NOTE ON (n,m)-GROUPS

Janez Ušan

Mathematica Moravica, Tome 3 (1999) no. 1.

Résumé

Among the results of the paper is the following proposition. Let $2m \leq n 3m$ and let $(Q,A)$ be an (n,m)-groupoid $(n,m \in N)$. Then, $(Q,A)$ is an (n,m)-group iff there are mappings $^{-1}$ and $e$ respectively of the sets $Q^{n-m}$ and $Q^{n-2m}$ into the set $Q^{m}$ such that the following laws hold in the algebra $(Q,A,^{-1},e)$: $A(A(x_1^n),x_{n+1}^{2n-m})=A(x_1,A(x_2^{n+1),x_{n+2}^{2n-m})$, $A(A(x_1^n),x_{n+1}^{2n-m})=A(x_1^{n-m},A(x_{n-m+1}^{2n-m}))$, $A(x_1^m, a_1^{n-2m}, e(a_1^{n-2m}))=x_1^m$ and $A(x_1^m, a_1^{n-2m}, (a_1^{n-2m},x_1^m)^{-1})=e(a_1^{n-2m})$.