Reduced and Cyclically Reduced Words in Unitary Letters

Let $\mathbb{N}$ be the set of natural numbers with its addition and order, and for $k\in\mathbb{N}$ let $[k]$ be the initial segment determined by $k$. Let $d\in\mathbb{N}$, and let $W_{2d}$, with empty word $\varnothing$, the letters $w_{1},\dots,w_{k}$ of a word $w$ of length $k\in\mathbb{N}$, and the inverse letters $l\mapsto l^{-1}$ of letters $l\in[2d]$ be as in Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words.

1. (Reduced words)¶ A word $w\in W_{2d}$ is reduced if $w=\varnothing$, or if $w$ has length $k\in\mathbb{N}$ and $w_{i+1}\neq(w_{i})^{-1}$ for every $i\in[k]$ with $i<k$. Here $i+1\in[k]$: by claim 7 of Properties of the Order on the Natural Numbers there is $m\in\mathbb{N}$ with $k=i+m$, and $1\le m$ by claim 4 of Properties of the Order on the Natural Numbers, so that $i+1\le i+m=k$ by claim 6 of Properties of the Order on the Natural Numbers.

2. (Cyclically reduced words)¶ A word $w\in W_{2d}$ is cyclically reduced if it is reduced and, in case $w$ has length $k\in\mathbb{N}$ with $1<k$, $w_{k}\neq(w_{1})^{-1}$. We write $W^{\circ}_{d}$ for the set of cyclically reduced words and, for $k\in\mathbb{N}$, $W^{\circ}_{d,k}$ for the set of cyclically reduced words of length $k$.