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The geometry of weakly minimal types

  • Steven Buechler (a1)

Let T be superstable. We say a type p is weakly minimal if R(p, L, ∞) = 1. Let MT be uncountable and saturated, H = p(M). We say DH is locally modular if for all X, YD with X = acl(X) ∩ D, Y = acl(Y) ∩ D and XY ≠ ∅,

Theorem 1. Let pS(A) be weakly minimal and D the realizations of stp(a/A) for some a realizing p. Then D is locally modular or p has Morley rank 1.

Theorem 2. Let H, G be definable over some finite A, weakly minimal, locally modular and nonorthogonal. Then for all aH∖acl(A), bG∖acl(A) there area′ ∈ H, b′ ∈ G such that a′ ∈ acl(abbA)∖acl(aA). Similarly when H and G are the realizations of complete types or strong types over A.

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[La] D. Lascar , Ranks and definability in superstable theories, Israel Journal of Mathematics, vol. 23 (1976), pp. 5387.

[Z1] B. I. Zil′ber , Strongly minimal countablty categorical theories, Siberian Mathematical Journal, vol. 21 (1980), pp. 219230.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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