\mathbf P-NDOP and \mathbf P-decompositions of ℵε-saturated models of superstable theories

2015 ◽  
Vol 229 (1) ◽  
pp. 47-81
Author(s):  
Saharon Shelah ◽  
Michael C. Laskowski
Keyword(s):  
Saturated Models ◽  
Superstable Theories

Saturated models

1998 ◽  
Vol 43 (7) ◽  
pp. 562-564
Author(s):  
Guolong Chen
Keyword(s):  
Saturated Models

Automorphism groups of arithmetically saturated models

2006 ◽  
Vol 71 (1) ◽  
pp. 203-216 ◽  
Author(s):  
Ermek S. Nurkhaidarov
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Peano Arithmetic ◽  
Standard System ◽  
Permutation Groups ◽  
Saturated Model ◽  
Saturated Models ◽  
Homogeneous Set ◽  
Image Position

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that if M is a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2 be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then SSy(M1) = SSy(M2).We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1, M2be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then for every n < ωHere RT2n is Infinite Ramsey's Theorem stating that every 2-coloring of [ω]n has an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic.

A Note on Saturated Models for Many-Valued Logics

2015 ◽  
Vol 65 (4) ◽  
Author(s):  
Tommaso Flaminio ◽  
Matteo Bianchi
Keyword(s):  
Representation Theorem ◽  
Amalgamation Property ◽  
Short Paper ◽  
Fuzzy Logics ◽  
Saturated Models ◽  
Joint Embedding ◽  
Joint Embedding Property

AbstractIn this short paper we will discuss on saturated and κ-saturated models of many-valued (t-norm based fuzzy) logics. Using these peculiar structures we show a representation theorem à la Di Nola for several classes of algebras including MV, Gödel, product, BL, NM and WNM-algebras. Then, still using (κ)-saturated algebras, we finally show that some relevant subclasses of algebras related to many-valued logics also enjoy the joint embedding property and the amalgamation property.

Notes on the stability of separably closed fields

1979 ◽  
Vol 44 (3) ◽  
pp. 412-416 ◽  
Author(s):  
Carol Wood
Keyword(s):  
Differential Algebra ◽  
Complete Theory ◽  
Model Completeness ◽  
Closed Fields ◽  
Superstable Theories ◽  
Basic Facts ◽  
The Stability ◽  
Differential Fields

AbstractThe stability of each of the theories of separably closed fields is proved, in the manner of Shelah's proof of the corresponding result for differentially closed fields. These are at present the only known stable but not superstable theories of fields. We indicate in §3 how each of the theories of separably closed fields can be associated with a model complete theory in the language of differential algebra. We assume familiarity with some basic facts about model completeness [4], stability [7], separably closed fields [2] or [3], and (for §3 only) differential fields [8].

scholarly journals On the uniqueness of cellular injectives

2018 ◽  
Vol 167 (3) ◽  
pp. 489-504 ◽  
Author(s):  
J. ROSICKÝ
Keyword(s):  
Banach Spaces ◽  
Category Theory ◽  
Existence And Uniqueness ◽  
Boolean Algebras ◽  
Small Object ◽  
Unified Approach ◽  
Basic Tool ◽  
Saturated Models ◽  
Small Object Argument

AbstractA. Avilés and C. Brech proved an intriguing result about the existence and uniqueness of certain injective Boolean algebras or Banach spaces. Their result refines the standard existence and uniqueness of saturated models. They express a wish to obtain a unified approach in the context of category theory. We provide this in the framework of weak factorisation systems. Our basic tool is the fat small object argument.

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