3D compatible ternary systems and Yang?Baxter maps

According to Shibukawa, ternary systems defined on quasi-groups and satisfying certain conditions provide a way of constructing dynamical Yang–Baxter (YB) maps. After noting that these conditions can be interpreted as three-dimensional compatibility of equations on quad-graphs, we investigate when the associated dynamical YB maps are in fact parametric YB maps. In some cases these maps can be obtained as reductions of higher dimensional maps through compatible constraints. Conversely, parametric YB maps on quasi-groups with an invariance condition give rise to three-dimensional compatible systems. The application of this method on spaces with certain quasi-group structures provides new examples of multi-parametric YB maps and three-dimensional compatible systems.