zbMATH — the first resource for mathematics

Two enumeration principles for finite free algebras in locally finite varieties, based respectively on essential dependence and linearity, are discussed and illustrated. A concept of linearly essential dependence of an algebraic operation on its arguments is then introduced, enabling these two principles to be combined. The technique obtained is applied to varieties of commutative Moufang loops. In particular, the free commutative exponent 3 Moufang loop of nilpotence 4 with n generators is shown to have cardinality $$3^{d(n)}$$, where $$d(n)=n+\left( \begin{matrix} n\\ 3\end{matrix} \right)+4\left( \begin{matrix} n\\ 4\end{matrix} \right)+14\left( \begin{matrix} n\\ 5\end{matrix} \right)+30\left( \begin{matrix} n\\ 6\end{matrix} \right)+20\left( \begin{matrix} n\\ 7\end{matrix} \right).$$