# Terminology for a cyclically ordered set of objects

If I have an ordered set of objects (for concreteness, say they're integers) $(x_1,\ldots,x_n)$, I might call it a tuple of integers.

Perhaps, though, I have an set of integers $(x_1,\ldots,x_n)$ but the order is only defined up to cyclical permutation (imagine they're sitting at distinct points along a circle); so $(x_1,\ldots,x_n)$ is the same as $(x_2,\ldots,x_n,x_1)$. I thus can't call this a "tuple of integers" because that would imply there is a canonical ordering. Is there some standard term I can use, besides the unweildy phrase "cyclically ordered set of integers"?

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A necklace $\$ – Gjergji Zaimi Jul 26 '12 at 2:22
Even with the term necklace one needs some care. It matters in some cases that the necklace not be worn upside down. I suggest a less common term like circlet, with a modifier in case reversals are NOT allowed. Gerhard "Doesn't Know The Standard Term" Paseman, 2012.07.26 – Gerhard Paseman Jul 26 '12 at 16:30
Perhaps clocklet? Gerhard "OK I'll Stop For Now" Paseman, 2012.07.26 – Gerhard Paseman Jul 26 '12 at 16:34

It certainly depends on the context. I've seen 'cycle' or 'necklace' used before for this, but you had best define it first as these aren't universal.

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Cyclic orderings are somewhat rare beasts but they do show up here and there. A set with a cyclic ordering is a "cyclically ordered set", just as a set with a total (or partial) ordering is a "totally (or partially) ordered set". The formal definition of a cyclic ordering is found in http://en.wikipedia.org/wiki/Cyclic_order. Its just a certain kind of ternary relation on a set.

I suppose if you wanted to mimic "poset" for "partially ordered set" you could say "coset", but I would not advise it :-) and for an exhaustive list of the most common alternatives see the link above.

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If your objects that sit at distinct points along a circle are all elements of some set $L$, and repetitions are allowed, the term "cyclic words in the alphabet $L$" is fairly commonly used. If no repetitions are allowed, so that you just have an equivalence relation on total orderings of some set $A$, one usually talks about "cyclic orderings of $A$".

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