# Cycle notation for cyclic orders

Is there a convenient cycle notation for cyclic orders (https://en.wikipedia.org/wiki/Cyclic_order)?

For example:

Definition. A set of four elements $$a, b, c, d$$ of a cyclically ordered set is a 4-cycle $$[a, b, c, d]$$ if $$[a,b,c] \land [c,d,a]$$.

Using transitivity it is easy to show that

$$[a, b, c, d] \implies [a,b,c]$$, $$[b,c,d]$$, $$[c,d,a]$$, $$[d,a,b]$$, and all the cyclic equivalents:
$$[b,c,a]$$, $$[c,a,b]$$, $$[c,d,b]$$, $$[d,b,c]$$, $$[a,c,d]$$, $$[d,a,c]$$, $$[a,b,d]$$, $$[b,d,a]$$.

Which also means $$[a, b, c, d] \iff [b, c, d, a] \iff [c, d, a, b] \iff [d, a, b, c]$$.

It could simplify the ternary arithmetic.

I am not aware of any standard notation, but your notation $$[a, b, c, d]$$ makes sense and could be extended to $$[a_1, a_2, \dotsm, a_n]$$ for any $$n$$.