptyset$.

See also

• intransitivity