The reduced positive section b_w, its length additivity, and the degree homomorphism

The product ${\sigma }_{s_1}\cdots {\sigma }_{s_k}$ is independent of the reduced expression: by Matsumoto's theorem (Matsumoto's theorem: braid connectivity of reduced expressions, with singleton detection in dihedral subgroups (1)), any two reduced expressions of an element $w\in W$ are connected by finitely many replacements of an alternating subword $sts\cdots$ of length $m(s,t)<\mathrm{\infty }$ by the other alternating subword $tst\cdots$, inside some context $x(\cdot )y$. Such a subword pair is exactly a braid pair of Artin monoid and Artin group presentations, and the canonical monoid-to-group map (2), so by the congruence property of ${\equiv }^{+}$ the two full words are equivalent and their classes in $A^{+}$ coincide; hence the product in $A^{+}$ depends only on $w$. A reduced expression of $w$ exists because $\mathrm{\ell }(w)$ is a minimum over a nonempty set of word lengths (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups).

The empty word is a reduced expression of $1$ because $\mathrm{\ell }(1)=0$, so $b_1=[\epsilon ]=1_{A^{+}}$ by [F1]. For the projection, [F4] gives ${\pi }^{+}({\sigma }_s)=s$ and $\pi \circ \gamma ={\pi }^{+}$ while [F9] makes $\gamma$ a monoid homomorphism, so for a reduced expression $w=s_1\cdots s_k$ one has ${\pi }^{+}(b_w)={\pi }^{+}({\sigma }_{s_1})\cdots {\pi }^{+}({\sigma }_{s_k})=s_1\cdots s_k=w$, and composing with $\pi \circ \gamma ={\pi }^{+}$ gives $\pi \circ \gamma \circ b={\pi }^{+}\circ b={\mathrm{i}\mathrm{d}}_W$.

The length map exists and is unique: apply the monoid universal property [F3] to $M:=(\mathbb{N},+,0)$ and the constant map $f(s):=1$, which is legitimate because the two words of a braid pair both have length $m(s,t)$, so their images under the word-length map agree; this gives a unique monoid homomorphism $L:A^{+}\to (\mathbb{N},+,0)$ with $L({\sigma }_s)=1$. For a word $u=s_1\cdots s_k$ one has $L([u])=L({\sigma }_{s_1}\cdots {\sigma }_{s_k})=1+\cdots +1=k$ by additivity [F6] and $L([\epsilon ])=0$ [F7], so $L$ is the word length on classes and in particular $L(b_w)=k=\mathrm{\ell }(w)$ for every $w$.

The degree map exists: apply the group universal property [F2] to $G:=(\mathbb{Z},+,0)$ of [F8] and the constant map $f(s):=1$, whose two values on a braid pair both equal $m(s,t)$; this gives a group homomorphism $\mathrm{deg}:A\to \mathbb{Z}$ with $\mathrm{deg}({\sigma }_s)=1$. Then $\mathrm{deg}\circ \gamma$ and $L$ are monoid homomorphisms $A^{+}\to \mathbb{Z}$ agreeing on every generator: $\mathrm{deg}(\gamma ({\sigma }_s))=\mathrm{deg}({\sigma }_s)=1=L({\sigma }_s)$, so by the uniqueness in [F3] they agree on all of $A^{+}$, that is, $\mathrm{deg}(\gamma (x))=L(x)$ for every $x\in A^{+}$.

The map $b$ is injective and a section: if $b_u=b_v$ then $u={\pi }^{+}(b_u)={\pi }^{+}(b_v)=v$, and ${\pi }^{+}\circ b={\mathrm{i}\mathrm{d}}_W$ and $\pi \circ \gamma \circ b={\mathrm{i}\mathrm{d}}_W$ were proved in step 2.1; a map with a left inverse is injective, so $b$ is a set-theoretic section of both maps.

For the forward direction of (4), suppose $\mathrm{\ell }(uv)=\mathrm{\ell }(u)+\mathrm{\ell }(v)$ and let $u=s_1\cdots s_p$, $v=t_1\cdots t_q$ be reduced expressions. The concatenated word $s_1\cdots s_p{t}_1\cdots t_q$ represents $uv$ and has length $p+q=\mathrm{\ell }(u)+\mathrm{\ell }(v)=\mathrm{\ell }(uv)$, so it is a reduced expression of $uv$; hence by step 1.1, $b_{uv}={\sigma }_{s_1}\cdots {\sigma }_{s_p}{\sigma }_{t_1}\cdots {\sigma }_{t_q}=b_u{b}_v$. Conversely, if $b_u{b}_v=b_{uv}$, then applying the additive map $L$ of step 2.2 gives $\mathrm{\ell }(u)+\mathrm{\ell }(v)=L(b_u)+L(b_v)=L(b_u{b}_v)=L(b_{uv})=\mathrm{\ell }(uv)$. In particular, if $\mathrm{\ell }(uv)<\mathrm{\ell }(u)+\mathrm{\ell }(v)$ then $L(b_u{b}_v)=\mathrm{\ell }(u)+\mathrm{\ell }(v)>\mathrm{\ell }(uv)=L(b_{uv})$, so $b_u{b}_v\mathrm{\ne }{b}_{uv}$.

If $S\mathrm{\ne }\mathrm{\varnothing }$, fix $s\in S$. By part 1 of Length parity, exchange, two-letter deletion, and faithfulness of the signed reflection action the sign character of $W$ satisfies $\mathrm{sgn}(s)=-1\mathrm{\ne }1=\mathrm{sgn}(1)$, so $s\mathrm{\ne }1$ in $W$; since $s$ is a word of length $1$ we have $\mathrm{\ell }(s)\le 1$, while $\mathrm{\ell }(s)\mathrm{\ne }0$ because the empty word is the only word of length $0$ and its value is $1$ (Coxeter matrices, the presented Coxeter group, reduced words, length, and standard parabolic subgroups), so $\mathrm{\ell }(s)=1$. Hence the one-letter word $s$ is a reduced expression and $b_s=[s]$ by step 1.1, while $s^2=1$ in $W$ with $\mathrm{\ell }(1)=0$ makes the empty word a reduced expression of $s^2$, so $b_{s^2}=b_1=[\epsilon ]$ by step 2.1. Hence $b_s{b}_s=[s][s]=[ss]$ by [F1], whereas $L([ss])=2\mathrm{\ne }0=L([\epsilon ])$ by step 2.2, so $b_s{b}_s\mathrm{\ne }{b}_{s^2}$ and $b$ is not a monoid homomorphism. For the composite, $(\gamma \circ b)(s{)}^2=\gamma ([ss])={\sigma }_s^2$ while $(\gamma \circ b)(s^2)=\gamma ([\epsilon ])=1_A$, and these differ because $\mathrm{deg}({\sigma }_s^2)=2\mathrm{\ne }0=\mathrm{deg}(1_A)$ by step 3.1, where $2=1+1\mathrm{\ne }0$ in $\mathbb{Z}$ is the nontriviality recorded in [F8], and a group homomorphism preserves the identity (Monoid homomorphism and group homomorphism); so $\gamma \circ b$ is not a group homomorphism.

Scope and choice: injectivity of $\gamma$ and of ${\pi }^{+}$, an Ore or Garside condition, an embedding of $A^{+}$ into $A$ and every topological statement are outside this result, and nothing here asserts them. No choice is used: $b_w$ is defined by the uniqueness proved in step 1.1, so it is a definite description rather than a selection among expressions, ${\equiv }^{+}$ is an intersection of a definable family of congruences, and all computations are finite.