Only recently I learned of E. Minguzzi's article
"Clocks' synchronization without round-trip conditions", [gr-qc: arXiv:1009.3005] ...
(Notably, the article available for download is dated May 23, 2022.)
In this article, Minguzzi ...
... introduce[s] a mathematical framework which will allow us to deal with the problem of synchronization [...]
It will prove particularly general [...] At first the mathematical framework may seem somewhat abstract ...
... and in terms of this abstract framework he goes on to propose ...
... a generalization of the Poincaré-Einstein method [of synchronizing distant clocks.]
I have two specific questions about the presentation ...
Summary of the Abstract Framework
For clarity, and summarizing my understanding, in the following I'd first like to iterate the whole "abstract framework" laid out in the article; in terms of Minguzzi's notation, and introducing the essential (but not all) terminology. It features:
set $S$ called the space, each element $s \in S$ being called a space point or clock;
spacetime $M$ , whose elements are called events, defined as the disjoint union $M = \bigcup_{s \in S} \mathbb E_s$ where $\mathbb E_s$ are one-dimensional affine spaces over ...
... one-dimensional vector spaces $T_s$,
that is given two elements $e_1, e_2 \in \mathbb E_s$ the difference makes sense and belongs to $T_s$;for each $s \in S$ a unit of measure $\tau_s \in T_s$ [...] such that $e_2 - e_1 = t_{12} \tau_s$ [... for] number $t_{12} \in \mathbb R$;
the space of units of measure is $T = \bigcup_{s \in S} T_s$, and $\pi_{TS} : T \rightarrow S$ is the canonical projection [...]
$T$ is time oriented in the sense that a choice of positive halve for $T_s$ has been made at each $s \in S$ (which makes the inequality $e^{\prime} − e \ge 0$ meaningful if $e$ and $e^{\prime}$ belong to the same fiber);a natural projection $\pi : M \rightarrow S $ which assigns to [event] $e \in M$ [...] the point $s$ such that $e \in \mathbb E_s$;
propagation map $P : M \times S \rightarrow M$ [...] given the event $e_{s_1} \in M, \pi[ \, e_{s_1} \, ] = s_1,$ and $s_2 \in S$, the map sends the pair $(e_{s_1}, s_2)$ to a new event $e_{s_2} = P[ \, e_{s_1}, s_2 \, ]$ which projects on $s_2$;
propagation map $p : T \times S \rightarrow T$ which for any given [...] $\tau_{s_1} \in T_{s_1}$ and point $s_2 \in S$ gives $p[ \, \tau_{s_1}, s_2 \, ] \in T_{s_2}$.
On those two types of propagation maps there are the following four groups of conditions imposed:
(a) Given a sequence of points $s_0$, $s_1$ and $s_2$, map $P$ satisfies
$$ P[ \, P[ \, e_{s_0}, s_1 \, ], s_2 \, ] - P[ \, e_{s_0}, s_2 \, ] \ge 0. $$
(b) It is $\qquad P[ \, e_{s_0}, s_0 \, ] = e_{s_0}, \qquad$ and (more generally):
Given a sequence $s_0, s_1, . . . s_j$ of $j \ge 1$ distinct points, the cyclic composition of maps $P$ satisfies
$$ \underbrace{P[ \, P[ \, ... P[ }_{(j + 1) \text{ times}} \, e_{s_0}, s_1 \, ], ..., s_j \, ], s_0 \, ] - e_{s_0} \ge 0.$$
(c) $P$ is an affine map that is for every $e_{s_1} \in \mathbb E_{s_1}, s_2 \in S,$ and $\tau_{s_1} \in T_{s_1}$ it is $$ P[ \, e_{s_1} + \tau_{s_1}, s_2 \, ] = P[ \, e_{s_1}, s_2 \, ] + p[ \, \tau_{s_1}, s_2 \, ]. \tag{*}$$
$\qquad$ Stated in another way, if $e_{s_1}, e^{\prime}_{s_1} \in \mathbb E_{s_1}$ and $s_2 \in S$, then $$ P[ \, e^{\prime}_{s_1}, s_2 \, ] - P[ \, e_{s_1}, s_2 \, ] = p[ \, e^{\prime}_{s_1} - e_{s_1}, s_2 \, ].$$
$\qquad$ Correspondingly, $p$ is an injective linear map which preserves the time orientation of $T$,
$\qquad$ that is for every $\tau_{s_1} \in T_{s_1}, s_2 \in S$, and $\alpha \in \mathbb R$: $\, p[ \, \tau_{s_1}, s_2 \, ]$ is positive iff $\tau_{s_1}$ is positive, and
$$p[ \, \alpha \, \tau_{s_1}, s_2 \, ] = \alpha \, p[ \, \tau_{s_1}, s_2 \, ].$$
(d) It is $\qquad p[ \, \tau_{s_0}, s_0 \, ] = \tau_{s_0}, \qquad$ and (more generally):
Given a sequence $s_0, s_1, . . . s_j$ of $j \ge 1$ distinct points, the cyclic composition of maps $p$ satisfies
$$ \underbrace{p[ \, p[ \, ... p[ }_{(j + 1) \text{ times}} \, \tau_{s_0}, s_1 \, ], ..., s_j \, ], s_0 \, ] = \tau_{s_0}.$$
(The equation marked $(*)$ above is in Minguzzi's article numbered as equation $(3)$.)
Question 1:
Proceding with a discussion of the related syntonization problem, Minguzzi presents as a solution the assignment $(5)$:
$$p[ \, \tau_{s_1}, s_2 \, ] = {\huge \tau_{s_2}}. \tag{5}$$ After some brief remark he continues:
As a consequence, throughout this work we shall omit reference to the application $p$ assuming that a section $\tau$ with property $(5)$ has been chosen. Thus [...] equations such as $(3)$
... i.e. correspondingly eq. $(*)$ above ...
can be written more sloppily
$$ P[ \, e_{s_1} + \tau_{s_1}, s_2 \, ] = P[ \, e_{s_1}, s_2 \, ] + {\huge \tau_{s_1}}. \tag{6}$$
Should this not much rather be instead:
$$ P[ \, e_{s_1} + \tau_{s_1}, s_2 \, ] = P[ \, e_{s_1}, s_2 \, ] + {\huge \tau_{s_2}} \tag{$6^{\prime}$}$$ ?
Might Minguzzi's equation $(6)$, as stated, possibly be a misprint ?? ...
Or, if this equation $(6)$ is indeed exactly as Minguzzi intended, then -- what's the purpose of such apparent "sloppy writing", namely of putting $\large \tau_{s_1}$ where eq. $(5)$ inserted in eq. $(3)$ a.k.a. $(*)$ so obviously suggests putting $\large \tau_{s_2}$ instead ?? ...
Question 2:
In the subsequent chapter Minguzzi turns to defining (among others)
function $r : S \times S \rightarrow [0, +\infty)$ [...] by
$$ r[ s_0, s_1 ] = P[ \, P[ \, e_{s_0}, s_1 \, ], s_0 \, ] - e_{s_0}. \tag{7} $$
Then he presents ...
Theorem 3.3 The function $r$ is symmetric. Proof. Recall that
$$ r[ s_1, s_0 ] = P[ \, P[ \, e_{s_1}, s_0 \, ], s_1 \, ] - e_{s_1}. \tag{**} $$
Since $P$ preserves the affine structure
$$ \begin{align} r[ s_1, s_0 ] & = & P[ \, P[ \, P[ \, e_{s_1}, s_0 \, ], s_1 \, ], s_0 \, ] - P[ \, e_{s_1}, s_0 \, ] \qquad & & \tag{***} \\ & = & P[ \, P[ \, (P[ \, e_{s_1}, s_0 \, ]), s_1 \, ], s_0 \, ] - (P[ \, e_{s_1}, s_0 \, ]) \, \, & = & r[ s_0, s_1 ]. & \tag{****} \end{align} $$
While I can accept the initial and the concluding part of this (supposed) proof by themselves, namely eq. $(**)$ and eq. $(* \! * \! * *)$ separately, as such, I'm doubtful about the intermediate part marked as eq. $(* \! * \!*)$.
Can eq. $(* \! * \! *)$ be derived explicitly from eq. $(**)$ and the relations (a) through (d), as laid out above ?
My suspicion to the contrary is motivated by setting
$$ P[ \, P[ \, e_{s_1}, s_0 \, ], s_1 \, ] \equiv e_{s_1} + \rho_{101} \, {\huge \tau_{s_1}} $$
for a certain suitable (generally non-zero) real number value $\rho_{101}$, and to conclude that thereby
$$ r[ s_1, s_0 ] = P[ \, P[ \, e_{s_1}, s_0 \, ], s_1 \, ] - e_{s_1} \equiv e_{s_1} + \rho_{101} \, \tau_{s_1} - e_{s_1} = \rho_{101} \, {\huge \tau_{s_1}}, $$
but using relations of (c), along with $(5)$ or $(6^{\prime})$:
$$ \begin{align} P[ \, P[ \, P[ \, e_{s_1}, s_0 \, ], s_1 \, ], s_0 \, ] - P[ \, e_{s_1}, s_0 \, ] & \equiv & ~ \\ P[ \, e_{s_1} + \rho_{101} \, \tau_{s_1}, s_0 \, ] - P[ \, e_{s_1}, s_0 \, ] \qquad & = & ~ \\ P[ \, e_{s_1}, s_0 \, ] + \rho_{101} \, p[ \, \tau_{s_1}, s_0 \, ] - P[ \, e_{s_1}, s_0 \, ] & = & \rho_{101} \, {\huge \tau_{s_0}}. \end{align}$$
