  According to Shibukawa, ternary systems defined on quasigroups and satisfying
certain conditions provide a way of constructing dynamical Yang-Baxter maps.
After noticing that these conditions can be interpreted as 3-dimensional
compatibility of equations on quad-graphs, we investigate when the associated
dynamical Yang-Baxter maps are in fact parametric Yang-Baxter maps. In some
cases these maps can be obtained as reductions of higher dimensional maps
through compatible constraints. Conversely, parametric YB maps on quasigroups
with an invariance condition give rise to 3-dimensional compatible systems. The
application of this method on spaces with certain quasigroup structures
provides new examples of multi-parametric YB maps and 3-dimensional compatible
systems.
