Model completeness of the new strongly minimal sets

1999 ◽  
Vol 64 (3) ◽  
pp. 946-962 ◽  
Author(s):  
Kitty L. Holland
Keyword(s):  
Algebraic Closure ◽  
Common Elements ◽  
Model Completeness ◽  
Elementary Class ◽  
Closed Fields ◽  
Completeness Result ◽  
Closure Relations ◽  
Algebraically Closed Fields ◽  
Special Case ◽  
Selection Of

Boris Zil'ber conjectured that all strongly minimal theories are bi-interpretable with one of the “classical” sorts: theories of algebraically closed fields, theories of infinite vector spaces over division rings and theories with trivial algebraic closure relations. Hrushovski produced the first two classes of counterexample to this conjecture in [10] and [9]. Subsequently, in [8], the author gave an explicit axiomatization of a special case of [9] from which model completeness could quickly be deduced. It was unclear at that writing whether the model completeness result was true in the general case or was due to peculiarities of the case under consideration. The main new result of this paper is model completeness, not only of the general case in [9], but also of the theories described in [10]. Specifically, we present a general framework in which producing a strongly minimal theory is reduced to finding an elementary class of theories satisfying certain requirements (see below). We present the theories of [10] and [9] as special instances of such theories, giving an explicit axiomatization from which model completeness immediately follows in each case.We hope by presenting these constructions in parallel, using common language and extracting common elements, to make easier both the exploitation of the ideas involved in their making and their comparison with other recent constructions of a similar flavor. For a selection of such constructions, see [6], [1], [2] and [3]. For more general background, see [2], [4] and [11].

ON THE CLASSIFICATION OF CENTRALLY FINITE ALTERNATIVE DIVISION RINGS SATISFYING ALGEBRAIC CLOSURE CONDITIONS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250088
Author(s):  
RICCARDO GHILONI
Keyword(s):  
Algebraic Closure ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Division Rings ◽  
Closed Fields ◽  
Closure Conditions ◽  
The One ◽  
Algebraically Closed Fields

In this paper, we prove that the rings of quaternions and of octonions over an arbitrary real closed field are algebraically closed in the sense of Eilenberg and Niven. As a consequence, we infer that some reasonable algebraic closure conditions, including the one of Eilenberg and Niven, are equivalent on the class of centrally finite alternative division rings. Furthermore, we classify centrally finite alternative division rings satisfying such equivalent algebraic closure conditions: up to isomorphism, they are either the algebraically closed fields or the rings of quaternions over real closed fields or the rings of octonions over real closed fields.

Recursion theory on orderings. I. A model theoretic setting

1979 ◽  
Vol 44 (3) ◽  
pp. 383-402 ◽  
Author(s):  
G. Metakides ◽  
J.B. Remmel
Keyword(s):  
Algebraic Closure ◽  
Recursion Theory ◽  
Boolean Algebras ◽  
Vector Spaces ◽  
Formal Definition ◽  
Recursively Enumerable ◽  
Closed Fields ◽  
Partial Orderings ◽  
Definition Of ◽  
Algebraically Closed Fields

In [6], Metakides and Nerode introduced the study of the lattice of recursively enumerable substructures of a recursively presented model as a means to understand the recursive content of certain algebraic constructions. For example, the lattice of recursively enumerable subspaces,, of a recursively presented vector spaceV∞has been studied by Kalantari, Metakides and Nerode, Retzlaff, Remmel and Shore. Similar studies have been done by Remmel [12], [13] for Boolean algebras and by Metakides and Nerode [9] for algebraically closed fields. In all of these models, the algebraic closure of a set is nontrivial. (The formal definition of the algebraic closure of a setS, denoted cl(S), is given in §1, however in vector spaces, cl(S) is just the subspace generated byS, in Boolean algebras, cl(S) is just the subalgebra generated byS, and in algebraically closed fields, cl(S) is just the algebraically closed subfield generated byS.)In this paper, we give a general model theoretic setting (whose precise definition will be given in §1) in which we are able to give constructions which generalize many of the constructions of classical recursion theory. One of the main features of the modelswhich we study is that the algebraic closure of setis just itself, i.e., cl(S) = S. Examples of such models include the natural numbers under equality 〈N, = 〉, the rational numbers under the usual ordering 〈Q, ≤〉, and a large class ofn-dimensional partial orderings.

Definable equivalence relations on algebraically closed fields

1989 ◽  
Vol 54 (3) ◽  
pp. 928-935 ◽  
Author(s):  
Lou van den Dries ◽  
David Marker ◽  
Gary Martin
Keyword(s):  
Equivalence Relation ◽  
Analogous Result ◽  
Equivalence Relations ◽  
Complex Numbers ◽  
Closed Field ◽  
Closed Fields ◽  
Finite Set ◽  
The Right ◽  
Algebraically Closed Fields ◽  
Special Case

This article was inspired by the question: is there a definable equivalence relation on the field of complex numbers, each of whose equivalence classes has exactly two elements? The answer turned out to be no, as we now explain in greater detail.Let Κ be an algebraically closed field and let E be a definable equivalence relation on Κ. [Note: By “definable” we will always mean “definable with parameters”.] Either E has one cofinite class, or all classes are finite and there is a number d such that all but a finite set of classes have cardinality d. In the latter case let B be the finite set of elements of Κ which are not in a class of size d. We prove the following result.Theorem 1. a) If char(Κ) = 0 or char(Κ) = p > d, then ∣B∣ ≡ 1 (mod d).b) If char(Κ) = 2 and d = 2, then ∣B∣ ≡ 0 (mod 2).c) If char(Κ) = p > 2 and d = p + s, where 1 ≤ s ≤ p/2, then ∣B∣ ≡ p + 1 (mod d).Furthermore, a)−c) are the only restrictions on ≡B≡.If one is in the right mood, one can view this theorem as saying that the “algebraic cardinality” of the complex numbers is congruent to 1 (mod n) for every n.§1 contains a reduction of the problem to the special case where E is induced by a rational function in one variable. §2 contains the main calculations and the proofs of a)−c). §3 contains eight families of examples showing that all else is possible. In §4 we prove an analogous result for real closed fields.

Recursion theory in a lower semilattice

1992 ◽  
Vol 57 (3) ◽  
pp. 892-911 ◽  
Author(s):  
Alex Feldman
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Algebraic Structure ◽  
Algebraic Closure ◽  
Recursion Theory ◽  
Vector Spaces ◽  
Algebraic Structures ◽  
Recursively Enumerable ◽  
Closed Fields ◽  
Algebraically Closed Fields

In §3 we construct a universal, ℵ0-categorical recursively presented partial order with greatest lower bound operator. This gives us the unique structure which embeds every countable lower semilattice. In §§5 and 6 we investigate the recursive and recursively enumerable substructures of this structure, in particular finding a suitable definition for the simple-maximal hierarchy and giving an example of an infinite recursively enumerable substructure which does not contain any infinite recursive substructure.The idea of looking at the lattice of recursively enumerable substructures of some recursive algebraic structure was introduced by Metakides and Nerode in [5], and since then many different kinds of algebraic structures have been studied in this way, including vector spaces, Boolean algebras, groups, algebraically closed fields, and equivalence relations. Since different algebraic structures have different recursion theoretic properties, one natural question is whether an algebraic structure with relatively little structure (such as a partial order or an equivalence relation) exhibits behavior more like classical recursion theory than one with more structure (such as vector spaces or algebraically closed fields).In [6] and [7], Metakides and Remmel studied recursion theory on orderings, and, as they point out in [6], orderings differ from most other algebraic structures in that the algebraic closure operation on orderings is trivial; but this does not present a problem for them, given the questions they explore. Moreover, they take an approach of proving general theorems which can then be applied to specific orderings. Our tack is different, although also well-established (see, for example, [3]), in which a “largest” structure is defined (in §3) which corresponds to the natural numbers in classical recursion theory. In order to distinguish substructures from subsets, a function symbol is added, namely greatest lower bound. The greatest lower bound function is fundamental to the study of orderings and occurs naturally in many of them, and thus is an appropriate addition to the theory of orderings. In §4 we redefine the concepts of simple and maximal in a manner appropriate to this structure, and prove several existence theorems.

RATIONAL JORDAN DECOMPOSITION OF BILINEAR FORMS

2005 ◽  
Vol 07 (06) ◽  
pp. 769-786 ◽  
Author(s):  
DRAGOMIR Ž. ĐOKOVIĆ ◽  
KAIMING ZHAO
Keyword(s):  
Vector Space ◽  
Algebraic Closure ◽  
Real Field ◽  
Closed Field ◽  
Bilinear Forms ◽  
Dimensional Vector ◽  
Jordan Decomposition ◽  
Dimensional Vector Space ◽  
Closed Fields ◽  
Algebraically Closed Fields

This is a continuation of our previous work on Jordan decomposition of bilinear forms over algebraically closed fields of characteristic 0. In this note, we study Jordan decomposition of bilinear forms over any field K0 of characteristic 0. Let V0 be an n-dimensional vector space over K0. Denote by [Formula: see text] the space of bilinear forms f : V0 × V0 → K0. We say that f = g + h, where f, g, [Formula: see text], is a rational Jordan decomposition of f if, after extending the field K0 to an algebraic closure K, we obtain a Jordan decomposition over K. By using the Galois group of K/K0, we prove the existence of rational Jordan decompositions and describe a method for constructing all such decompositions. Several illustrative examples of rational Jordan decompositions of bilinear forms are included. We also show how to classify the unimodular congruence classes of bilinear forms over an algebraically closed field of characteristic different from 2 and over the real field.

Functorial properties of algebraic closure and Skolemization

1981 ◽  
Vol 31 (2) ◽  
pp. 136-141 ◽  
Author(s):  
C. J. Ash ◽  
A. Nerode
Keyword(s):  
Model Theory ◽  
Algebraic Closure ◽  
Closed Field ◽  
Ordered Sets ◽  
Algebraically Closed Field ◽  
Closed Fields ◽  
Algebraically Closed Fields

AbstractIt is shown that no functor F exists from the category of sets with injections, to the category of algebraically closed fields of given characteristic, with monomorphisms, having the properties that for all sets A. F(A) is an algebraically closed field having transcendence base A and for all injections f. F(f) extends f. There does exist such a functor from the category of linearly-ordered sets with order monomorphisms.An application to model-theory using the same methods is given showing that while the theory of algebraically closed fields is ω-stable, its Skolemization is not stable in any power.

On the representation of Herbrand functions in algebraically closed fields

1957 ◽  
Vol 22 (2) ◽  
pp. 187-204 ◽  
Author(s):  
A. H. Lightstone ◽  
A. Robinson
Keyword(s):  
Algebraic Closure ◽  
A Priori ◽  
Choice Functions ◽  
Commutative Field ◽  
Algebraic Functions ◽  
Closed Fields ◽  
Definition Of ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Algebraically Closed Fields

1. Let X be a statement which is formulated in the lower predicate calculus in terms of the relations of addition, multiplication, and equality, and — possibly — of the elements of a given commutative field M. Suppose moreover that X is in prenex normal form, e.g.where Z does not include any further quantifiers. In order that X be satisfied by the algebraic closure M* of M, it is necessary and sufficient that there exist Herbrand functions (or choice functions) φ1(x1, x2), φ2(x1, x2), φ3(x1, x2, x3, x4) with arguments ranging over M* and taking values in M* such thatholds for all x1, x2, x3, x4 in M*. In general the definition of these functions is far from being unique, and a priori they bear no relation to the functions which are defined ‘naturally’ in M* i.e. the rational, and more generally the algebraic, functions with coefficients in M or M*. However, we shall show in the present paper that the entire domain of variation of the arguments x1, x2, x3, x4 — regarded as the affine space S4 over M* — can be divided up into a finite number of regions Di such that in each Di the functions ϕk can be chosen as algebraic functions of their arguments. Indeed, our complete result (§ 3) proves rather more than that. In particular, it turns out that the regions Di may be taken as differences of algebraic varieties in M*.

scholarly journals Joint universal modular plasmids (JUMP): a flexible vector platform for synthetic biology

2021 ◽  
Vol 6 (1) ◽  
Author(s):  
Marcos Valenzuela-Ortega ◽  
Christopher French
Keyword(s):  
Copy Number ◽  
Life Science ◽  
Insertion Site ◽  
Golden Gate ◽  
Common Elements ◽  
Modern Life ◽  
Specific Host ◽  
Modular Cloning ◽  
Golden Gate Cloning ◽  
Selection Of

Abstract Generation of new DNA constructs is an essential process in modern life science and biotechnology. Modular cloning systems based on Golden Gate cloning, using Type IIS restriction endonucleases, allow assembly of complex multipart constructs from reusable basic DNA parts in a rapid, reliable and automation-friendly way. Many such toolkits are available, with varying degrees of compatibility, most of which are aimed at specific host organisms. Here, we present a vector design which allows simple vector modification by using modular cloning to assemble and add new functions in secondary sites flanking the main insertion site (used for conventional modular cloning). Assembly in all sites is compatible with the PhytoBricks standard, and vectors are compatible with the Standard European Vector Architecture (SEVA) as well as BioBricks. We demonstrate that this facilitates the construction of vectors with tailored functions and simplifies the workflow for generating libraries of constructs with common elements. We have made available a collection of vectors with 10 different microbial replication origins, varying in copy number and host range, and allowing chromosomal integration, as well as a selection of commonly used basic parts. This design expands the range of hosts which can be easily modified by modular cloning and acts as a toolkit which can be used to facilitate the generation of new toolkits with specific functions required for targeting further hosts.

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