ON TOPOLOGICAL n-GROUPS

Janez Ušan

Mathematica Moravica, Tome 2 (1998) no. 1.

Résumé

Let $(Q,A)$ be an n-group, $^{-1}$ its inversing operation [:[13,16],1.3], $n \geq 2$ and let $Q$ be equipped with a topology $O$. Then, in this paper, we say that $Q,A,O$ is topological n-group iff: a) the n-ary operation $A$ is continuous in $O$, and b) the (n-1)-ary operation $^{-1}$ is continuous in $O$. The main result of the paper is the following proposition. Let $(Q,A)$ be an n-group, $n \geq 3$ and let $(Q,{\cdot, \varphi,b})$ be an arbitrary nHG-algebra associated to the n-group $(Q,A)$ [:[15],1.5]. Also, $Q$ is equipped with a topology $O$. Then, $Q,A,O$ is a topological n-group iff the following statements hold: 1) $(Q,\cdot, O)$ is a topological group [:e.g. [7]], and 2) the unary operation $\varphi$ is continuous in $O$.