Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words

In the setting of The Real Numbers: Standing Notation and Background, let $\mathbb{N}$ be the set of natural numbers with its addition and order, let $\mathbb{R}$ be the real numbers as fixed there, into which $\mathbb{N}$ is carried by the natural-number image, and for $k\in\mathbb{N}$ let $[k]$ be the initial segment determined by $k$. Let $d\in\mathbb{N}$, write $2d$ for the natural number $d+d$, and let $W_{2d}$ be the set of words in the letters $1,\dots,2d$, with empty word $\varnothing$, concatenation $(u,v)\mapsto uv$ and reversal $w\mapsto w^{\mathrm{rev}}$. A letter $l\in[2d]$ is identified with the word $(l)$ of length $1$, and the letters of a word $w$ of length $k$ are its values $w_{1},\dots,w_{k}$. The letter $j\in[d]$ is to be read as a unitary $u_{j}$, and the letter $d+j$ as its inverse.

1. (Generators, signs and inverse letters)¶ Let $l\in[2d]$, so that $l\le 2d$. If $l\le d$, put $g(l)=l$, $\varepsilon(l)=1$ and $l^{-1}=d+l$; here $d+l\in[2d]$, because $d+l\le d+d$ by claim 6 of Properties of the Order on the Natural Numbers. If $d<l$, then by claim 7 of Properties of the Order on the Natural Numbers there is exactly one $j\in\mathbb{N}$ with $l=d+j$, and $j\le d$: otherwise $d<j$ by claim 3 of Properties of the Order on the Natural Numbers, that is, $j=d+m$ for some $m\in\mathbb{N}$ by Order on the Natural Numbers, so that $l=d+(d+m)=(d+d)+m$ by claim 3 of Arithmetic of Addition on the Natural Numbers and hence $2d<l$, which together with $l\le2d$ contradicts claim 2 of Properties of the Order on the Natural Numbers. Thus $j\in[d]$; put $g(l)=j$, $\varepsilon(l)=-1$ and $l^{-1}=j$. The number $g(l)\in[d]$ is the generator, $\varepsilon(l)\in\mathbb{R}$ the sign and $l^{-1}\in[2d]$ the inverse letter of $l$.

2. (Lengths)¶ The length of $w\in W_{2d}$ is the real number $|w|$ equal to $0$ if $w=\varnothing$, and to the image in $\mathbb{R}$ of $k$ if $w$ is a word of length $k\in\mathbb{N}$.

3. (Adjoint words)¶ The adjoint of $w\in W_{2d}$ is the word $w^{*}\in W_{2d}$ given by $\varnothing^{*}=\varnothing$ and, if $w$ has length $k\in\mathbb{N}$, by the word $w^{*}$ of length $k$ with

$$(w^{*})_{i}=\bigl((w^{\mathrm{rev}})_{i}\bigr)^{-1}\qquad(i\in[k]).$$