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Existence of many L∞,λ-equivalent, non- isomorphic models of T of power λ

Saharon Shelah
1987 Annals of Pure and Applied Logic  
If A > ITI1 is regular or strong limit, we construct 2" non-isomorphic, pairwise L", -equivalent models of T of power A, which are reducts of models of TI.  ...  Note, however, that the proof applies to the class of models of T, T (superstable but) with dop or otop and even to appropriate non-elementary classes as well.  ...  Introduction This paper has a place in two lines of research: existence of L,,*-equivalent non-isomorphic models of power A and classification theory. On the history of construction of L.  ... 
doi:10.1016/0168-0072(87)90005-4 fatcat:4tpnjluwpfcx3mipm4rrt4meoy

Distributions of countable models of theories with continuum many types [article]

Roman A. Popkov, Sergey V. Sudoplatov
2012 arXiv   pre-print
We present distributions of countable models and correspondent structural characteristics of complete theories with continuum many types: for prime models over finite sets relative to Rudin-Keisler preorders  ...  , for limit models over types and over sequences of types, and for other countable models of theory.  ...  Since the process of extension of models M p by continuum many models M q can be continued unboundedly many times, there are continuum many pairwise non-isomorphic limit models, i. e., L(T ) = 2 ω . ✷  ... 
arXiv:1210.4043v1 fatcat:ghimqf6cz5hwrgmdqnrn2qng64

Characterizing all models in infinite cardinalities

Lauri Keskinen
2013 Annals of Pure and Applied Logic  
We can ask the question: what kind of a logic L is needed to characterize all models of cardinality κ (in a finite vocabulary) up to isomorphism by their L-theories?  ...  In other words: for which logics L it is true that if any models A and B of cardinality κ satisfy the same L-theory then they are isomorphic?  ...  Acknowledgements This article is based on the author's Ph.D. thesis titled "Characterizing All Models in Infinite Cardinalities".  ... 
doi:10.1016/j.apal.2012.10.012 fatcat:32lrhxfaeba65axevlr45rk3um

Forcing isomorphism [article]

John T. Baldwin, Michael C. Laskowski, Saharon Shelah
1993 arXiv   pre-print
Hence, if two models of a classifiable theory of power lambda are non-isomorphic, they are non-isomorphic after a lambda-complete forcing.  ...  In particular, we show that non-isomorphism of models of a classifiable theory need not be preserved by ccc forcings.  ...  That is, if two models of a classifiable theory of power λ are non-isomorphic, they remain non-isomorphic after a λ-complete forcing.  ... 
arXiv:math/9301208v1 fatcat:baydltafo5bmleniv5w4cpfl2u

An instance of Vaught's conjecture using algebraic logic [article]

Mohammed Assem, Tarek Sayed Ahmed
2013 arXiv   pre-print
In all case we also count the number of non isomorphic models omitting < covK many non isolated types (the latter is the least cardinal such that the Baire Category theorem fails and also the largest cardinal  ...  We show that the equivalence relation induced by such (indistinguishable) models is a Borel subset of the product of the Stone Polish space (consisting of ultarfilters corresponding to models) with itself  ...  Suppose that T is a theory, |T | = λ, λ regular, then there exist models M i : i < χ = λ 2, each of cardinality λ, such that if i(1) = i(2) < χ, a i(l) ∈ |M i(l) |, l = 1, 2,, tp(ā l(1) ) = tp(ā l(2) )  ... 
arXiv:1304.0883v1 fatcat:tp2ckxuadbeojdaqf4pzz4cfdq

Recovering function fields from their decomposition graphs [chapter]

Florian Pop
2011 Number Theory, Analysis and Geometry  
We develop the global theory of a strategy to tackle a program initiated by Bogomolov in 1990.  ...  program aims at giving a group theoretical recipe by which one can reconstruct function fields K|k with td(K|k) > 1 and k algebraically closed from the maximal pro-abelian-by-central Galois group Π c K of  ...  Hence finally ε is a power of the characteristic exponent of k and l. Equivalently, ı is a Frobenius twist of ı.  ... 
doi:10.1007/978-1-4614-1260-1_24 fatcat:sp6fauxp55cjheqvqwz3islbhu

Isoperimetric inequalities and the Friedlander-Milnor conjecture [article]

Tibor Beke
2004 arXiv   pre-print
discrete, are equivalent to the existence of an isoperimetric inequality in the homological bar complex of G(F), where F is the algebraic closure of a finite field.  ...  We prove that Friedlander's generalized isomorphism conjecture on the cohomology of algebraic groups, and hence the Isomorphism Conjecture for the cohomology of the complex algebraic Lie group G(C) made  ...  Identifying the l th -power roots of unity in F p with Z/l ∞ , one gets an injection Z/l ∞ → F × p such that (Z/l ∞ ) r → T(F p ) induces isomorphism on H * (−, Z/l).  ... 
arXiv:math/0403426v2 fatcat:gudf6vskize63dcb47zedh7lhu

Model theory without choice: Categoricity [article]

Saharon Shelah
2008 arXiv   pre-print
Another central result here is, in this context: the number of models of a countable first order T of cardinality aleph_alpha is either >=|alpha| for every aleph_\alpha or it has a small upper bound (close  ...  We prove Los conjecture = Morley theorem in ZF, with the same characterization (of first order countable theories categorical in aleph_alpha for some (equivalently for every) ordinal alpha>0.  ...  classes (c) the number of models of T in µ up to isomorphism is equal to the number of E T,µ -equivalence classes.  ... 
arXiv:math/0504196v3 fatcat:klrhlohghngntbmvjf6vrm6k5i

Combinations of structures [article]

Sergey V. Sudoplatov
2016 arXiv   pre-print
We investigate combinations of structures by families of structures relative to families of unary predicates and equivalence relations.  ...  The notions of e-spectra are introduced and possibilities for e-spectra are described.  ...  As usual we denote by I(T, λ) the number of pairwise non-isomorphic models of T having the cardinality λ.  ... 
arXiv:1601.00041v1 fatcat:2ejf63ynkbhodc5tiohp4m6gcm

On Σ^1_1-complete Equivalence Relations on the Generalized Baire Space [article]

Tapani Hyttinen, Vadim Kulikov
2012 arXiv   pre-print
Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if V = L, then many of them are \Sigma^1_1-complete, in particular  ...  the isomorphism relation of dense linear orders.  ...  The research was partially supported by the Academy of Finland through its grant WBS 1251557 and the second author was funded by the Science Foundation of the University of Helsinki.  ... 
arXiv:1209.3932v1 fatcat:if5yx5puivd5tnbb5mrqrj7vpe

Elementary equivalences and accessible functors

T. Beke, J. Rosický
2018 Annals of Pure and Applied Logic  
This notion specializes to Lλ -equivalence and Lλ -elementary embedding for categories of structures in a language of arity less than λ, and interacts well with functors and λ-directed colimits.  ...  We introduce the notion of λ-equivalence and λ-embeddings of objects in suitable categories.  ...  Karp's theorem, however, equates Lλ -equivalence of structures with the existence of a set of partial isomorphisms satisfying certain extension properties.  ... 
doi:10.1016/j.apal.2018.03.004 fatcat:ts3bphrykzgyvhhtbx7gqkf6re

Elementary equivalences and accessible functors [article]

Tibor Beke, Jiri Rosicky
2016 arXiv   pre-print
We introduce the notion of λ-equivalence and λ-embeddings of objects in suitable categories.  ...  This notion specializes to L_∞λ-equivalence and L_∞λ-elementary embedding for categories of structures in a language of arity less than λ, and interacts well with functors and λ-directed colimits.  ...  Karp's theorem, however, equates Lλ -equivalence of structures with the existence of a set of partial isomorphisms satisfying certain extension properties.  ... 
arXiv:1603.02500v1 fatcat:hqd6wmfxmfcklphzcaoqvouw4u

Introduction to twisted commutative algebras [article]

Steven V Sam, Andrew Snowden
2012 arXiv   pre-print
This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries  ...  The third part develops some basic properties of twisted commutative algebras from the perspective of classical commutative algebra and summarizes some of the results of the authors.  ...  The theorem that H M belongs to Q[t, e t ] can be stated equivalently using differential equations: it amounts to the existence of a polynomial p, whose roots are non-negative integers, such that p(d/dt  ... 
arXiv:1209.5122v1 fatcat:2dsvs653lff45kxmpeo5e3m3im

Combinations of structures

S. V. Sudoplatov
2018 Известия Иркутского государственного университета: Серия "Математика"  
We investigate combinations of structures by families of structures relative to families of unary predicates and equivalence relations.  ...  not approximate infinitely many n-types for n ∈ ω.  ...  As usual we denote by I(T, λ) the number of pairwise non-isomorphic models of T having the cardinality λ.  ... 
doi:10.26516/1997-7670.2018.24.82 fatcat:uoam6x74affg5alxi3zwmtvw6u

Non-computable models of certain first order theories

Gabor Sagi, Ramon Horvath
2017 2017 IEEE 14th International Scientific Conference on Informatics  
We show, that if T is a first order theory with at least one uncountable Stone space then T has a countable model not isomorphic to any D-presented one.  ...  We also show that there is a countable ℵ0-categorical structure in a finite language which is not isomorphic to any D-presented structure; in addition, there exists a consistent first order theory in a  ...  Suppose H is an uncountable family of pairwise non-isomorphic countable L-structures.  ... 
doi:10.1109/informatics.2017.8327265 fatcat:izfm4x6lp5hgjffxu7raivci7i
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