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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.

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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 $n$.

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