Submission date: 10 January 2022

Abstract:

We investigate the following model-theoretic independence relation: b ⊥^{bu}_A c if and only if bdd^u(Ab) ∩ bdd^u(Ac) = bdd^u(A), where bdd^u(X) is the class of all ultraimaginaries bounded over X. In particular, we sharpen a result of Wagner to show that b ⊥^{bu}_A c if and only if ⟨ Autf(𝕄/Ab) ∪ Autf(𝕄/Ac) ⟩ = Autf(𝕄/A), and we establish full existence over hyperimaginary parameters (i.e., for any set of hyperimaginaries A and ultraimaginaries b and c, there is a b' ≡_A b such that b' ⊥^{bu}_A c). Extension then follows as an immediate corollary.
We also study total ⊥^{bu}-Morley sequences (i.e., A-indiscernible sequences I satisfying J ⊥^{bu}_A K for any J and K with J + K ≡^{EM}_A I), and we prove that an A-indiscernible sequence I is a total ⊥^{bu}-Morley sequence over A if and only if whenever I and I' have the same Lascar strong type over A, I and I' are related by the transitive, symmetric closure of the relation 'J+K is A-indiscernible.' This is also equivalent to I being 'based on' A in a sense defined by Shelah in his early study of simple unstable theories.
Finally, we show that for any A and b in any theory T, if there is an Erdös cardinal κ(α) with |Ab|+|T| < κ(α), then there is a total ⊥^{bu}-Morley sequence (b_i)_{i<ω} over A with b_0 = b.

Mathematics Subject Classification: 03C45

Keywords and phrases:

Full text arXiv 2201.03631: pdf, ps.