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:
    

 1. $f_i(x) \le f_j(x)$ for all $x \in X$ and $i \preceq j$; 
 2. 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 $i \in I$.

In particular, 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 $I$. 

>**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 $\bf N$).