# Cyclic order

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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*:

The **reverse cyclic order** is given by σ^{−1} (where σ^{−1}(*s*) is the element preceding *s*).

**Remarks:**

- If the condition holds for one element then it holds for all elements.
- All cyclic orders of
*n*elements are isomorphic. - 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*. - 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*:

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.