Cyclic order

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.

The abstract concept analogous to this situation is, 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 describes a cyclic order if (and only if) for some element s the orbit under σ is the whole set S:

$S=\{\sigma ^{k}(s)\mid k=1,\dots ,n\}$

Remarks:

1. If this 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) describe the same cyclic (seating) order.
(Alice, Don, Celia, Bob) describes the reverse cyclic order, and (Alice, Celia, Bob, Don) describe a different cyclic order.

The hours on a clock are in cyclic order: one o'clock follows twelve o'clock.