A Comment on (n, m)–Groups for $n\geq 3m$

Janez Ušan

In the present paper the following proposition is proved. Let $n\geq 3m$ and let $(Q,A)$ be an $(n, m)$-groupoid. Then, $(Q,A)$ is an $(n, m)$-group if for some $i\in \{m + 1,\dots, n − 2m + 1\}$ the following conditions hold: (a) the $\langle i − 1,i\rangle$-associative law holds in $(Q,A)$; (b) the $\langle i, i + 1\rangle$-associative law holds in $(Q,A)$; and (c) for every $a_{1}^{n}\in Q$ there is exactly one $x_{1}^{m}\in Q$ such that the following equality holds $A(a_{1}^{i-1}, x_{1}^{m}, a_{i}^{n−m}) = a_{n-m+1}^{n}$.