This is basically another reference request.

Let $X$ be a set, and $\mathscr{F} = (f_i)_{i \in I}$ an indexed family of functions $X \to \bf R$. If $\preceq$ is a partial order on $I$, we say that the family $\mathscr{F}$ is *linearly range independent wrt to $\preceq$* if the following hold:

- $f_i(x) \le f_j(x)$ for all $x \in X$ and $i \preceq j$;
- However we consider an indexed set $(a_i)_{i \in I}$ of real numbers with the property that $a_i \in f_i[X] := \{f_i(x): x \in X\}$ for each $i \in I$ and $a_i \le a_j$ whenever $i \preceq j$, there exists $x \in X$ such that $f_i(x) = a_i$ for all $i \in I$.

Accordingly, we say that $\mathscr F$ is a *range independent family* if it is linearly range independent wrt to the discrete order on $I$, and a *linearly range independent* family if it is linearly range independent wrt to $\preceq$ under the additional condition that $I$ is an ordinal number and $\preceq$ the natural well-order on $I$ (which includes the case of a nonempty $n$-tuple $(f_1, \ldots, f_n)$ of functions $X \to \bf R$).

Question.Is there a more standard name for either of the properties defined here above?

Of course, there is nothing too special about $\bf R$, but the current formulation is already more general than the case in which I'm interested (where $X$ is, say, the power set of $\bf N$ and $\mathscr{F}$ is an $n$-tuple).

**Added later.** Just in case, let me try to explain my motivation for this stuff. The above notions are essentially inspired by a kind of problems that are typical of the ``theory of densities'', some concrete examples in this direction having been already discussed on MO, see e.g. Question 206801: *On the independence of lower and upper asymptotic and Banach densities* and references therein.

In this context, one basic situation occurs when $\mathscr{F}$ is just a pair $(f_\ast, f^\ast)$ of (set) functions $\mathcal P({\bf N}) \to \bf R$, and we assume that $f_\ast$ is conjugate to $f^\ast$, viz. $f_\ast(X) := 1 - f^\ast({\bf N} \setminus X)$ for every $X \subseteq \bf N$, and enforce some conditions on $f^\ast$ so that the image of $f^\ast$ is an interval (most typically, the interval $[0,1]$) and $f_\ast(A) \le f^\ast(A)$ for every $A \subseteq \bf N$; for instance, this is the case when $f^\ast$ is the upper asymptotic density or the upper Schnirelmann density.

However, there are situations in which the above conditions are not satisfied, and still a notion of independence, say, for a pair $(f,g)$ in the lines of the one drawn in this post can be studied, either because it is interesting *per se*, or to assembly some weird counterexamples and benchmark the logical strength of certain theorems.