Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$.
Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: {\bf H} \to {\bf H}$ such that $\theta[X] \in \mathcal D$ and $f(\theta[X]) = f(X)$ for all $X \in \mathcal D$, where $\theta[X] := \{\theta(x): x \in X\}$ is the direct image of $X$ under $\theta$.
It is easily seen that the identity function on ${\bf H}$ belongs to $\mathcal M(f)$ and $\mathcal M(f)$ is closed under the composition of functions.
We call $\mathcal M(f)$ the Lévy monoid of $f$, by analogy with what is usually called the Lévy group after Hida [H], who was in turn motivated by earlier work of Lévy [L, Part III] (I don't have access to a copy of Lévy's book, so this is second-hand information). In addition, we set $$\mathcal G(f) := \mathcal M(f) \cap {\rm aut}({\bf H}),$$ where ${\rm aut}({\bf H})$ is the set of all permutations of ${\bf H}$, and we refer to $\mathcal G(f)$ as the Lévy group of $f$ if, to nobody's surprise, $\mathcal G(f)$ is a subgroup of $\mathcal M(f)$, i.e. $f^{-1} \in \mathcal G(f)$ for every $f \in \mathcal G(f)$.
Obviously, it is hopeless to try to study $\mathcal M(f)$ and $\mathcal G(f)$ without further assumptions on $f$, but there is a class of cases, among many others, in which this looks doable (and interesting, to some degree).
Namely, assume in what follows that $\bf H$ is either $\mathbf Z$, $\mathbf N$, or $\mathbf N^+$, and let $f$ be the quasi-density induced by an upper quasi-density $f^\ast$ on $\bf H$. This means that $f^\ast$ is a function $\mathcal P(\mathbf H) \to \mathbf R$ such that, for all $X, Y \subseteq \bf H$ and $h,k \in \mathbf N^+$, the following hold:
- $f^\ast(\mathbf H) = 1$;
- $f^\ast(X \cup Y) \le f^\ast(X) + f^\ast(Y)$;
- $f^\ast(k \cdot X + h) = \frac{1}{k} f^\ast(X)$, where $k \cdot X + h := \{kx+h: x \in X\}$;
and $f$ is the restriction of $f^\ast$ to the set $$\{X \in \mathcal P(\mathbf H): f^\ast(X) + f^\ast(\mathbf H \setminus X) = 1\}.$$
So, here come my questions:
Q1. Is $\mathcal M(f)$ infinite? Q2. Is $\mathcal G(f)$ infinite? Q3. Is $\mathcal G(f)$ a group under composition? Q4. And what about Q1 and Q2 with $f^\ast$ in place of $f$? (Of course, $\mathcal G(f^\ast)$ does always form a group under the operation of composition.)
Both Q1 and Q2 can be answered in the affirmative if $f^\ast$ is monotone (nondecreasing), viz. $f^\ast(X) \le f^\ast(Y)$ whenever $X \subseteq Y \subseteq \bf H$, in which case $f^\ast$ is called an upper density on $\bf H$ and $f$ an induced density.
Moreover, $\mathcal M(f^\ast)$ is always infinite, as any function of the form $\mathbf H \to \mathbf H: x \mapsto x+h$, with $h \in \bf N$, belongs to $\mathcal M(f^\ast)$, and the same is true of $\mathcal G(f^\ast)$ when $\mathbf H = \mathbf Z$ (regardless of whether or not $f^\ast$ is monotone), which marks a difference, as minor as it may be, among the cases parametrized by $\bf H$. On the other hand, it's completely unclear to me if there is any (substantial) difference between the cases ${\bf H} = \bf N$ and ${\bf H} = \bf N^+$, when my feeling is that the question should boil down to deciding whether or not there exists a unique extension of an upper quasi-density on ${\bf N}^+$ to a quasi-density on $\bf N$ (which is certainly true in the presence of monotonicity).
Lastly, it is folklore that the answer to Q3 is positive if $f$ is the asymptotic density (though I ignore where this was first noted).
I don't know much more on other cases, which leads me to the following:
Q5. What is there in the literature about $\mathcal G(f)$ in the special case where $f^\ast$ is, say, the upper Banach density, the upper logarithmic density, or the upper analytic density (on $\bf N$)? (All of these are upper densities.)
For the record: In spite of a few articles dealing with the Lévy group of the asymptotic density on $\bf N$ and/or measure densities extending the asymptotic density to a finitely additive probability measure $\mathcal P(\mathbf N) \to \bf R$, see e.g. [NP], [SZ] and references therein, I can't even mention a single paper picking up with any of the cases mentioned in Q5.
Bibliography.
[H] T. Hida, Analysis of Brownian Functionals, Lecture Notes, IMA: University of Minnesota, 1986.
[L] P. Lévy, Problémes Concrets d'Analyse Fonctionnelle, Gauthier-Villars: Paris, 1951.
[NP] M. B. Nathanson and R. Parikh, Density of sets of natural numbers and the Lévy group, J. Number Theory 124 (2007), No. 1, 151-158.
[SZ] M. Sleziak and M. Ziman, Lévy group and density measures, J. Number Theory 128 (2008), No. 12, 3005-3012.
