Transitivity falls out of associativity: a Choice theory
Trying to build a theory of choice from primitives. First thing I can prove properly.
Axioms
Choice operator #, binary, returns one of its two arguments.
Idempotence: a # a = a
Associativity: a # (b # c) = (a # b) # c
Theorem: transitivity follows.
Given a # b = a and b # c = b:
b # (b # c) = b # b = b
so a # (b # (b # c)) = a # b = a
regroup: a # (b # (b # c)) = (a # (b # b)) # c = (a # b) # c = a # c
therefore a # c = a
Converse (where the pairs are comparable): transitivity gives associativity back. Case analysis on which argument each comparison returns:
a#b = a, b#c = b: both groupings give a
a#b = a, b#c = c: both give a # c
a#b = b, b#c = b: both give b
a#b = b, b#c = c: both give c, using transitivity in the second
Transitivity is not an extra rationality axiom. It is downstream of associativity. What this leads to is the following: non-associativity implies state, or what was compared before affects what comes after.
What is open right now for me is whether idempotence smuggles in totality in some way. Totality is like a way of saying that given these subset of variables, the operator is defined on every pair in the cross product of said subset with itself. Put simply, it's just can I compare everything with everything within this set? The proof for transitivity implying associativity needed totality. Not sure about idempotence