Im trying to understand the relationship between lyndon words and necklaces. The sequence for lyndon words is: $$ ε, 0, 1, 01, 001, 011, 0001, 0011, 0111, 00001, 00011, 00101, 00111, 01011, 01111 $$
For Necklaces the following sequence includes 00, 01, and 11:
Why don't lyndon words contain 00 and 11? I have read that
A Lyndon word is an aperiodic notation for representing a necklace.
but clearly the necklace of 00 and 11 is not being represented. Also why is 0101 not included but 00101 is?
On
By definition, a Lyndon word is a word that is lexicographically smaller than every of its rotations (where $z$ is a rotation of $z'$ iff there exist words $u,v$ such that $z=uv$ and $z'=vu$.). If a word is a power (i.e., a concatenation of copies of a shorter word), then it cannot be Lyndon. Indeed, in this case there is a rotation that equals the original word, so that the original word is not strictly smaller than every of its rotations.
If you relax the condition in the definition by imposing that a word must be smaller or equal than every of its rotations, then you get the definition of a necklace. In fact, necklaces are just powers of Lyndon words (meaning that a necklace is a word of the form $u^n$, for a Lyndon word $u$ and an integer $n\geq 0$).
It's because both $00$ and $11$ are periodic, and are covered already by the inclusion of $0$ and $1$.