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Prime types and geometric completeness
[article]

2012
*
arXiv
*
pre-print

This framework contains a logical equivalent

arXiv:1210.0679v1
fatcat:aktt5mzixvadzdi7gdvm6yxe5u
*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*. ...##
###
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]

2021
*
arXiv
*
pre-print

As a result we see that every pair

arXiv:2105.09520v1
fatcat:wdlt46b2fnablhkmmefq4trlve
*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*. ...##
###
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. ...

##
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Existentially closed Leibniz algebras and an embedding theorem
[article]

2021
*
arXiv
*
pre-print

In this paper we introduce

arXiv:2003.03509v3
fatcat:iemqivtggffynhnvhvls226e3a
*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. ...##
###
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

1984
*
Bulletin of the Australian Mathematical Society
*

It is shown that

doi:10.1017/s0004972700021456
fatcat:jh2dcqzklbbwxlt27szqusyipi
*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. ...##
###
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

1971
*
Annals of Mathematical Logic
*

Let K be a theory (i.e. a deductively-

doi:10.1016/0003-4843(71)90016-7
fatcat:3zkha4h2ojarhe6tapmgfofrti
*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. ...##
###
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. ...

##
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The uniform companion for large differential fields of characteristic 0

2005
*
Transactions of the American Mathematical Society
*

As an application, we introduce differentially

doi:10.1090/s0002-9947-05-03981-4
fatcat:lykrzyswhbbc3kwj6hgqygnmcm
*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 ...##
###
Existentially closed models of the theory of artinian local rings

1999
*
Journal of Symbolic Logic (JSL)
*

We show that its

doi:10.2307/2586504
fatcat:bxvjuqrxcnhddoynh5jvtsgzfe
*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. ...##
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Geometrically closed rings
[article]

2013
*
arXiv
*
pre-print

We develop

arXiv:1309.5572v1
fatcat:jjpvqt2mrvbrvprc7rate7f2me
*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 ...##
###
Existentially closed fields with finite group actions
[article]

2017
*
arXiv
*
pre-print

We study

arXiv:1604.03581v2
fatcat:ju6ikzayejfo7pves5jsooaiea
*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. ...
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