Edit: I updated question 1 and added a link in Section 3.1. I also explained the intuition in section 4.4.1.

I have yet to understand amenability and group theory, but the questions in my attachment might be of interest.

I would be glad if all the questions were answered; however, if you wish for a particular question, read section 4 of my paper and the question in section 5.

I want an elegant choice function where, for specific A in 4.4, gives the structure (defined in 4.2) that I'm looking for.

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