Corrigendum toF. Point, Existentially closed ordered difference fields and rings

2015 ◽  
Vol 61 (1-2) ◽  
pp. 117-119
Author(s):  
Françoise Point
Keyword(s):  
Existentially Closed ◽  
Difference Fields

A note on existentially closed difference fields with algebraically closed fixed field

2001 ◽  
Vol 66 (2) ◽  
pp. 719-721 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Model Companion ◽  
Fixed Field ◽  
Existentially Closed ◽  
Difference Fields

AbstractWe point out that the theory of difference fields with algebraically closed fixed field has no model companion.

Existentially closed central extensions of locally finite p-groups

1986 ◽  
Vol 100 (2) ◽  
pp. 281-301 ◽  
Author(s):  
Felix Leinen ◽  
Richard E. Phillips
Keyword(s):  
Central Extensions ◽  
Locally Finite ◽  
Existentially Closed ◽  
Image Position

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we letwhere ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

Finitely generic models of TUH, for certain model companionable theories T

1985 ◽  
Vol 50 (3) ◽  
pp. 604-610
Author(s):  
Francoise Point
Keyword(s):  
Discriminator Variety ◽  
Generic Models ◽  
Elementary Class ◽  
Ordered Groups ◽  
Existentially Closed ◽  
Closed Element ◽  
Starting Point ◽  
Joint Embedding ◽  
Joint Embedding Property ◽  
Generic Elements

The starting point of this work was Saracino and Wood's description of the finitely generic abelian ordered groups [S-W].We generalize the result of Saracino and Wood to a class ∑UH of subdirect products of substructures of elements of a class ∑, which has some relationships with the discriminator variety V(∑t) generated by ∑. More precisely, let ∑ be an elementary class of L-algebras with theory T. Burris and Werner have shown that if ∑ has a model companion then the existentially closed models in the discriminator variety V(∑t) form an elementary class which they have axiomatized. In general it is not the case that the existentially closed elements of ∑UH form an elementary class. For instance, take for ∑ the class ∑0 of linearly ordered abelian groups (see [G-P]).We determine the finitely generic elements of ∑UH via the three following conditions on T:(1) There is an open L-formula which says in any element of ∑UH that the complement of equalizers do not overlap.(2) There is an existentially closed element of ∑UH which is an L-reduct of an element of V(∑t) and whose L-extensions respect the relationships between the complements of the equalizers.(3) For any models A, B of T, there exists a model C of TUH such that A and B embed in C.(Condition (3) is weaker then “T has the joint embedding property”. It is satisfied for example if every model of T has a one-element substructure. Condition (3) implies that ∑UH has the joint embedding property and therefore that the class of finitely generic elements of ∑UH is complete.)

scholarly journals Automated Proofs for Some Stirling Number Identities

10.37236/726 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Manuel Kauers ◽  
Carsten Schneider
Keyword(s):  
Stirling Number ◽  
Eulerian Numbers ◽  
Difference Fields

We present computer-generated proofs for some summation identities for ($q$-)Stirling and ($q$-)Eulerian numbers that were obtained by combining a recent summation algorithm for Stirling number identities with a recurrence solver for difference fields.

Existentially closed ${\rm II}_1$ factors

2015 ◽  
pp. 1-24 ◽  
Author(s):  
Ilijas Farah ◽  
Isaac Goldbring ◽  
Bradd Hart ◽  
David Sherman
Keyword(s):  
Existentially Closed
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