Lyndon bases of split $\mathrm{ı}$quantum groups

We introduce and study Lyndon bases of split $\mathrm{ı}$quantum groups ${\mathbf{U}}^{\mathrm{ı}}(\mathfrak{g})$. A relationship between the Lyndon bases and PBW-type bases was provided. As an application, we establish the existence of canonical bases for the type A split $\mathrm{ı}$quantum groups ${\mathbf{U}}^{\mathrm{ı}}({\mathfrak{s}\mathfrak{l}}_n)$.