Positive braid monoid

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 $n$.