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{\bf Christopher Degni and Arthur A.~Drisko }
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{\bf Gray-ordered Binary Necklaces}
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A $k$-ary {\em necklace\/} of order $n$ is an equivalence class of
strings of length $n$ of symbols from $\{0,1,\ldots,k-1\}$ under
cyclic rotation. In this paper we define an ordering on the free
semigroup on two generators such that the binary strings of length $n$
are in Gray-code order for each $n$. We take the binary necklace
class representatives to be the least of each class in this
ordering. We examine the properties of this ordering and in
particular prove that all binary strings factor canonically as
products of these representatives. We conjecture that stepping from
one representative of length $n$ to the next in this ordering requires
only one bit flip, except at easily characterized steps.
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