page_title: Cyclic order
section_title: Finite cycles
hierarchical_section_title: Cyclic order / Finite cycles
context_page_description: In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "a < b". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation [a, b, c], meaning "after a, one reaches b before c". For example, [June, October, February]. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and total. Dropping the "total" requirement results in a partial cyclic order.
  A set with a cyclic order is called a cyclically ordered set or simply a cycle. Some familiar cycles are discrete, having only a finite number of elements: there are seven days of the week, four cardinal directions, twelve notes in the chromatic scale, and three plays in rock-paper-scissors. In a finite cycle, each element has a "next element" and a "previous element". There are also continuously variable cycles with infinitely many elements, such as the oriented unit circle in the plane.
  Cyclic orders are closely related to the more familiar linear orders, which arrange objects in a line.
context_section_description: A cyclic order on a set X with n elements is like an arrangement of X on a clock face, for an n-hour clock. Each element x in X has a "next element" and a "previous element", and taking either successors or predecessors cycles exactly once through the elements as x(1), x(2), ..., x(n).
  There are a few equivalent ways to state this definition. A cyclic order on X is the same as a permutation that makes all of X into a single cycle. A cycle with n elements is also a Zₙ-torsor: a set with a free transitive action by a finite cyclic group. Another formulation is to make X into the standard directed cycle graph on n vertices, by some matching of elements to vertices.
  It can be instinctive to use cyclic orders for symmetric functions, for example as in
  xy + yz + zx
  where writing the final monomial as xz would distract from the pattern.
  A substantial use of cyclic orders is in the determination of the conjugacy classes of free groups. Two elements g and h of the free group F on a set Y are conjugate if and only if, when they are written as products of elements y and y⁻¹ with y in Y, and then those products are put in cyclic order, the cyclic orders are equivalent under the rewriting rules that allow one to remove or add adjacent y and y⁻¹.
  A cyclic order on a set X can be determined by a linear order on X, but not in a unique way. Choosing a linear order is equivalent to choosing a first element, so there are exactly n linear orders that induce a given cyclic order. Since there are n! possible linear orders, there are (n − 1)! possible cyclic orders.
caption: A 5-element cycle