Based on entropy and symmetrical uncertainty (SU), we define a metric for categorical random variables and show that this metric can be promoted into an appropriate quotient space of categorical random variables. Moreover, we also show that there is a natural commutative monoid structure in the same quotient space, which is compatible with the topology induced by the metric, in the sense that the monoid operation is continuous.

翻译：基于熵和对称不确定性(SU)，我们为分类随机变量定义了一种度量，并证明该度量可以提升至分类随机变量的适当商空间。此外，我们还证明在同一商空间中存在自然的交换幺半群结构，该结构与度量诱导的拓扑相容，即幺半群运算是连续的。