Quasigroups as Boolean functions, their equation systems and Groebner bases

In this short note we represent quasigroups of order $2^n$ as vector valued Boolean functions $f:\{0,1\}^{2n}\rightarrow \{0,1\}^n$. We use this representation to give a natural classification of the class of finite quasigroups in two subclasses: the subclass of linear quasigroups and the subclass of nonlinear quasigroups. Also, the representation of finite quasigroups as vector valued Boolean functions allows us to solve systems of quasigroup equations by using Gr\"{o}bner bases.