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:

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

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:
${\displaystyle \dots \rightarrow 1\rightarrow 2\rightarrow 3\rightarrow \dots \rightarrow (n-1)\rightarrow n\rightarrow 1\rightarrow \dots }$
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.