Definition 2

Given two functions $f, g$ of $\mathscr{H}(\mathfrak{F}, V)$ we say that $f$ and $g$ are similar (along $\mathfrak{F}$), and write $f \simeq g$, if $f \preccurlyeq g$ and $g \preccurlyeq f$.

For every scalar $\lambda \neq 0$ the relation $f \simeq g$ is equivalent to $f \simeq g \lambda$. It implies the existence of a set $X \in \mathfrak{F}$ such that that the subset of $X$ of points where $f(x) = 0$ is identical with the subset of $X$ of points where $g(x) = 0$.