Online premeans and their computation complexity

We extend some approach to a family of symmetric means (i.e. symmetric functions $\mathcal{M}\text{\hspace{0.17em}⁣}:{\bigcup }_{n=1}^{\mathrm{\infty }}{I}^n\to I$ 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 $\mathcal{M}(x_1,\dots ,x_n):=F(f(x_1)+\cdots +f(x_n))$, where $f\text{\hspace{0.17em}⁣}:I\to G$ and $F\text{\hspace{0.17em}⁣}:G\to I$ ($G$ 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 celebrated families of quasi-arithmetic means ($G=$\mathbb{R}$\times $\mathbb{Z}$$)andBajraktarevi\acute{c}means($G=$\mathbb{R}$^2$ under some additional assumptions). Moreover, we establish certain estimations of complexity for several other classical families.