# Cyclic order  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

The typical example of a cyclic order are people seated at a (round) table: Each person has a right-hand and a left-hand neighbour, and no position is distinguished from the others. The seating order can be described by listing the persons, starting from any arbitrary position, in clockwise (or counterclockwise) order.

## Mathematical formulation

The abstract concept analogous to sitting around a table can be described in mathematical terms as follows:

On a finite set S of n elements, consider a function σ that defines for each element s its successor σ(s).
This gives rise to a cyclic order if (and only if) for some element s the orbit under σ is the whole set S:

$S=\{\sigma ^{k}(s)\mid k=1,\dots ,n\}$ The reverse cyclic order is given by σ−1 (where σ−1(s) is the element preceding s).

Remarks:

1. If the condition holds for one element then it holds for all elements.
2. All cyclic orders of n elements are isomorphic.
3. Cyclic orders cannot be considered as order relations because both s < t and t < s would hold for any two distinct elements s and t.
4. Cyclic orders occur naturally in number theory (residue sets and group theory (cyclic groups, permutations).

## Examples

• (Alice, Bob, Celia, Don), (Bob, Celia, Don, Alice), (Celia, Don, Alice, Bob), and (Don, Alice, Bob, Celia) all describe the same cyclic (seating) order.
(Alice, Don, Celia, Bob) describes the reverse cyclic order, and (Alice, Celia, Bob, Don) describes a different cyclic order.
• The hours on a clock are in cyclic order: one o'clock follows twelve o'clock.
The counterclockwise order, two o'clock, one o'clock, twelve o'clock, etc., is also a cyclic order, the corresponding reverse cyclic order.
• The numbers 1,2,..,n taken in their natural order are in cyclic order if, in addition, 1 is considered as successor of n:
$\dots \rightarrow 1\rightarrow 2\rightarrow 3\rightarrow \dots \rightarrow (n-1)\rightarrow n\rightarrow 1\rightarrow \dots$ Assuming this definition (for n=3), all of (123), (231), (312) are in cyclic order, while (132), (213), (321) are in cyclic order reverse to it.