Garside Structure, Normal Forms, and the Center

Let $n\in \mathbb{N}$ (The natural numbers $\mathbb{N}$ (von Neumann)). Put

${\mathrm{\Sigma }}_n:=\{{\sigma }_1,\dots ,{\sigma }_{n-1}\},$

an alphabet of $n-1$ symbols for $n\ge 2$, with ${\mathrm{\Sigma }}_n=\mathrm{\varnothing }$ for $n\le 1$. A positive braid word on $n$ strands, or simply a positive word, is a finite string of letters from ${\mathrm{\Sigma }}_n$ only, with no formal inverse letters. Thus it is a word in the sense of Words in an alphabet with formal inverses, elementary cancellation, and reduced words restricted to the original alphabet; its letters are read from left to right, its length $\mathrm{\mid }w\mathrm{\mid }$ is the number of its letters, and the empty word is denoted $\epsilon$. Concatenation of words makes the set ${\mathrm{\Sigma }}_n^{\ast }$ of all positive words into a monoid with identity $\epsilon$ (Semigroup and monoid).

The defining relation pairs. Let $R_n$ be the set of pairs of positive words consisting of

$({\sigma }_i{\sigma }_{i+1}{\sigma }_i,{\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1})(1\le i\le n-2),({\sigma }_i{\sigma }_j,{\sigma }_j{\sigma }_i)(\mathrm{\mid }i-j\mathrm{\mid }>1)\mathrm{.}$

These are the same two families of words that occur in the Artin presentation of The braid group by Artin presentation, with each relation now read as a pair of words rather than as an equation between group elements; the index sets are empty when the indicated range contains no integer, so that for $n\le 2$ the second family is the only one, and for $n\le 1$ both families are empty.

The congruence ${\equiv }^{+}$. A congruence on ${\mathrm{\Sigma }}_n^{\ast }$ is an equivalence relation $\sim$ on ${\mathrm{\Sigma }}_n^{\ast }$ (Equivalence relation, equivalence class, and the quotient set $A\mathrm{/}\sim$) such that $u\sim v$ implies $xuy\sim xvy$ for all words $x,y\in {\mathrm{\Sigma }}_n^{\ast }$. The intersection of any nonempty family of congruences is again a congruence, and the total relation is a congruence, so there is a smallest congruence containing any prescribed set of pairs of words. Let ${\equiv }^{+}$ be the smallest congruence on ${\mathrm{\Sigma }}_n^{\ast }$ containing every pair in $R_n$, that is, containing $w$ and $w^{\mathrm{\prime }}$ whenever $w=xuy$, $w^{\mathrm{\prime }}=xvy$ and $(u,v)\in R_n$ or $(v,u)\in R_n$ for some words $x,y$.

The monoid. The positive braid monoid on $n$ strands is the quotient monoid

$B_n^{+}:={\mathrm{\Sigma }}_n^{\ast }\mathrm{/}\text{\hspace{0.17em}⁣}{\equiv }^{+},$

with elements written $[w]$ for $w\in {\mathrm{\Sigma }}_n^{\ast }$ and with product $[u]\cdot [v]:=[uv]$. This product is well defined, because ${\equiv }^{+}$ is compatible with concatenation, and it is associative with two-sided identity $[\epsilon ]$, since concatenation has these properties on words (Semigroup and monoid). The elements of $B_n^{+}$ are called positive braids, and ${\bar{\sigma }}_i:=[{\sigma }_i]$ are the Artin generators of $B_n^{+}$. For $n\le 1$ no generators occur and $B_n^{+}$ is the trivial monoid $\{[\epsilon ]\}$.

Universal property. $B_n^{+}$ is generated as a monoid by ${\bar{\sigma }}_1,\dots ,{\bar{\sigma }}_{n-1}$. Moreover, if $M$ is any monoid and $a_1,\dots ,a_{n-1}\in M$ satisfy $a_i{a}_{i+1}{a}_i=a_{i+1}{a}_i{a}_{i+1}$ for $1\le i\le n-2$ and $a_i{a}_j=a_j{a}_i$ for $\mathrm{\mid }i-j\mathrm{\mid }>1$, then there is exactly one monoid homomorphism $\phi \text{\hspace{0.17em}⁣}:B_n^{+}\to M$ with $\phi ({\bar{\sigma }}_i)=a_i$: evaluating a positive word letter by letter defines a homomorphism ${\mathrm{\Sigma }}_n^{\ast }\to M$ which identifies the two words of every pair in $R_n$ and therefore identifies ${\equiv }^{+}$-equivalent words (by the minimality of ${\equiv }^{+}$), so it descends to the quotient; uniqueness holds because the ${\bar{\sigma }}_i$ generate the quotient monoid.

Comparison with $B_n$. The presentation of The braid group by Artin presentation uses the same symbols and the same relations, but it is a group presentation: there the symbols are invertible and the whole group $B_n$ is the quotient of the free group on $\{{\sigma }_1,\dots ,{\sigma }_{n-1}\}$. Here no inverse symbols occur at all: a positive braid is a class of words in the generators only, and $B_n^{+}$ is a monoid that is not a priori a group, nor a priori a submonoid of $B_n$. That $B_n^{+}$ is cancellative, that it embeds into its group of fractions (which is $B_n$), and that ${\bar{\sigma }}_i$ is not invertible in $B_n^{+}$, are proved on this page, in The positive braid monoid is left and right cancellative and The group of fractions of the positive braid monoid is the Artin braid group; until those results are available, "positive braid" always means an element of $B_n^{+}$ as defined above, not a braid that happens to be expressible by a positive word.

• The empty word and the identity are both written $1$ when no confusion is possible; $B_n^{+}$ is generated by the ${\bar{\sigma }}_i$, and every element is a product ${\bar{\sigma }}_{i_1}\cdots {\bar{\sigma }}_{i_k}$ for some $k\ge 0$.
• Length is at present a function of words, not of elements: no length on $B_n^{+}$ is defined here, because it is not yet known that ${\equiv }^{+}$-equivalent words have the same length. That invariance, together with the finiteness of the set of words of each fixed length, is the subject of Positive artin relations preserve homogeneous length.
• All relations in $R_n$ are positive and homogeneous: both sides of each pair are nonempty and have the same number of letters. No relation of the form $u=\epsilon$ with $u$ nonempty occurs, which is why the quotient is expected to have no nontrivial invertible element; this is proved as conicality in Positive artin relations preserve homogeneous length.
• The construction above applies to every $n\in \mathbb{N}$: for $n=0$ and $n=1$ the monoid $B_n^{+}$ is trivial and there are no Artin generators. The half twist $\mathrm{\Delta }$ is defined in The Garside half twist and simple positive braids, where $\mathrm{\Delta }=1$ for these two values of $n$.

Let $n\in \mathbb{N}$ and let $B_n^{+}$ be the positive braid monoid of Positive braid monoid, with generators ${\bar{\sigma }}_i$ and defining pairs $R_n$. Then:

(a) Any two ${\equiv }^{+}$-equivalent positive words have the same length. Consequently there is a well-defined function

$\mathrm{\ell }\text{\hspace{0.17em}⁣}:B_n^{+}⟶\mathbb{N},\mathrm{\ell }([w]):=\mathrm{\mid }w\mathrm{\mid },$

which is a monoid homomorphism: $\mathrm{\ell }(1)=0$ and $\mathrm{\ell }(xy)=\mathrm{\ell }(x)+\mathrm{\ell }(y)$ for all $x,y\in B_n^{+}$.

(b) For every $k\in \mathbb{N}$ the set $\{x\in B_n^{+}:\mathrm{\ell }(x)=k\}$ is finite; more precisely there are exactly $\mathrm{\mid }{\mathrm{\Sigma }}_n{\mathrm{\mid }}^k$ positive words of length $k$ over the alphabet ${\mathrm{\Sigma }}_n=\{{\sigma }_1,\dots ,{\sigma }_{n-1}\}$, and $B_n^{+}$ contains at most $\mathrm{\mid }{\mathrm{\Sigma }}_n{\mathrm{\mid }}^k$ elements of length $k$. For $n\ge 2$ the alphabet has $n-1$ letters and the bound reads $(n-1{)}^k$; for $n\le 1$ the alphabet is empty, so the word count is $1$ for $k=0$ and $0$ for $k\ge 1$, and $B_n^{+}=\{1\}$.

(c) Conicality. $\mathrm{\ell }(x)=0$ if and only if $x=1$. If $x_1,\dots ,x_r\in B_n^{+}$ and $x_1\cdots x_r=1$, then $x_1=\cdots =x_r=1$; in particular $xy=1$ forces $x=y=1$, so the only invertible element of $B_n^{+}$ is $1$.

(d) $\mathrm{\ell }({\bar{\sigma }}_{i}x)=\mathrm{\ell }(x)+1>\mathrm{\ell }(x)$ for every $i$ and every $x\in B_n^{+}$.

No choice principle is used; all arguments are finite inductions on word length.

• Part (a) is the invariance of homogeneous length: the two sides of every defining relation have the same number of letters, so the congruence cannot change length. This is exactly the property that makes the length of a word a function of its class.
• Part (b) is the "locally finite" input for later arguments: at each length only finitely many elements exist, so a search over positive words of a fixed length is a finite search.
• Part (d) says in the language of Positive braid monoid that the word-length function $w\mapsto \mathrm{\mid }w\mathrm{\mid }$ is a right-Noetherianity witness for the Artin presentation: it does not decrease when a generator is appended, and it strictly increases in the presence of a generator because no generator is invertible (step 4.1).

Let $n\in \mathbb{N}$, with the positive braid monoid $B_n^{+}$ of Positive braid monoid, its alphabet ${\mathrm{\Sigma }}_n=\{{\sigma }_1,\dots ,{\sigma }_{n-1}\}$, and its defining pairs $R_n$. Throughout, $s,t$ range over letters of ${\mathrm{\Sigma }}_n$ and $u,v,w$ over positive words.

The syntactic right complement. Define a function $\theta$ on pairs of letters by

$\theta (s,s):=\epsilon ,\theta ({\sigma }_i,{\sigma }_j):={\sigma }_j{\sigma }_i(\mathrm{\mid }i-j\mathrm{\mid }=1),\theta ({\sigma }_i,{\sigma }_j):={\sigma }_j(\mathrm{\mid }i-j\mathrm{\mid }\ge 2)\mathrm{.}$

Then $s\theta (s,t)$ and $t\theta (t,s)$ are the two sides of a defining pair of $R_n$ when $s\mathrm{\ne }t$, and are equal words when $s=t$. Indeed: if $s=t$ both words are the one-letter word $s$; if $s={\sigma }_i$, $t={\sigma }_j$ with $\mathrm{\mid }i-j\mathrm{\mid }=1$ they are ${\sigma }_i{\sigma }_j{\sigma }_i$ and ${\sigma }_j{\sigma }_i{\sigma }_j$, the two sides of the braid pair; if $\mathrm{\mid }i-j\mathrm{\mid }\ge 2$ they are ${\sigma }_i{\sigma }_j$ and ${\sigma }_j{\sigma }_i$, the two sides of the commutation pair. Consequently

$[s\theta (s,t)]=[t\theta (t,s)]\text{in }{B}_n^{+}\text{ for all letters }s,t,$

and for $s\mathrm{\ne }t$ the pair $\{s\theta (s,t),t\theta (t,s)\}$ is the unique pair of $R_n$ whose two sides begin with $s$ and with $t$ respectively. In the terminology of the source, the presentation of $B_n^{+}$ is right-complemented with syntactic right complement $\theta$.

The complement recursion. $\theta$ is extended to a map $\mathrm{\Theta }$ on pairs of positive words, written $\mathrm{\Theta }(u,v)$, by evaluating the following recursion in the order described. For a letter $s$ and a word $v$:

$\mathrm{\Theta }(s,\epsilon )=\epsilon ,\mathrm{\Theta }(s,tv)=\theta (s,t)\mathrm{\Theta }(\theta (t,s),v)(t\in {\mathrm{\Sigma }}_n,v\in {\mathrm{\Sigma }}_n^{\ast });$

and for words $u,v$:

$\mathrm{\Theta }(\epsilon ,v)=v,\mathrm{\Theta }(u,\epsilon )=\epsilon ,\mathrm{\Theta }(s{u}^{\mathrm{\prime }},v)=\mathrm{\Theta }(u^{\mathrm{\prime }},\mathrm{\Theta }(s,v))\mathrm{.}$

These rules are not a description by induction on the pair: the rule $\mathrm{\Theta }(s,tv)=\theta (s,t)\mathrm{\Theta }(\theta (t,s),v)$ expresses $\mathrm{\Theta }$ at $(s,tv)$ through its value at $(\theta (t,s),v)$, whose first entry $\theta (t,s)$ may itself be a two-letter word and is therefore not smaller. The rules are the recursion rules of the source, whose well-definedness is the content of its Lemma 4.32: in the right-complemented case the squares of the grid are filled in a unique way, the reversing procedure terminates or not independently of the order in which the steps are enumerated, and the rules above describe the resulting terminal pair. We therefore take $\mathrm{\Theta }(u,v)$ to be the partial map so defined, exactly as in the source, its agreement with the recursion rules being read off from the terminal pair by induction on the number of reversing steps (The principle of mathematical induction): $\mathrm{\Theta }(u,v)$ is defined if and only if the reversing of the negative--positive signed path $u^{-1}v$ (the letters of $u$ read negatively, then those of $v$ positively) reaches a terminal pair of blocks, and it is then the first block of that pair, while $\mathrm{\Theta }(v,u)$ is the second. Its four defining rules, and the fact that it is the least extension of $\theta$ satisfying them, are established as part (a) of Artin positive word reversing is complete ↗; the same item shows that $\mathrm{\Theta }$ is undefined exactly on those pairs of words that admit no common right multiple in $B_n^{+}$, so it is defined on every pair as soon as $\mathrm{\Delta }$-power divisibility is available (Every positive braid divides a power of the half twist on both sides): the totality of $\mathrm{\Theta }$ is a theorem, not part of the definition. The defining rules of the source are recovered as $\mathrm{\Theta }({\sigma }_i,{\sigma }_j)=\theta ({\sigma }_i,{\sigma }_j)$, $\mathrm{\Theta }(u_1{u}_2,v)=\mathrm{\Theta }(u_2,\mathrm{\Theta }(u_1,v))$ for $u_1$ nonempty, $\mathrm{\Theta }(u,{\epsilon }_x)=\epsilon$ and $\mathrm{\Theta }({\epsilon }_x,u)=u$.

Word reversing. A signed path is a finite word whose letters are signed copies $s^{+}$ or $s^{-}$ of letters $s\in {\mathrm{\Sigma }}_n$. These are formal words in the signed alphabet of Words in an alphabet with formal inverses, elementary cancellation, and reduced words, with concatenation as the word operation; negative letters are not morphisms of the positive monoid. For $u=s_1\cdots s_k$, the notation $u^{-1}$ means the formal word $s_k^{-1}\cdots s_1^{-1}$, in reversed order. A right-reversing step replaces a negative--positive subpath $s^{-1}t$ by $\theta (s,t)\theta (t,s{)}^{-1}$, using the defining relation $s\theta (s,t)=t\theta (t,s)$, and deletes $s^{-1}s$. This is the source's syntactic transformation on signed words, not an equality in $B_n^{+}$; it preserves the represented element in the presented group, although the number of signed letters may change. In the source's convention the pattern is a negative--positive pair: right-reversing acts on the signed path $u^{-1}v$, in which all letters of $u$ are read negatively and all letters of $v$ positively, and, when it terminates, it reaches a terminal positive--negative path $v^{\mathrm{\prime }}(u^{\mathrm{\prime }}{)}^{-1}$ whose two positive blocks satisfy $u{v}^{\mathrm{\prime }}{\equiv }^{+}v{u}^{\mathrm{\prime }}$. (Feeding the opposite orientation $u{v}^{-1}$ instead would already be terminal: it is the negative--positive path $u^{-1}v$ that encodes the comparison of the two positive words.)

The intended use. The pair $(\mathrm{\Theta }(u,v),\mathrm{\Theta }(v,u))$ is the pair that reversing is meant to compute: the source's Lemma II.4.32 identifies the terminal blocks of the reversing of $u^{-1}v$ with $v^{\mathrm{\prime }}=\mathrm{\Theta }(u,v)$ and $u^{\mathrm{\prime }}=\mathrm{\Theta }(v,u)$, and consequently $u\mathrm{\Theta }(u,v){\equiv }^{+}v\mathrm{\Theta }(v,u)$: the common word $u\mathrm{\Theta }(u,v)=v\mathrm{\Theta }(v,u)$ in $B_n^{+}$ is a common right multiple of $[u]$ and $[v]$, and it is their least common right multiple whenever a common right multiple exists. Both statements, together with the coherence of the recursion rules under the other evaluation order, are the content of Artin positive word reversing is complete ↗; no lcm property is used in this definition.

A worked value. By the recursion,

$\mathrm{\Theta }({\sigma }_2{\sigma }_1,{\sigma }_3)=\mathrm{\Theta }({\sigma }_1,\mathrm{\Theta }({\sigma }_2,{\sigma }_3))=\mathrm{\Theta }({\sigma }_1,{\sigma }_3{\sigma }_2)=\theta ({\sigma }_1,{\sigma }_3)\mathrm{\Theta }(\theta ({\sigma }_3,{\sigma }_1),{\sigma }_2)={\sigma }_3\mathrm{\Theta }({\sigma }_1,{\sigma }_2)={\sigma }_3{\sigma }_2{\sigma }_1,$

and likewise $\mathrm{\Theta }({\sigma }_1{\sigma }_2,{\sigma }_3{\sigma }_2)={\sigma }_3{\sigma }_2{\sigma }_1$ by the source's Example 4.11. Both values are used on the companion examples page.

• Only the two words $\mathrm{\Theta }(u,v)$ and $\mathrm{\Theta }(v,u)$ are needed on this page; they are the "two sides" of a rectangle whose vertical side carries $u$ and whose horizontal side carries $v$. Each of the two words records how far the other side has to be extended so that the two extensions match.
• The recursion rules are the algebraic transcription of the square-filling process of the source: the square on the letters $s,t$ has lower side $\theta (s,t)$ and right side $\theta (t,s)$, and the identity $s\theta (s,t)=t\theta (t,s)$ in $B_n^{+}$ is the commutativity of that square. The coherence of the two evaluation orders ("first the first letter of $u$, then the rest" versus "split the second argument") is the technical content of the source's Lemma II.4.32 and is established in Artin positive word reversing is complete ↗.
• No choice principle occurs: $\mathrm{\Theta }$ is computed on positive words by the four recursion rules above, and every verification below is a finite computation. Negative letters occur only in the signed paths that witness the reversing, and they are not elements of $B_n^{+}$; the recursion is partial in general, and its totality for the Artin presentation is the theorem of Every positive braid divides a power of the half twist on both sides together with Artin positive word reversing is complete ↗.

Let $n\in \mathbb{N}$ and let $\mathrm{\Theta }$ be the right complement of Artin right complements and word reversing, with ${\equiv }^{+}$ the congruence of Positive braid monoid. For letters $u,v,w\in {\mathrm{\Sigma }}_n$ put

${\mathrm{\Theta }}_3(u,v,w):=\mathrm{\Theta }(\mathrm{\Theta }(u,v),\mathrm{\Theta }(u,w)),{\mathrm{\Theta }}_3(v,u,w):=\mathrm{\Theta }(\mathrm{\Theta }(v,u),\mathrm{\Theta }(v,w))\mathrm{.}$

Then, for every triple of letters $u,v,w\in {\mathrm{\Sigma }}_n$, the two words ${\mathrm{\Theta }}_3(u,v,w)$ and ${\mathrm{\Theta }}_3(v,u,w)$ are defined and ${\equiv }^{+}$-equivalent; that is, the $\theta$-cube condition of the source holds for every triple of generators of the Artin presentation. In the case of three consecutive indices the values are, for $1\le i\le n-3$,

${\mathrm{\Theta }}_3({\sigma }_i,{\sigma }_{i+1},{\sigma }_{i+2})={\sigma }_{i+2}{\sigma }_{i+1}{\sigma }_i={\mathrm{\Theta }}_3({\sigma }_{i+1},{\sigma }_i,{\sigma }_{i+2}),$

${\mathrm{\Theta }}_3({\sigma }_{i+1},{\sigma }_{i+2},{\sigma }_i)={\sigma }_i{\sigma }_{i+1}{\sigma }_{i+2}={\mathrm{\Theta }}_3({\sigma }_{i+2},{\sigma }_{i+1},{\sigma }_i),$

${\mathrm{\Theta }}_3({\sigma }_{i+2},{\sigma }_i,{\sigma }_{i+1})={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+2}{\sigma }_{i+1}{\equiv }^{+}{\sigma }_{i+1}{\sigma }_{i+2}{\sigma }_i{\sigma }_{i+1}={\mathrm{\Theta }}_3({\sigma }_i,{\sigma }_{i+2},{\sigma }_{i+1}),$

where the last equivalence uses the commutation ${\sigma }_i{\sigma }_{i+2}={\sigma }_{i+2}{\sigma }_i$. No choice principle is used and every value is obtained by finitely many applications of the recursion of Artin right complements and word reversing.