A NOTE ON TOPOLOGICAL n-GROUPS

Janez Ušan

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

Résumé

In the present paper is proved the following proposition. Let $(Q,A)$ be an n-group, $^{-1}$ its nversing operation, $n \geq 2$ and $Q$ is equipped with a topology $O$. Also let $^{-1}A(x,a_1^{n-2},y)=z \iff A(z,a_1^{n-2},y)=x$ (def) and $^{-1}A(x,a_1^{n-2},y)=z \iff A(x,a_1^{n-2},z)=y$ (def) for all $x,y,z \in Q$ and for every sequence $a_1^{n-2}$ over $Q$. Then the following statements are equivalent: (i) the n-ary operation $A$ is continuous in $O$ and the (n-1)-ary operation $^{-1}$ is continuous in $O$; (ii) the n-ary operation $^{-1}A$ is continuous in $O$; and (iii) the n-ary operatin $A^{-1}$ is continuous in $O$. [See, also Remark 2.2.]