experimental physics - How to determine the order of indications of a clock?

Given the description of a clock $\mathcal A$, as

• 
  

  (1) a set $A$ of all (more than 2) distinct indications of this clock, in no particular order
  (where the individual indications contained in set $A$ are denoted below as "$A_J$", "$A_Q$", "$A_V$" etc.), together with

  
  


• 
  

  (2) a function $t_{\mathcal A} : A \rightarrow \mathbb R$
  (such that we may speak of set $A$ and function $t_{\mathcal A}$ together as "a clock, $\mathcal A$" at all),

  
  

and, in line with Einstein's assertion that:

"All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more {... participants}",
given

• (3) a sufficient suitable account, in no particular order, of participants (such as $A$, $J$, or $Q$) and their encounters (where $A_J$ is $A$'s indication at the encounter of $A$ and $J$, $A_Q$ is $A$'s indication at the encounter of $A$ and $Q$, and so on) --

How can the order of indications of clock $\mathcal A$ be derived?

Or to consider the simplest case:
How can be determined whether
$A_J$ was between $A_Q$ and $A_V$, or
$A_Q$ was between $A_V$ and $A_J$, or

$A_V$ was between $A_J$ and $A_Q$
?

Note:
It should not be assumed that clock $\mathcal A$ was "monotonous";
in other words: it is not an acceptable answer to argue
"if $(t_{\mathcal A}[ A_V ] - t_{\mathcal A}[ A_Q ]) \times (t_{\mathcal A}[ A_Q ] - t_{\mathcal A}[ A_J ]) > 0$ then $A_Q$ was between $A_V$ and $A_J$".

Another note in response to a comment by Jim:
The encounters (coincidence events) in which the participant took part whose set of indications at those encounters is called $A$ (and who is therefore him/her/itself conveniently called $A$ as well) may be denoted as
"$\mathscr E_{AJ}$", "$\mathscr E_{AQ}$", "$\mathscr E_{AV}$",
among others;

corresponding to the indications "$A_J$", "$A_Q$", "$A_V$" of $A$ (and, together with function $t_{\mathcal A}$, therefore indications of clock $\mathcal A$) which had been named explicitly above.

As far as a Lorentzian manifold may be assigned to those events in which $A$ took part (together with additional events, as may be required) and as far as the corresponding Causal structure would be determined, those events in which $A$ took part were all elements of one chronological (or timelike) Curve.

My question is consequently, in other words, whether and how elements of such a chronological (or timelike) curve may be ordered, given their parametrization only by some arbitrary function $t_{\mathcal A} : A \rightarrow \mathbb R$,
together, of course, with a sufficient suitable account (3, above) of all participants involved, and their encounters, if any, in no particular order.