Prologue
For convenience, I have numbered the references listed in this answer following the ones listed in the question.
The words "Picone convergence" and also "Shatunovskii convergence" may not sound familiar, but perhaps "Moore-Smith convergence" does: these terms characterize the same basic general theory of convergence, precisely one (the older one) of the two modern theories, the other being Cartan's theory of filters [3a],[3b].
First of all, let's consider the definition given by Picone ([7], chapter I, §1.4, pp. 8-9, which is exactly the same given in [1], chapter II, §25, p. 100, and also [4] chapter II, §7, pp. 17-18) in a "modernized" language, trying nevertheless to adhere as strictly as possible to his notation.
A couple $([\mathrm{O}], \preccurlyeq)$, where $[\mathrm{O}]$ is a set (which we assume non empty) and $\preccurlyeq$ is a binary relation, is said to be a ordered set of operations if
- if, for every three objects $\mathrm{O}^\prime, \mathrm{O}^{\prime\prime}, \mathrm{O}^{\prime\prime\prime}\in [\mathrm{O}]$, we have $\mathrm{O}^\prime\preccurlyeq \mathrm{O}^{\prime\prime}$ and $\mathrm{O}^{\prime\prime}\preccurlyeq \mathrm{O}^{\prime\prime\prime}$ then it is also $\mathrm{O}^\prime\preccurlyeq \mathrm{O}^{\prime\prime\prime}$, and
- for every two objects $\mathrm{O}^\prime, \mathrm{O}^{\prime\prime}\in [\mathrm{O}]$ there exists an object $\mathrm{O}^{\prime\prime\prime}$ such that $\mathrm{O}^\prime\preccurlyeq \mathrm{O}^{\prime\prime\prime}$ and $\mathrm{O}^{\prime\prime}\preccurlyeq \mathrm{O}^{\prime\prime\prime}$.
Picone's definition is nearly equivalent to the standard definition of directed set: indeed Picone does not require $\preccurlyeq$ to be reflexive. i.e. he does not require that
- for every $\mathrm{O}\in [\mathrm{O}]$ then $\mathrm{O}\preccurlyeq\mathrm{O}$,
as this property is not needed for the main applications possibly he (and the other conceivers of this theory) have in mind. Nevertheless, when the reflexivity property is assumed (as customarily happens), the relation $\preccurlyeq$ is technically a directed preorder.
Now let's try to answer to the asked questions.
Q1
But what exactly this ordering does?
A1. The ordering relation $\preccurlyeq$ simply defines what, in this generalized context, means "to have successive values or instances of a object $\mathrm{O}$": in allowing this, it allows to extend to (far) more general cases what is customarily done with continuous, as $x\in\Bbb R$, or discrete, as $i\in\Bbb N$, totally ordered (directed) variables.
Simply stated, the directed preorder (or directed "pre-preorder" as we could call this relation if we decide to follow the pat of Picone and his school and drop requirement 3 on it) is a far reaching generalization of both the concept of a real variable and of an integer index, beyond the limit of standard total order and, as we see below, cardinality.
Q2
Are the operations we are talking about all the operations that we use to in order to calculate a limit or operations that provide the definition of a limit?
A2. No, they are simply a way to generalize the argument of the functions ($x\in\Bbb R$ in classical analysis) or the index set of a sequence ($\Bbb N$ or $\Bbb Z$ in classical analysis). A standard example is the theory of set limits, used in measure theory, for example in the proof of Cafiero's theorem: let's see how this works.
Consider an indexed family of sets $\{A_i\}_{{i}\in I}$ where the index set $I$ is directed, and assume for the sake of simplicity that there exists a least index (i.e. an element $i_o\in I$ such that $i_o\preccurlyeq i$ for all $i\in I$): then you can define effortlessly its limit inferior and limit superior relative to the index $i$ as
$$
\begin{align}
\liminf_I A_i &=\bigcup_{{i_0 \preccurlyeq i}} \bigcap_{j\preccurlyeq i} A_j\\
\limsup_I A_i &=\bigcap_{{i_0 \preccurlyeq i}} \bigcup_{j\preccurlyeq i} A_j
\end{align}
$$
whatever the cardinality of $I$, and moreover these limits always exists and their "value" is not a value, but any kind of abstract set.
Frankly speaking, even if the directed set $[\mathrm{O}]$ may be, for example, a set of linear differential operators and their inverses, calling its elements "operations" seems to me somewhat misguiding and confusing.
Epilogue: notes and observations.
- More abstract directed sets and associated $\liminf/\limsup$ concept may be introduced: the formal structure of the theory does not change. This is perhaps the strongest feature of this theory.
- A more modern treatment of directed sets and of Moore and Smith-Picone-Shatunovskii (for questions of attribution, please see the following point) convergence can be found in [5], chapter 2, pp. 62-83.
- The theory development of the theory of limits by the use of nets (directed sets) is usually attributed to Moore and Smith, who published it in their 1922 paper [6]. As a matter of fact, Picone's and Shatunovskii's books were published in 1923, nearly a year later, and moreover the work of Shatunovskii was spread only later by the works and words of Andrei Kolmogorov. It should however be noted that while [6] is a research paper, [8] is a textbook and [7] is the second edition of the 1919 lecture notes of a course on infinitesimal calculus held by Picone at the University of Catania: this means that the contents of the latter two has already circulated for some years before the publication of [7]. It seems therefore not unfair to share the attribution of discovery to all the four scientists involved.
- It is customarily affirmed that, while seemingly different, the convergence theories of Moore and Smith, Picone, Shatunovskii on one side and Cartan [3a],[3b] on the other are exactly equivalent. This is however true apart for one detail: in the former theory there is not a concept which is equivalent to the one of ultrafilter. This implies that, while every Cartan's filter is contained in a maximal one, this does not happen for nets (directed sets).
References
[3a] Henri Cartan, Théorie des filtres, (French),
Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Paris, 205, 595-598 (1937), JFM 63.0569.03, Zbl 0017.24305.
[3b] Henri Cartan, "Filtres et ultrafiltres" (French) Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Paris, 205, 777-779 (1937), JFM 63.0569.03Zbl 0018.00302.
[4] Gaetano Fichera, Lezioni sulle trasformazioni lineari. I.: Introduzione all’analisi lineare (Lectures on linear transformations, I.: Introduction to linear analysis, (Italian) (Terza ristampa, 1962) Trieste: Istituto Matematico-Università XIX, 502 p. (1954), MR67346, Zbl 0057.33601.
[5] John L. Kelley, General topology, 2nd ed.,
Graduate Texts in Mathematics, 27, New York - Heidelberg - Berlin: Springer-Verlag, pp. XIV+298 (1975), MR370454, Zbl 0306.54002.
[6] Eliakim Hastings Moore, Hermann Lyle Smith, "A general theory of limits", American Journal of Mathematics 44, 102-121 (1922), JFM 48.1254.01.
[7] Mauro Picone, Lezioni di analisi infinitesimale. Vol. 1. Parte 1 e 2. La derivazione e l’integrazione (Italian), Catania: Circolo matematico. IV, II u. 742 (1923), JFM 49.0172.07.
[8] Samuil Osipovich Shatunovskiĭ, Введение в анализ (Introduction to analysis), (Russian), Odessa: Mathesis (1923).
