Note on Polyagroups

Janez Ušan ; Mališa Žižović

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

Résumé

In the paper the following proposition is proved. Let $k>1$, $s>1$, $n=k-s+1$ and let $(Q,A)$ be an n-groupoid. Then, $(Q,A)$ is a polyagroup of the type $(s,n-1)$ iff the following statements hold: (i) $(Q,A)$ is an $\langle i, s+i\rangle$-associative n-groupoid for all $i\in\{1,\dots,s\}$; $\langle l,n\rangle$-associative n-groupoid; (iii) for every $a_{1}^{n}\in Q$ there is at least one $x\in Q$ and at least one $y\in Q$ such that the following equalities hold $A(x,a_{1}^{n-1})=a_{n}$ and $A(a_{1}^{n-1}) = a_{n}$; and (iv) for every $a_{1}^{n}\in Q$ and for all $i\in \{2,\dots,s\}\cup \{(k-1)\cdot s+2,\dots,k\cdot s\}$ there is exactly one $X_{i}\in Q$ such that the following equality holds $A(a_{1}^{i-1},x_{i},a_{i}^{n-1}) = a_{n}$. [The case $s=1$ (: (i) – (iii)) is discribed in [4].]

Détail

Mots-clés : n-groupoid, n-semigroup, n-quasigroup, iPs-associative n-groupoid, polyagroup, near-P-polyagroup