Note on n-Groups

Janez Ušan ; Mališa Žižović

Mathematica Moravica, Tome 6 (2002) no. 1.

Résumé

Among the results of the paper is the following proposition. Let $n\geq 3$ and let $(Q,A)$ be an n-grupoid. Then: $(Q,A)$ is an n-group iff there are mappings $\alpha$ and $\beta$, respectively, of the sets $Q^{n-2}$ and $Q$ into the set $Q$ such that the laws $A(A(x_{1}^{n}),x_{n+1}^{2n-1}) = A(x_{1},A(x_{2}^{n+1}),x_{n+2}^{2n-1})$, $\beta A(x_{1}^{n}) = A(x_{1}^{n-1},\beta(x_{n})) = A(x_{1}^{n-2},\beta(x_{n-1}),x_{n})$, $A(x,a_{1}^{n-2},\alpha(a_{1}^{n-2})) = A(b_{1}^{n-2},\alpha(b_{1}^{n-2}),x)$ and $\beta A(x,c_{1}^{n-2},\alpha(c_{1}^{n-2})) = x$ hold in the algebra $(Q,\{A,\alpha,\beta\})$ [:3.1].

Détail

Mots-clés : n-semigroup, n-quasigroup, n-group, {1;n}-neutral operation on n-groupoid, central operation on n-group