10 found
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  1.  12
    On the Consistency of Some Partition Theorems for Continuous Colorings, and the Structure of ℵ 1 -Dense Real Order Types.J. Steprans, Uri Abraham, Matatyahu Rubin & Saharon Shelah - 2002 - Bulletin of Symbolic Logic 8 (2):303.
    We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ℵ 1 -dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ℵ 1 -dense subsets of R are isomorphic, is consistent. We e.g. prove Con). Let K H , be the set of order types of ℵ 1 -dense homogeneous subsets of (...)
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  2.  6
    On Well-Generated Boolean Algebras.Robert Bonnet & Matatyahu Rubin - 2000 - Annals of Pure and Applied Logic 105 (1-3):1-50.
    A Boolean algebra B that has a well-founded sublattice L which generates B is called a well-generated Boolean algebra. If in addition, L is generated by a complete set of representatives for B , then B is said to be canonically well-generated .Every WG Boolean algebra is superatomic. We construct two basic examples of superatomic non well-generated Boolean algebras. Their cardinal sequences are 1,0,1,1 and 0,0,20,1.Assuming MA , we show that every algebra with one of the cardinal sequences , α<1, (...)
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  3.  24
    On the Elementary Equivalence of Automorphism Groups of Boolean Algebras; Downward Skolem Löwenheim Theorems and Compactness of Related Quantifiers.Matatyahu Rubin & Saharon Shelah - 1980 - Journal of Symbolic Logic 45 (2):265-283.
    THEOREM 1. (⋄ ℵ 1 ) If B is an infinite Boolean algebra (BA), then there is B 1 such that $|\operatorname{Aut} (B_1)| \leq B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut} (B_1)\rangle \equiv \langle B, \operatorname{Aut}(B)\rangle$ . THEOREM 2. (⋄ ℵ 1 ) There is a countably compact logic stronger than first-order logic even on finite models. This partially answers a question of H. Friedman. These theorems appear in §§ 1 and 2. THEOREM 3. (a) (⋄ ℵ 1 ) If (...)
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  4.  38
    On the Expressibility Hierarchy of Magidor-Malitz Quantifiers.Matatyahu Rubin & Saharon Shelah - 1983 - Journal of Symbolic Logic 48 (3):542-557.
    We prove that the logics of Magidor-Malitz and their generalization by Rubin are distinct even for PC classes. Let $M \models Q^nx_1 \cdots x_n \varphi(x_1 \cdots x_n)$ mean that there is an uncountable subset A of |M| such that for every $a_1, \ldots, a_n \in A, M \models \varphi\lbrack a_1, \ldots, a_n\rbrack$ . Theorem 1.1 (Shelah) $(\diamond_{\aleph_1})$ . For every n ∈ ω the class $K_{n + 1} = \{\langle A, R\rangle \mid \langle A, R\rangle \models \neg Q^{n + 1} (...)
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  5.  64
    Elementary Embedding Between Countable Boolean Algebras.Robert Bonnet & Matatyahu Rubin - 1991 - Journal of Symbolic Logic 56 (4):1212-1229.
    For a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B1 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then $\langle M_T, \leq\rangle$ is well-quasi-ordered. ■ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such that $B\upharpoonright (...)
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  6.  17
    On Poset Boolean Algebras of Scattered Posets with Finite Width.Robert Bonnet & Matatyahu Rubin - 2003 - Archive for Mathematical Logic 43 (4):467-476.
    We prove that the poset algebra of every scattered poset with finite width is embeddable in the poset algebra of a well ordered poset.
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  7.  9
    On L Α,Ω Complete Extensions of Complete Theories of Boolean Algebras.Matatyahu Rubin - 2004 - Archive for Mathematical Logic 43 (5):571-582.
    For a complete first order theory of Boolean algebras T which has nonisomorphic countable models, we determine the first limit ordinal α = α(T) such that We show that for some and for all other T‘s, A nonprincipal ideal I of B is almost principal, if a is a principal ideal of B} is a maximal ideal of B. We show that the theory of Boolean algebras with an almost principal ideal has complete extensions and characterize them by invariants similar (...)
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  8.  12
    On Essentially Low, Canonically Well-Generated Boolean Algebras.Robert Bonnet & Matatyahu Rubin - 2002 - Journal of Symbolic Logic 67 (1):369-396.
    Let B be a superatomic Boolean algebra (BA). The rank of B (rk(B)), is defined to be the Cantor Bendixon rank of the Stone space of B. If a ∈ B - {0}, then the rank of a in B (rk(a)), is defined to be the rank of the Boolean algebra $B b \upharpoonright a \overset{\mathrm{def}}{=} \{b \in B: b \leq a\}$ . The rank of 0 B is defined to be -1. An element a ∈ B - {0} is (...)
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  9.  5
    A Superatomic Boolean Algebra with Few Automorphisms.Matatyahu Rubin & Sabine Koppelberg - 2001 - Archive for Mathematical Logic 40 (2):125-129.
    Assuming GCH, we prove that for every successor cardinal μ > ω1, there is a superatomic Boolean algebra B such that |B| = 2μ and |Aut B| = μ. Under ◊ω1, the same holds for μ = ω1. This answers Monk's Question 80 in [Mo].
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  10.  10
    On the Consistency of Some Partition Theorems for Continuous Colorings, and the Structure of ℵ1-Dense Real Order Types.Uri Abraham, Matatyahu Rubin & Saharon Shelah - 1982 - Annals of Pure and Applied Logic 29 (2):123-206.
    We present some techniques in c.c.c. forcing, and apply them to prove consistency results concerning the isomorphism and embeddability relations on the family of ℵ 1 -dense sets of real numbers. In this direction we continue the work of Baumgartner [2] who proved the axiom BA stating that every two ℵ 1 -dense subsets of R are isomorphic, is consistent. We e.g. prove Con). Let K H, be the set of order types of ℵ 1 -dense homogeneous subsets of R (...)
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