A horizontal mesh algorithm for posets with positive Tits form

Following our paper [Fund. Inform. 136 (2015), 345–379], we define a horizontal mesh algorithm that constructs a $\widehat{\Phi}_I$-mesh translation quiver $\Gamma(\widehat{\mathcal{R}}_I,\widehat{\Phi}_I)$ consisting of $\widehat{\Phi}_I$-orbits of the finite set $\widehat{\mathcal{R}}_I=\{v\in\mathbb{Z}^I\; ;\;\widehat{q}_I(v)=1\}$ of Tits roots of a poset $I$ with positive definite Tits quadratic form $\widehat q_I:\mathbb{Z}^I \to \mathbb{Z}$. Under the assumption that $\widehat q_I:\mathbb{Z}^I \to \mathbb{Z}$ is positive definite, the algorithm constructs $\Gamma(\widehat{\mathcal{R}}_I,\widehat{\Phi}_I)$ such that it is isomorphic with the $\widehat{\Phi}_D$-mesh translation quiver $\Gamma({\mathcal{R}}_D,{\Phi}_D)$ of $\widehat{\Phi}_D$-orbits of the finite set ${\mathcal{R}}_D$ of roots of a simply laced Dynkin quiver $D$ associated with $I$.