scholarly journals An Institution-independent Generalization of Tarski's Elementary Chain Theorem

2007 ◽  
Vol 17 (3) ◽  
pp. 605-605
Keyword(s):  
Elementary Chain

scholarly journals Elementary chain composition of guinea pig thyroglobulin.

1975 ◽  
Vol 250 (18) ◽  
pp. 7294-7299 ◽  
Author(s):  
A Haeberli ◽  
J Bilstad ◽  
H Edelhoch ◽  
J E Rall
Keyword(s):  
Guinea Pig ◽  
Elementary Chain ◽  
Chain Composition

scholarly journals COUNTABLE MODELS OF THE THEORIES OF BALDWIN–SHI HYPERGRAPHS AND THEIR REGULAR TYPES

2019 ◽  
Vol 84 (3) ◽  
pp. 1007-1019
Author(s):  
DANUL K. GUNATILLEKA
Keyword(s):  
Large Body ◽  
Countable Model ◽  
Stable Theory ◽  
Generic Structure ◽  
Elementary Chain ◽  
Regular Type ◽  
Image Position ◽  
Original Class ◽  
Countable Models

AbstractWe continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $ if and only if the dimension of $\mathfrak{M}_\beta $ is at most the dimension of $\mathfrak{M}_\gamma $ and that each countable model is isomorphic to some $\mathfrak{M}_\beta $. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω-stable theory with a nonlocally modular regular type, answering a question of Pillay in [11].

Enthalpies of elementary chain propagation steps in the oxidation of organic compounds

1999 ◽  
Vol 68 (2) ◽  
pp. 285-289
Author(s):  
I. M. Borisov ◽  
D. A. Mikhaylov ◽  
S. L. Khursan
Keyword(s):  
Organic Compounds ◽  
Chain Propagation ◽  
Elementary Chain ◽  
Oxidation Of Organic Compounds

Elementary epimorphisms

2005 ◽  
Vol 70 (2) ◽  
pp. 473-487 ◽  
Author(s):  
Philipp Rothmaler
Keyword(s):  
Inverse Limits ◽  
Elementary Chain ◽  
Flat Modules ◽  
Inverse Systems

AbstractThe concept of elementary epimorphism is introduced. Inverse systems of such maps are considered, and a dual of the elementary chain lemma is found (Cor. 4.2). The same is done for pure epimorphisms (Cor. 4.3 and 4.4). Finally, this is applied to certain inverse limits of flat modules (Thm. 6.4) and certain inverse limits of absolutely pure modules (Cor. 6.3).

Elementary Chains of Invariant Subspaces of a Banach Space

1995 ◽  
Vol 47 (2) ◽  
pp. 290-301
Author(s):  
Jon M. Clauss
Keyword(s):  
Banach Space ◽  
Compact Operator ◽  
Polynomial Identity ◽  
Invariant Subspaces ◽  
Compact Operators ◽  
Operator Norm ◽  
Spectrum Of An Operator ◽  
Elementary Chain ◽  
Closed Subalgebra ◽  
Simple Chain

AbstractWe will generalize Ringrose's notion of a simple chain of closed invariant subspaces of a compact operator acting on a Banach space, to that of an elementary chain of invariant subspaces of a subalgebra of compact operators. With this we expand the notion of diagonal coefficients to that of diagonal representations and subsequently generalize Ringrose's theorem equating the spectrum of an operator to the collection of diagonal coefficients. This in turn, in conjunction with some results from the theory of Polynomial Identity algebras, allows us to generalize Murphy's theorem which states that a closed subalgebra of compact operators is simultaneously triangularizable if and only if / rad () is commutative. Let be an algebra of compact operators acting on a Banach space with a norm ‖ · ‖ which dominates the operator norm, and under which 𝒜 is complete. Then has an elementary chain of invariant subspaces of bound n if and only if / rad () satisfies the standard polynomial .

A note on finitely generated models

1983 ◽  
Vol 48 (1) ◽  
pp. 163-166
Author(s):  
Anand Pillay
Keyword(s):  
Countable Model ◽  
Stable Theory ◽  
Prime Model ◽  
Finitely Generated ◽  
First Order ◽  
Elementary Chain ◽  
Skolem Functions ◽  
Order Language ◽  
Finite Set

A model M (of a countable first order language) is said to be finitely generated if it is prime over a finite set, namely if there is a finite tuple ā in M such that (M, ā) is a prime model of its own theory. Similarly, if A ⊂ M, then M is said to be finitely generated overA if there is finite ā in M such that M is prime over A ⋃ ā. (Note that if Th(M) has Skolem functions, then M being prime over A is equivalent to M being generated by A in the usual sense, that is, M is the closure of A under functions of the language.) We show here that if N is ā model of an ω-stable theory, M ≺ N, M is finitely generated, and N is finitely generated over M, then N is finitely generated. A corollary is that any countable model of an ω-stable theory is the union of an elementary chain of finitely generated models. Note again that all this is trivial if the theory has Skolem functions.The result here strengthens the results in [3], where we show the same thing but assuming in addition that the theory is either nonmultidimensional or with finite αT. However the proof in [3] for the case αT finite actually shows the following which does not assume ω-stability): Let A be atomic over a finite set, tp(ā / A) have finite Cantor-Bendixson rank, and B be atomic over A ⋃ ā. Then B is atomic over a finite set.

scholarly journals An Institution-independent Generalization of Tarski's Elementary Chain Theorem

2006 ◽  
Vol 16 (6) ◽  
pp. 713-735 ◽  
Author(s):  
D. Gaina ◽  
A. Popescu
Keyword(s):  
Elementary Chain

There are reasonably nice logics

1991 ◽  
Vol 56 (1) ◽  
pp. 300-322 ◽  
Author(s):  
Wilfrid Hodges ◽  
Saharon Shelah
Keyword(s):  
Interpolation Theorem ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Elementary Chain ◽  
Uncountable Cardinal ◽  
Joint Embedding ◽  
Preservation Property ◽  
Strongly Compact Cardinals

A well-known question of Feferman asks whether there is a logic which extends the logic , is ℵ0-compact and satisfies the interpolation theorem. (Cf. Makowsky [M] for background and terminology.)The same question was open when ℵ1 in is replaced by any other uncountable cardinal κ. We shall show that when κ is an uncountable strongly compact cardinal and there is a strongly compact cardinal > κ, then there is such a logic. It is impossible to prove the existence of uncountable strongly compact cardinals in ZFC. However, the logic that we describe has a simple and natural definition, together with several other pleasant properties. For example it satisfies Robinson's lemma, PPP (pair preservation property, viz. the theory of the sum of two models is the sum of their theories), versions of the elementary chain lemma for chains of length < λ, and isomorphism of (suitable) ultralimits.This logic is described in §2 below; we call it 1. It is not a new logic—it was introduced in [Sh, Part II, §3] as an example of a logic which has the amalgamation and joint embedding properties. See the transparent presentation in [M]. But we shall repeat all the definitions. In [HS] we presented a logic with some of the same properties as 1, also based on a strongly compact cardinal λ; but unlike 1, it was not a sublogic of λ,λ.

A new proof of a theorem of Shelah

1972 ◽  
Vol 37 (1) ◽  
pp. 133-134 ◽  
Author(s):  
John W. Rosenthal
Keyword(s):  
Model Theory ◽  
Continuum Hypothesis ◽  
Order Theory ◽  
Transfinite Induction ◽  
Original Proof ◽  
First Order ◽  
Elementary Chain ◽  
First Order Theory ◽  
Definable Subset ◽  
Generalized Continuum

In [10, §0, E), 5)] Shelah states using the proofs of 7.9 and 6.9 in [9] it is possible to prove that if a countable first-order theory T is ℵ0-stable (totally transcendental) and not ℵ1-categorical, then it has at least ∣1 + α∣ models of power ℵα.In this note we will give a new proof of this theorem using the work of Baldwin and Lachlan [1]. Our original proof used the generalized continuum hypothesis (GCH). We are indebted to G. E. Sacks for suggesting that the notions of ℵ0-stability and ℵ1-categoricity are absolute, and that consequently our use of GCH was eliminable [8]. Routine results from model theory may be found, e.g. in [2].Proof (with GCH). In the proof of Theorem 3 of [1] Baldwin and Lachlin show of power ℵα such that there is a countable definable subset in . Let B0 be such a subset. Say . We will give by transfinite induction an elementary chain of models of T of power ℵα such that B[i1 … in] has power ℵβ and such that every infinite definable subset of has power ≥ℵβ. This clearly suffices.

Elementary Chain Counting Problems

2000 ◽  
pp. 25-34
Keyword(s):  
Counting Problems ◽  
Elementary Chain
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