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In 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.
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.
It can be instinctive to use cyclic orders for symmetric
functions, for example as in
- xy + yz + zx
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.
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