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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:

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 $i \in I$.

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.

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:

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 $i \in I$.

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.

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:

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 $i \in I$.

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. Let me try to explain my motivation for this notion, just in case it can make the question and the whole stuff a little bit more meaningful.

  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 is when $\mathscr{F}$ is just a pair $(f_\ast, f^\ast)$ of (set) functions $\mathcal P(S) \to \bf R$, and we assume that $f_\ast$ is conjugate to $f^\ast$, viz. $f_\ast(X) := 1 - f^\ast(S \setminus X)$ for every $X \subseteq S$, 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 $A \subseteq S$.

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 used, e.g., to assembly some weird counterexamples and benchmark the logical strength of certain hypotheses.

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:

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 $i \in I$.

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.

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:

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 $i \in I$.

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. Let me try to explain my motivation for this notion, just in case it can make the question and the whole stuff a little bit more meaningful.

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 is when $\mathscr{F}$ is just a pair $(f_\ast, f^\ast)$ of (set) functions $\mathcal P(S) \to \bf R$, and we assume that $f_\ast$ is conjugate to $f^\ast$, viz. $f_\ast(X) := 1 - f^\ast(S \setminus X)$ for every $X \subseteq S$, 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 $A \subseteq S$.

However, there are some situations in which the above conditions are not satisfied, but still a notion of independence, say, for the pair $(f,g)$ in the lines of the one drawn in this post can be used, e.g., to assembly some weird counterexamples and benchmark the logical strength of certain hypotheses.

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:

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 $i \in I$.

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. Let me try to explain my motivation for this notion, just in case it can make the question and the whole stuff a little bit more meaningful.

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 is when $\mathscr{F}$ is just a pair $(f_\ast, f^\ast)$ of (set) functions $\mathcal P(S) \to \bf R$, and we assume that $f_\ast$ is conjugate to $f^\ast$, viz. $f_\ast(X) := 1 - f^\ast(S \setminus X)$ for every $X \subseteq S$, 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 $A \subseteq S$.

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 used, e.g., to assembly some weird counterexamples and benchmark the logical strength of certain hypotheses.