Conjugating trivial automorphisms of π«(β)/`ch
Will Brian, Ilijas Farah Β· Transactions of the American Mathematical Society Β· 2025
A trivial automorphism of the Boolean algebra P ( N ) / F i n \mathcal P(\mathbb N) / \mathrm {Fin} is an automorphism induced by the action of some function N β N \mathbb N \rightarrow \mathbb N . The forcing axiom O C A T \mathsf {OCA}_{\mathrm {T}} implies all automorphisms are trivial, and therefore two trivial automorphisms are conjugate if and only if they have the same (modulo finite) orbit structure. We show that the Continuum Hypothesis implies that two trivial automorphisms are conjugate if and only if there are no first-order obstructions to their conjugacy and their indices have the same parity, if and only if the given trivial automorphisms are conjugate in some forcing extension of the universe. To each automorphism Ξ± \alpha of P ( N ) / F i n \raise 1.5pt\hbox {\(\scriptstyle \mathcal {P}(\mathbb {N})\)}\mkern -1mu/\mkern -1mu{\scriptstyle \mathrm {Fin}} we associate the first-order structure A Ξ± = ( P ( N )