Cyclic order
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In combinatorial mathematics, a cyclic order on a set X with n elements is an arrangement of X as on a clock face, for an n-hour clock. That is, rather than an order relation on X, we define on X just functions 'element immediately before' and 'element immediately following' any given x, in such a way that taking predecessors, or successors, cycles once through the elements as x(1), x(2), ..., x(n). Another way to put it is to say that we make X into the standard n-cycle directed graph on n vertices, by some matching of elements to vertices. A popular example is Rock, Paper, Scissors.
Any such cyclic ordering corresponds to n different total orders on X, considered as 'biting their tails'. There are therefore (n − 1)! cyclic orders on X.
An infinite set can also be ordered cyclically. The precise definition is more involved, but the idea is the same: arrange the (infinitely many) members of the set around a circle. The examples below refer to finite cyclic orders, however.
It can be instinctive to use cyclic orders for symmetric functions, for example as in
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.
