A note on rosy theories

Hans Scheuermann

July 7, 2003

Theorem 1. Let T be a simple theory that eliminates hyperimaginaries, C ⊆ B. If tp(a/B) does not þ-fork over C, then tp(a/B) does not fork over C.

Proof. Let (a_{i})_{i<ω} be a þ-Morley sequence over C s.t. a_{i} ≡_{B} a for all i < ω. It suffices to prove that (a_{i})_{i<ω} is in fact a Morley sequence over C.

Let (a_{i})_{i<ω·3} be a C-indiscernible extension of (a_{i})_{i<ω} and write

b_{j} = (a_{i})_{ω·j≤i<ω·(j+1)}.

Let d = cb stp(b_{0}/b_{1}b_{2}). Since tp(b_{0}/b_{1}b_{2}) is an heir of tp(b_{0}/b_{1}) and also an heir of tp(b_{0}/b_{2}), we have d = cb stp(b_{0}/b_{1}) = cb stp(b_{0}/b_{2}), so d ∈ acl^{eq} b_{1} ∩ acl^{eq} b_{2} ⊆ acl^{eq} C.

Thus tp(b_{0}/Cb_{1}b_{2}) does not fork over C. By indiscernibility, (a_{i})_{i<ω} is a Morley sequence. □

This is a condensed version of my original argument, which used Theorems 2 and 3 below and the existence of (hyperimaginary) canonical bases for forking in simple theories.

Theorem 2. A complete theory with infinite models is rosy if and only if it has a notion of independence that satisfies the reflexivity axiom for imaginaries. In a rosy theory þ-forking is the coarsest such notion of independence.

Proof sketch. It suffices to prove that the ternary relation induced by þforking is always coarser than a given reflexive notion of independence. For if such a notion of independence exists, þ-forking satisfies the local character axiom, so the theory is rosy by [3], Theorem 2.1.1.

So suppose a is independent from B over C, C ⊆ B, with respect to some reflexive notion of independence. For any b ∈ B there is a Morley sequence (b_{i})_{i<ω} over C such that b_{i} ⊧ tp(b/Ca). These Morley sequences witness that no formula in tp(a/B) strongly divides over C. By extension and monotonicity of the reflexive notion of independence, tp(a/B) does not even þ-fork over C. □

Theorem 3. If a notion of independence satisfies the reflexivity axiom for imaginaries and has imaginary canonical bases, then it is the coarsest notion of independence satisfying the reflexivity axiom for imaginaries.

Theorem 3 is Satz 1.7.4 in [4].

References

[1] B. Hart, B. Kim, A. Pillay, Coordinatisation and canonical bases in simple theories, Journal of Symbolic Logic 65 (1), 293–309 (2000).

[2] A. Onshuus, Properties and consequences of thorn-independence, Preprint (2003), arXiv:math.LO/0205004 v2.

[3] A. Onshuus, Th-forking, algebraic independence and examples of rosy theories, Preprint (2003), arXiv:math.LO/0306003 v1.

[4] H. Scheuermann, Unabhängigkeitsrelationen, Diplomarbeit, Freiburg, 1996.