12/15/2017 - 12:53 PM US EDST Please note: as a result of the decision by
mercio, Math_QED, user8795, HK Lee, kjetil, and b halvorsen to put the question on hold because it is unclear to them, I have added a further clarification which should address their concern. I hope it does, and I want to thank them for their help in making sure that this question is as clear as possible.
12/14/2017 - 11:54 US EDST - Please note: as a result of previous discussion with @Joppy in comments, I have completely reworded this question - I think the question is much clearer this way.
In standard group-theory, the inverse operation is always defined in terms of just one bijection, so far as I know (and I am the first to admit that I don't know very much!).
Is there a "probabilistic" flavor of group-theory in which the inverse operation is defined in terms of an EQUIVALENCE CLASS of bijections from which one bijection must be chosen at random?
Here is an alternative way to ask the same question, which may be clearer.
I'm asking if there exists anything like a non-standard probabilistic version of standard group-theory in which:
1) the bijection which establishes a group's inverse can be selected from a set of equally acceptable alternative bijections which are part of the definition of the group;
2) the definition of the group also includes an instruction for how to choose the operative bijection when instantiating a copy of the group.
With respect to (2), there might be two different kinds of instructions - one kind for a single "one-off" construction of the group, and one kind for repeated constructions of the group such as might occur during a Monte Carlo run or other type of experiment.
For "one-off" constructions of the group, the instruction would simply be:
2)a. "randomly choose one of the alternative bijections provided in the definition of the group".
For repeated constructions of the group, the instruction would be something like the following for a case in which the definition of the group specified four different alternative bijections $B_1$, $B_2$, $B_3$, $B_4$.
2b). "for repeated constructions of the group 8k times, use the four alternative bijections $B_1$, $B_2$, $B_3$, $B_4$ in the ratios 1:3:3:1 respectively."
I hope the above makes my question clearer.
Edited 12/14/2017 07:19 US EDST to add:
Actually, in the case I'm looking at, there would be an "(super)class of equivalence classes", because there would be different equivalence classes for different subsets of the entire set. But the (super)class would not itself be an equivalence class, except indirectly by virtue of the equivalence classes which it contains.