A note on CM-triviality and the geometry of forking

2000 ◽  
Vol 65 (1) ◽  
pp. 474-480 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Finite Rank ◽  
Algebraic Closure ◽  
Simple Groups ◽  
General Context ◽  
Stable Theory ◽  
Geometric Properties ◽  
Regular Type ◽  
Superstable Theory ◽  
Minimal Sets ◽  
The Right

CM-triviality of a stable theory is a notion introduced by Hrushovski [1]. The importance of this property is first that it holds of Hrushovski's new non 1-based strongly minimal sets, and second that it is still quite a restrictive property, and forbids the existence of definable fields or simple groups (see [2]). In [5], Frank Wagner posed some questions about CM-triviality, asking in particular whether a structure of finite rank, which is “coordinatized” by CM-trivial types of rank 1, is itself CM-trivial. (Actually Wagner worked in a slightly more general context, adapting the definitions to a certain “local” framework, in which algebraic closure is replaced by P-closure, for P some family of types. We will, however, remain in the standard context, and will just remark here that it is routine to translate our results into Wagner's framework, as well as to generalise to the superstable theory/regular type context.) In any case we answer Wagner's question positively. Also in an attempt to put forward some concrete conjectures about the possible geometries of strongly minimal sets (or stable theories) we tentatively suggest a hierarchy of geometric properties of forking, the first two levels of which correspond to 1-basedness and CM-triviality respectively. We do not know whether this is a strict hierarchy (or even whether these are the “right” notions), but we conjecture that it is, and moreover that a counterexample to Cherlin's conjecture can be found at level three in the hierarchy.

scholarly journals COUNTABLE MODELS OF THE THEORIES OF BALDWIN–SHI HYPERGRAPHS AND THEIR REGULAR TYPES

2019 ◽  
Vol 84 (3) ◽  
pp. 1007-1019
Author(s):  
DANUL K. GUNATILLEKA
Keyword(s):  
Large Body ◽  
Countable Model ◽  
Stable Theory ◽  
Generic Structure ◽  
Elementary Chain ◽  
Regular Type ◽  
Image Position ◽  
Original Class ◽  
Countable Models

AbstractWe continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $ if and only if the dimension of $\mathfrak{M}_\beta $ is at most the dimension of $\mathfrak{M}_\gamma $ and that each countable model is isomorphic to some $\mathfrak{M}_\beta $. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω-stable theory with a nonlocally modular regular type, answering a question of Pillay in [11].

On the biological basis of human laterality: II. The mechanisms of inheritance

1978 ◽  
Vol 1 (2) ◽  
pp. 270-277 ◽  
Author(s):  
Michael J. Morgan ◽  
Michael C. Corballis
Keyword(s):  
Genetic Control ◽  
Left Hemisphere ◽  
Right Hemisphere ◽  
Dominant Role ◽  
Preliminary Evidence ◽  
General Context ◽  
Cerebral Lateralization ◽  
Clear Cut ◽  
Random Fashion ◽  
The Right

AbstractThis paper focuses on the inheritance of human handedness and cerebral lateralization within the more general context of structural biological asymmetries. The morphogenesis of asymmetrical structures, such as the heart in vertebrates, depends upon a complex interaction between information coded in the cytoplasm and in the genes, but the polarity of asymmetry seems to depend on the cytoplasmic rather than the genetic code. Indeed it is extremely difficult to find clear-cut examples in which the direction of an asymmetry is under genetic control. As one possible case, there is some evidence that the direction, clockwise or counterclockwise, of rotation of the abdomen in certain mutant strains of Drosophila is controlled by a particular gene locus, although there appears to be some degree of confusion on this point. By contrast, it is much easier to find examples in which the degree but not the direction of asymmetry is under genetic control. For instance, there is a mutant strain of mice in which half of the animals display situs inversus of the viscera. The proportion has remained at one half despite many years of inbreeding, suggesting that the mutant allele effectively cancels the normal situs and allows the asymmetry to be specified in random fashion.Although this account does not deny that the right hemisphere of humans may be the more specialized for certain functions, it does attribute a leading or dominant role to the left hemisphere (at least in most individuals). We suggest that so-called “right-hemisphere” functions are essentially acquired by default, due to the left hemisphere's prior involvement with speech and skilled motor acts; we note, for instance, that these right-hemisphere functions include rather elementary perceptual processes. But perhaps the more critical prediction from our account is that the phenomenon of equipotentiality should be unidirectional: the right (lagging) hemisphere should be more disposed to take over left-hemisphere functions following early lesions than is the left (leading) hemisphere to take over right-hemisphere functions. We note preliminary evidence that this may be so.

On Some Matrix Theorems of Frobenius and McCoy

1953 ◽  
Vol 5 ◽  
pp. 332-335 ◽  
Author(s):  
J. K. Goldhaber ◽  
G. Whaples
Keyword(s):  
Algebraic Closure ◽  
Arbitrary Field ◽  
Characteristic Values ◽  
Image Position ◽  
The Right

McCoy, following Frobenius, studied a problem which can be described as follows. Let k be an arbitrary field, kc its algebraic closure, and any algebra of n ⨯ n matrices over k which contains the identity I . Define a canonical ordering to be a set of n mappings λ of , or of a subset of , into kc such that the sequence λ1(A),λ2(A), …, λn(A), for each A ∈ , consists of the characteristic values (roots of det(A — xI) = 0) of A, each with the right multiplicity. Define a canonical ordering to be a Frobenius ordering if, for all non-commutative polynomials f(x1, x2, … , xm) and all finite subsets A1, A2, …, Am of ,

On the number of nonisomorphic models of size |T|

1994 ◽  
Vol 59 (1) ◽  
pp. 41-59 ◽  
Author(s):  
Ambar Chowdhury
Keyword(s):  
Finite Rank ◽  
Superstable Theory

AbstractLet T be an uncountable, superstable theory. In this paper we proveTheorem A. If T has finite rank, then I(|T|, T) ≥ ℵ0.Theorem B. If T is trivial, then I(|T|, T) ≥ ℵ0.

The classification of small weakly minimal sets. III: Modules

1988 ◽  
Vol 53 (3) ◽  
pp. 975-979 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Finite Field ◽  
Algebraic Closure ◽  
Morley Rank ◽  
Minimal Sets

AbstractTheorem A. Let M be a left R-module such that Th(M) is small and weakly minimal, but does not have Morley rank 1. Let A = acl(∅) ⋂ M and I = {r ∈ R: rM ⊂ A}. Notice that I is an ideal.(i) F = R/Iis a finite field.(ii) Suppose that a, b0,…,bn, ∈ M and . Then there are s, ri ∈ R, i ≤ n, such that sa + Σi≤nribi ∈ A and s ∉ I.It follows from Theorem A that algebraic closure in M is modular. Using this and results in [B1] and [B2], we obtainTheorem B. Let M be as in Theorem A. Then Vaught's conjecture holds for Th(M).

scholarly journals Totally disconnected locally compact groups locally of finite rank

2015 ◽  
Vol 158 (3) ◽  
pp. 505-530 ◽  
Author(s):  
PHILLIP WESOLEK
Keyword(s):  
Lie Groups ◽  
Finite Rank ◽  
Open Subgroup ◽  
Simple Groups ◽  
Locally Compact ◽  
Compact Open Subgroup ◽  
Elementary Group ◽  
Finite Set ◽  
Finite Direct Product ◽  
Totally Disconnected

AbstractWe study totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contain a compact open subgroup with finite rank. We show such groups that additionally admit a pro-π compact open subgroup for some finite set of primes π are virtually an extension of a finite direct product of topologically simple groups by an elementary group. This result, in particular, applies to l.c.s.c. p-adic Lie groups. We go on to obtain a decomposition result for all t.d.l.c.s.c. groups containing a compact open subgroup with finite rank. In the course of proving these theorems, we demonstrate independently interesting structure results for t.d.l.c.s.c. groups with a compact open pro-nilpotent subgroup and for topologically simple l.c.s.c. p-adic Lie groups.

L-fuzzy cosets in universal algebras

2021 ◽  
Vol 71 (3) ◽  
pp. 573-594
Author(s):  
Gezahagne Mulat Addis
Keyword(s):  
General Theory ◽  
Universal Algebra ◽  
Fuzzy Set ◽  
Fuzzy Systems ◽  
Algebraic Closure ◽  
Sufficient Conditions ◽  
General Context ◽  
Variety Of Algebras ◽  
Universal Algebras ◽  
Necessary And Sufficient

Abstract In this paper, we introduce the notion of fuzzy costs in a more general context, in universal algebra by the use of coset terms. We study the structure of fuzzy cosets by applying the general theory of algebraic fuzzy systems. Fuzzy cosets generated by a fuzzy set are characterized in different ways. It is also proved that the class of fuzzy cosets determined by an element forms an algebraic closure fuzzy set system. Finally, we give a set of necessary and sufficient conditions for a given variety of algebras to be congruence permutable by applying the theory of fuzzy cosets.

On one-based theories

1994 ◽  
Vol 59 (2) ◽  
pp. 579-595 ◽  
Author(s):  
E. Bouscaren ◽  
E. Hrushovski
Keyword(s):  
Similar Type ◽  
Stable Theory ◽  
Dependence Relation ◽  
Regular Type

We know from [H1], [H2] that in a stable theory, given a nontrivial locally modular regular type q, one can define a group with generic domination equivalent to q, and that the dependence relation on q can be analyzed in terms of this group. In a stable one-based theory, every regular type is locally modular; hence, this result holds for every nontrivial regular type. We show here that, in fact, in a stable one-based theory, a similar type of construction can be done without the assumption of regularity. More precisely, we show that for any type q, the nontrivial part of q can be analyzed by generics of groups and that any nontrivial relation can be described by affine relations (Theorem A).This construction is then used to answer a question about homogeneity in pairs of models which is still open in the case of arbitrary stable theories (Theorem C).

scholarly journals On the lattice of right ideals of the endomorphism ring of an abelian group

1988 ◽  
Vol 38 (2) ◽  
pp. 273-291 ◽  
Author(s):  
Theodore G. Faticoni
Keyword(s):  
Abelian Group ◽  
Finite Index ◽  
Endomorphism Ring ◽  
Finite Rank ◽  
Ideal Structure ◽  
Left Module ◽  
Linear Topology ◽  
Closed Submodules ◽  
The Right ◽  
Torsion Free

Let A be an abelian group, let ∧ = End (A), and assume that A is a flat left ∧-module. Then σ = { right ideals I ⊂ ∧ | IA = A} generates a linear topology oil ∧. We prove that Hom(A,·) is an equivalence from the category of those groups B ⊂ An satisfying B = Hom(A, B)A, onto the category of σ-closed submodules of finitely generated free right ∧-modules. Applications classify the right ideal structure of A, and classify torsion-free groups A of finite rank which are (nearly) isomorphic to each A-generated subgroup of finite index in A.

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