Positive Artin and parabolic monoids

PDF p. 21: Definition 5.1, second bullet

Let $(W,S)$ be a finite-rank Coxeter system with $S=\{s_i:i\in {\mathrm{\Gamma }}_0\}$, let $A$ be its Artin–Tits group with Artin generators ${\sigma }_i$, and let $I\subseteq {\mathrm{\Gamma }}_0$. Write $A_I$ for the standard parabolic subgroup generated by the ${\sigma }_i$ with $i\in I$.

A monoid is a set with an associative binary operation and a two-sided identity. A submonoid of a monoid is a subset containing the identity and closed under the operation. The positive Artin submonoid $A^{+}$ is the submonoid of $A$ generated by all its Artin generators: explicitly, it consists of all finite products ${\sigma }_{i_1}\cdots {\sigma }_{i_n}$ with $n\ge 0$ and $i_k\in {\mathrm{\Gamma }}_0$. The positive parabolic submonoid $A_I^{+}$ is the submonoid of $A_I$ generated by $\{{\sigma }_i:i\in I\}$: explicitly, it consists of the same finite products with every $i_k\in I$. In both cases the empty product is the identity of $A$.

These definitions do not require spherical type, meaning that $W$ is finite. The article imposes spherical type in Sections 5 and 6.

Coxeter-system, roots, length and weak-order conventions

Artin–Tits group, pure subgroup and positive lifts

Standard parabolic subgroups and parabolic roots

Semigroup and monoid

The Artin group and its generators are supplied by Artin–Tits group, pure subgroup and positive lifts, and Standard parabolic subgroups and parabolic roots supplies $A_I$ as the subgroup generated by the indicated elements. A group is a monoid under its multiplication, and the submonoid convention is that of Semigroup and monoid.

Each displayed set contains the identity as the empty product and is closed under multiplication: concatenate two finite lists of indices. Associativity and the identity law are inherited from the ambient group. Every product with indices in $I$ belongs to the subgroup $A_I$, so the second set is a submonoid of $A_I$. Conversely, any submonoid containing the indicated generators contains every such finite product, by repeated closure under multiplication, as well as the empty product. Thus each displayed set is the unique smallest submonoid containing its stated generators. In particular, $A_I^{+}\subseteq A^{+}$, and when $I=\mathrm{\varnothing }$ it consists only of the identity.