We extend some approach to the family of symmetric means (i.e. symmetric functions \(\mathscr {M} :\bigcup _{n=1}^\infty I^n \rightarrow I\) with \(\min \le \mathscr {M}\le \max \); I is an interval). Namely, it is known that every symmetric mean can be written in a form \(\mathscr {M}(v_1,\dots ,v_n):=F(f(v_1)+\cdots +f(v_n))\), where \(f :I \rightarrow G\) and \(F :G \rightarrow I\) (G is a commutative semigroup). For \(G=\mathbb {R}^k\) or \(G=\mathbb {R}^k \times \mathbb {Z}\) (\(k \in \mathbb {N}\)) and continuous functions f and F we obtain two series of families (depending on k). It can be treated as a measure of complexity in a family of means (this idea is inspired by theory of regular languages and algorithmics). As a result we characterize the celebrated families of quasi-arithmetic means (\(G=\mathbb {R}\times \mathbb {Z}\)) and Bajraktarević means (\(G=\mathbb {R}^2\) under some additional assumptions). Moreover, we establish certain estimations of complexity for several other classical families.

我们将一些方法扩展到对称均值族（即对称函数\(\mathscr {M} :\bigcup _{n=1}^\infty I^n \rightarrow I\）与\(\min \le \mathscr {M}\le \max \) ; I是一个区间）。即，已知每个对称均值可以写成形式\(\mathscr {M}(v_1,\dots ,v_n):=F(f(v_1)+\cdots +f(v_n))\)，其中\(f :I \rightarrow G\)和\(F :G \rightarrow I\)（G 是交换半群）。对于\(G=\mathbb {R}^k\)或\(G=\mathbb {R}^k \times \mathbb {Z}\) ( \(k \in \mathbb {N}\) ) 和连续函数f和F我们得到两个系列的家庭（取决于k）。它可以被视为一系列手段中复杂性的度量（这个想法受到常规语言和算法理论的启发）。因此，我们描述了著名的拟算术均值族 ( \(G=\mathbb {R}\times \mathbb {Z}\) ) 和 Bajraktarević 均值 ( \(G=\mathbb {R}^2\)在一些额外的假设下）。此外，我们对其他几个经典家族的复杂性进行了某些估计。