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Prime types and geometric completeness [article]

Jean Berthet
2012 arXiv   pre-print
This framework contains a logical equivalent of the algebraic theory of prime and radical ideals, as well as the basics of an "affine algebraic geometry" in quasivarieties.  ...  The geometric form of Hilbert's Nullstellensatz may be understood as a property of "geometric saturation" in algebraically closed fields.  ...  algebras, and the "ideals" of an algebra A are its differential ideals.  ... 
arXiv:1210.0679v1 fatcat:aktt5mzixvadzdi7gdvm6yxe5u

Page 1761 of Mathematical Reviews Vol. 51, Issue 6 [page]

1976 Mathematical Reviews  
Simmons, H. 12518 Existentially closed structures. J. Symbolic Logic 37 (1972), 293-310. : This paper is devoted to a study of existentially closed structures.  ...  We say that % is [uniformly] existentially closed if it is [uniformly] existentially closed over Th(@). A structure & is said to be symmetric (a notion due to P.  ... 

Nullstellensatz for relative existentially closed groups [article]

Mohammad Shahryari
2021 arXiv   pre-print
As a result we see that every pair of G-existentially closed elements in an arbitrary variety of G-groups generate the same quasi-variety and if both of them are q_ω-compact, they are geometrically equivalent  ...  We prove that in every variety of G-groups, every G-existentially closed element satisfies nullstellensatz for finite consistent systems of equations. This will generalize Theorem G of .  ...  paly the role of existentially closed structures.  ... 
arXiv:2105.09520v1 fatcat:wdlt46b2fnablhkmmefq4trlve

Page 35 of Mathematical Reviews Vol. 58, Issue 1 [page]

1979 Mathematical Reviews  
The author proves that every algebraically closed model is existentially complete and thereby obtains from the Baumslag-Levin construction of the algebraically closed count- able models the existence of  ...  rings R; whose unique maximal ideal M, is nilpotent, and R,/M, is an algebraically closed field.  ... 

Existentially closed Leibniz algebras and an embedding theorem [article]

Chia Zargeh
2021 arXiv   pre-print
In this paper we introduce the notion of existentially closed Leibniz algebras. Then we use HNN-extensions of Leibniz algebras in order to prove an embedding theorem.  ...  The following description of existentially closed Leibniz algebras is provided based on the common definition of existentially closed algebras (see, e.g., [5]).  ...  Shahryari, Existentially closed structures and some embedding theorems, Mathematical Notes 101 6, (2017), 1023–1032. 12. S. Silvestrov, C.  ... 
arXiv:2003.03509v3 fatcat:iemqivtggffynhnvhvls226e3a

Page 1800 of Mathematical Reviews Vol. 50, Issue 6 [page]

1975 Mathematical Reviews  
is again existentially closed. {In the definition of “local homomorphism” of fields on p. 9, f should be required to annihilate the maximal ideal of Ko.  ...  The author shows here that these concepts lead to noncommuta- tive versions of the Nullstellensatz, with existentially closed fields taking the place of the algebraically closed fields of the commuta-  ... 

HNN-extensions of algebras and applications

Hans-Christian Mez
1984 Bulletin of the Australian Mathematical Society  
It is shown that the corresponding is false for existentially closed associative A-algebras but true for existentially universal nonassociative ft-algebras.  ...  The main part of the paper deals with applications of these results. For example, i t is known that every existentially closed group is w-homogeneous.  ...  Let us discuss the corresponding for algebras. Algebraically closed, existentially closed and existentially universal structures are currently investigated for many classes.  ... 
doi:10.1017/s0004972700021456 fatcat:jh2dcqzklbbwxlt27szqusyipi

Page 2851 of Mathematical Reviews Vol. , Issue 87f [page]

1987 Mathematical Reviews  
This paper gives numerous results on the ideal structure of existentially closed and algebraically closed algebras.  ...  2851 Eklof, Paul C. (1-CA3); Mez, Hans-Christian (1-CA3) The ideal structure of existentially closed algebras. J. Symbolic Logic 50 (1985), no. 4, 1025-1043 (1986).  ... 

Model-completions and modules

Paul Eklof, Gabriel Sabbagh
1971 Annals of Mathematical Logic  
Let K be a theory (i.e. a deductively-closed consistent set of sentences) in a f'trst-order language L.  ...  The model-co]npanion of K, if it exists, is unique ([4] Theorem 5.3); the model-completion of K, if it exists, is the model-companion.  ...  Any elementary substructure cj an algebraically (resp. existentially) closed structure is an algebraically (resp. existentially) closed structure. Corollary 7.9.  ... 
doi:10.1016/0003-4843(71)90016-7 fatcat:3zkha4h2ojarhe6tapmgfofrti

Page 3324 of Mathematical Reviews Vol. , Issue 82h [page]

1982 Mathematical Reviews  
In this paper the author describes various properties of algebraically or existentially closed commutative Archimedean semigroups with idempotents.  ...  A semigroup S is said to be algebraically closed [existentially closed] if every finite system of equations [and negations of equations] which has a solution in a semigroup extension of S has a solution  ... 

Page 3859 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews  
The theory ACFA (algebraically closed fields K with an automorphism a) of these universal domains is axiomatized (the model (K,c) is algebraically closed and every prime c-ideal has a K-rational point)  ...  Moreover, if L consists of only binary symbols then a countable %-structure A is existentially closed in H# if and only if it is 1l-existentially closed in % and has the pigeonhole property.  ... 

The uniform companion for large differential fields of characteristic 0

Marcus Tressl
2005 Transactions of the American Mathematical Society  
As an application, we introduce differentially closed ordered fields, differentially closed p-adic fields and differentially closed pseudofinite fields.  ...  a model complete theory of pure fields.  ...  (iv) For every n ∈ N and every prime ideal p ⊆ F [X], X = (X 1 , ..., X n ), if V (p) :=(the zeroes of p in the algebraic closure of F ) has a regular, Frational point, then F is existentially closed in  ... 
doi:10.1090/s0002-9947-05-03981-4 fatcat:lykrzyswhbbc3kwj6hgqygnmcm

Existentially closed models of the theory of artinian local rings

Hans Schoutens
1999 Journal of Symbolic Logic (JSL)  
We show that its existentially closed models are Gorenstein. of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form.  ...  The theory oτ l of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory τt l is companionable, with model-companion oτ l .  ...  For example, the existentially closed models of the theory of fields are precisely the algebraically closed fields.  ... 
doi:10.2307/2586504 fatcat:bxvjuqrxcnhddoynh5jvtsgzfe

Geometrically closed rings [article]

Jean Berthet
2013 arXiv   pre-print
We develop the basic theory of geometrically closed rings as a generalisation of algebraically closed fields, on the grounds of notions coming from positive model theory and affine algebraic geometry.  ...  algebraic version of coherent logic for rings.  ...  An affine algebraic set in the affine space A X is the set Z (I) of zeros of an ideal of A[X] and its coordinate algebra is the ring A[X]/I (Z (I)) understood with its A-algebra structure and embedded  ... 
arXiv:1309.5572v1 fatcat:jjpvqt2mrvbrvprc7rate7f2me

Existentially closed fields with finite group actions [article]

Daniel Max Hoffmann, Piotr Kowalski
2017 arXiv   pre-print
We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group.  ...  Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a (finite) group scheme action.  ...  In Section 3, we prove algebraic properties of existentially closed G-transformal fields and characterize them as G-closed perfect pseudo-algebraically closed fields.  ... 
arXiv:1604.03581v2 fatcat:ju6ikzayejfo7pves5jsooaiea
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