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Steven Buechler [22]Steven M. Buechler [1]
  1. Pseudoprojective Strongly Minimal Sets Are Locally Projective.Steven Buechler - 1991 - Journal of Symbolic Logic 56 (4):1184-1194.
    Let D be a strongly minimal set in the language L, and $D' \supset D$ an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T' be the theory of the structure (D', D), where D interprets the predicate D. It is known that T' is ω-stable. We prove Theorem A. If D is not locally modular, then T' has Morley rank ω. We say that a strongly minimal set D is pseudoprojective (...)
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  2. Annual Meeting of the Association for Symbolic Logic: Notre Dame, 1993.Steven Buechler - 1994 - Journal of Symbolic Logic 59 (2):696-719.
  3.  2
    Vaught’s Conjecture for Superstable Theories of Finite Rank.Steven Buechler - 2008 - Annals of Pure and Applied Logic 155 (3):135-172.
    In [R. Vaught, Denumerable models of complete theories, in: Infinitistic Methods, Pregamon, London, 1961, pp. 303–321] Vaught conjectured that a countable first order theory has countably many or 20 many countable models. Here, the following special case is proved.
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  4.  10
    Lascar Strong Types in Some Simple Theories.Steven Buechler - 1999 - Journal of Symbolic Logic 64 (2):817-824.
    In this paper a class of simple theories, called the low theories is developed, and the following is proved. Theorem. Let T be a low theory. A set and a, b elements realizing the same strong type over A. Then, a and b realized the same Lascar strong type over A.
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  5. One Theorem of Zil'ber's on Strongly Minimal Sets.Steven Buechler - 1985 - Journal of Symbolic Logic 50 (4):1054-1061.
    Suppose $D \subset M$ is a strongly minimal set definable in M with parameters from C. We say D is locally modular if for all $X, Y \subset D$ , with $X = \operatorname{acl}(X \cup C) \cap D, Y = \operatorname{acl}(Y \cup C) \cap D$ and $X \cap Y \neq \varnothing$ , dim(X ∪ Y) + dim(X ∩ Y) = dim(X) + dim(Y). We prove the following theorems. Theorem 1. Suppose M is stable and $D \subset M$ is strongly minimal. (...)
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  6.  4
    Locally Modular Theories of Finite Rank.Steven Buechler - 1986 - Annals of Pure and Applied Logic 30 (1):83-94.
  7.  4
    The Geometry of Weakly Minimal Types.Steven Buechler - 1985 - Journal of Symbolic Logic 50 (4):1044-1053.
    Let T be superstable. We say a type p is weakly minimal if R(p, L, ∞) = 1. Let $M \models T$ be uncountable and saturated, H = p(M). We say $D \subset H$ is locally modular if for all $X, Y \subset D$ with $X = \operatorname{acl}(X) \cap D, Y = \operatorname{acl}(Y) \cap D$ and $X \cap Y \neq \varnothing$ , dim(X ∪ Y) + dim(X ∩ Y) = dim(X) + dim(Y). Theorem 1. Let p ∈ S(A) be weakly (...)
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  8. Maximal Chains in the Fundamental Order.Steven Buechler - 1986 - Journal of Symbolic Logic 51 (2):323-326.
    Suppose T is superstable. Let ≤ denote the fundamental order on complete types, [ p] the class of the bound of p, and U(--) Lascar's foundation rank (see [LP]). We prove THEOREM 1. If $q and there is no r such that $q , then U(q) + 1 = U(p). THEOREM 2. Suppose $U(p) and $\xi_1 is a maximal descending chain in the fundamental order with ξ κ = [ p]. Then k = U(p). That the finiteness of U(p) in (...)
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  9.  2
    Simple Stable Homogeneous Expansions of Hilbert Spaces.Alexander Berenstein & Steven Buechler - 2004 - Annals of Pure and Applied Logic 128 (1-3):75-101.
    We study simplicity and stability in some large strongly homogeneous expansions of Hilbert spaces. Our approach to simplicity is that of Buechler and Lessmann 69). All structures we consider are shown to have built-in canonical bases.
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  10.  2
    On the Existence of Regular Types.Saharon Shelah & Steven Buechler - 1989 - Annals of Pure and Applied Logic 45 (3):277-308.
    The main results in the paper are the following. Theorem A. Suppose that T is superstable and M ⊂ N are distinct models of T eq . Then there is a c ϵ N⧹M such that t is regular. For M ⊂ N two models we say that M ⊂ na N if for all a ϵ M and θ such that θ ≠ θ , there is a b ∈ θ ⧹ acl . Theorem B Suppose that T is (...)
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  11.  8
    On Nontrivial Types of U-Rank.Steven Buechler - 1987 - Journal of Symbolic Logic 52 (2):548-551.
    Theorem A. Suppose that T is superstable and p is a nontrivial type of U-rank 1. Then R(p, L, ∞) = 1. Theorem B. Suppose that T is totally transcendental and p is a nontrivial type of U-rank 1. Then p has Morley rank 1.
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  12. On Nontrivial Types of U-Rank 1.Steven Buechler - 1987 - Journal of Symbolic Logic 52 (2):548-551.
  13.  8
    The Classification of Small Weakly Minimal Sets. II.Steven Buechler - 1988 - Journal of Symbolic Logic 53 (2):625-635.
    The main result is Vaught's conjecture for weakly minimal, locally modular and non-ω-stable theories. The more general results yielding this are the following. THEOREM A. Suppose that T is a small unidimensional theory and D is a weakly minimal set, definable over the finite set B. Then for all finite $A \subset D$ there are only finitely many nonalgebraic strong types over B realized in $\operatorname{acl}(A) \cap D$ . THEOREM B. Suppose that T is a small, unidimensional, non-ω-stable theory such (...)
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  14.  8
    Isolated Types in a Weakly Minimal Set.Steven Buechler - 1987 - Journal of Symbolic Logic 52 (2):543-547.
    Theorem A. Let T be a small superstable theory, A a finite set, and ψ a weakly minimal formula over A which is contained in some nontrivial type which does not have Morley rank. Then ψ is contained in some nonalgebraic isolated type over A. As an application we prove Theorem B. Suppose that T is small and superstable, A is finite, and there is a nontrivial weakly minimal type p ∈ S(A) which does not have Morley rank. Then the (...)
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  15.  13
    The Classification of Small Types of Rank Ω, Part I.Steven Buechler & Colleen Hoover - 2001 - Journal of Symbolic Logic 66 (4):1884-1898.
    Certain basic concepts of geometrical stability theory are generalized to a class of closure operators containing algebraic closure. A specific case of a generalized closure operator is developed which is relevant to Vaught's conjecture. As an application of the methods, we prove THEOREM A. Let G be a superstable group of U-rank ω such that the generics of G are locally modular and Th(G) has few countable models. Let G - be the group of nongeneric elements of G, G + (...)
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  16.  8
    Kueker's Conjecture for Superstable Theories.Steven Buechler - 1984 - Journal of Symbolic Logic 49 (3):930-934.
    We prove that if every uncountable model of a first-order theory T is ω-saturated and T is superstable then T is categorical in some infinite power.
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  17.  5
    The Classification of Small Weakly Minimal Sets. III: Modules.Steven Buechler - 1988 - Journal of Symbolic Logic 53 (3):975-979.
    Theorem A. Let M be a left R-module such that Th(M) is small and weakly minimal, but does not have Morley rank 1. Let $A = \mathrm{acl}(\varnothing) \cap M$ and $I = \{r \in R: rM \subset A\}$ . Notice that I is an ideal. (i) F = R/I is a finite field. (ii) Suppose that a, b 0 ,...,b n ∈ M and a b̄. Then there are s, r i ∈ R, i ≤ n, such that sa + (...)
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  18.  7
    Expansions of Models of Ω-Stable Theories.Steven Buechler - 1984 - Journal of Symbolic Logic 49 (2):470-477.
    We prove that every relation-universal model of an ω-stable theory is saturated. We also show there is a large class of ω-stable theories for which every resplendent model is homogeneous.
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  19.  5
    Book Review: S. Shelah. Classification Theory and the Number of Non-Isomorhic Models. [REVIEW]Steven Buechler - 1991 - Notre Dame Journal of Formal Logic 33 (1):154-158.