Gray-ordered Binary Necklaces

A $k$-ary 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.