Hyperimaginaries and automorphism groups

2001 ◽  
Vol 66 (1) ◽  
pp. 127-143 ◽  
Author(s):  
D. Lascar ◽  
A. Pillay
Keyword(s):  
Equivalence Relation ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Infinite Length ◽  
Abstract Group ◽  
Simple Theories ◽  
Bounded Number ◽  
Compact Topology ◽  
The One ◽  
Number Of Classes

A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [1], mainly with reference to simple theories. It was pointed out there how hyperimaginaries still remain in a sense within the domain of first order logic. In this paper we are concerned with several issues: on the one hand, various levels of complexity of hyperimaginaries, and when hyperimaginaries can be reduced to simpler hyperimaginaries. On the other hand the issue of what information about hyperimaginaries in a saturated structure M can be obtained from the abstract group Aut(M).In Section 2 we show that if T is simple and canonical bases of Lascar strong types exist in Meq then hyperimaginaries can be eliminated in favour of sequences of ordinary imaginaries. In Section 3, given a type-definable equivalence relation with a bounded number of classes, we show how the quotient space can be equipped with a certain compact topology. In Section 4 we study a certain group introduced in [5], which we call the Galois group of T, develop a Galois theory and make the connection with the ideas in Section 3. We also give some applications, making use of the structure of compact groups. One of these applications states roughly that bounded hyperimaginaries can be eliminated in favour of sequences of finitary hyperimaginaries. In Sections 3 and 4 there is some overlap with parts of Hrushovski's paper [2].

scholarly journals Parametric Matroid of Rough Set

2015 ◽  
Vol 23 (06) ◽  
pp. 893-908 ◽  
Author(s):  
Yanfang Liu ◽  
Hong Zhao ◽  
William Zhu
Keyword(s):  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Independent Set ◽  
Approximation Operator ◽  
Approximation Number ◽  
Lower Approximation ◽  
The One

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a generalization of linear algebra and graph theory. Recently, a matroidal structure of rough sets is established and applied to the problem of attribute reduction which is an important application of rough set theory. In this paper, we propose a new matroidal structure of rough sets and call it a parametric matroid. On the one hand, for an equivalence relation on a universe, a parametric set family, with any subset of the universe as its parameter, is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore a matroid is generated, and we call it a parametric matroid of the rough set. Through the lower approximation operator, three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, partition-circuit matroids are well studied through the lower approximation number, and then we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.

scholarly journals Quotienting the delay monad by weak bisimilarity

2017 ◽  
Vol 29 (1) ◽  
pp. 67-92 ◽  
Author(s):  
JAMES CHAPMAN ◽  
TARMO UUSTALU ◽  
NICCOLÒ VELTRI
Keyword(s):  
Computer Science ◽  
Equivalence Relation ◽  
Type Theory ◽  
Partial Functions ◽  
Homotopy Type Theory ◽  
Inductive Type ◽  
Alternative Approach ◽  
The One ◽  
Axiom Of Countable Choice

The delay datatype was introduced by Capretta (Logical Methods in Computer Science, 1(2), article 1, 2005) as a means to deal with partial functions (as in computability theory) in Martin-Löf type theory. The delay datatype is a monad. It is often desirable to consider two delayed computations equal, if they terminate with equal values, whenever one of them terminates. The equivalence relation underlying this identification is called weak bisimilarity. In type theory, one commonly replaces quotients with setoids. In this approach, the delay datatype quotiented by weak bisimilarity is still a monad–a constructive alternative to the maybe monad. In this paper, we consider the alternative approach of Hofmann (Extensional Constructs in Intensional Type Theory, Springer, London, 1997) of extending type theory with inductive-like quotient types. In this setting, it is difficult to define the intended monad multiplication for the quotiented datatype. We give a solution where we postulate some principles, crucially proposition extensionality and the (semi-classical) axiom of countable choice. With the aid of these principles, we also prove that the quotiented delay datatype delivers free ω-complete pointed partial orders (ωcppos).Altenkirch et al. (Lecture Notes in Computer Science, vol. 10203, Springer, Heidelberg, 534–549, 2017) demonstrated that, in homotopy type theory, a certain higher inductive–inductive type is the free ωcppo on a type X essentially by definition; this allowed them to obtain a monad of free ωcppos without recourse to a choice principle. We notice that, by a similar construction, a simpler ordinary higher inductive type gives the free countably complete join semilattice on the unit type 1. This type suffices for constructing a monad, which is isomorphic to the one of Altenkirch et al. We have fully formalized our results in the Agda dependently typed programming language.

The finite inner automorphism groups of division rings

1995 ◽  
Vol 118 (2) ◽  
pp. 207-213 ◽  
Author(s):  
M. Shirvani
Keyword(s):  
Finite Groups ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Commutative Field ◽  
Division Rings ◽  
Inner Automorphism ◽  
Projective Representations ◽  
Inner Automorphisms ◽  
A Finite Group

Let G be a finite group of automorphisms of an associative ring R. Then the inner automorphisms (x↦ u−1xu = xu, for some unit u of R) contained in G form a normal subgroup G0 of G. In general, the Galois theory associated with the outer automorphism group G/G0 is quit well behaved (e.g. [7], 2·3–2·7, 2·10), while little group-theoretic restriction on the structure of G/G0 may be expected (even when R is a commutative field). The structure of the inner automorphism groups G0 does not seem to have received much attention so far. Here we classify the finite groups of inner automorphisms of division rings, i.e. the finite subgroups of PGL (1, D), where D is a division ring. Such groups also arise in the study of finite collineation groups of projective spaces (via the fundamental theorem of projective geometry, cf. [1], 2·26), and provide examples of finite groups having faithful irreducible projective representations over fields.

scholarly journals On the number of ergodic measures for minimal shifts with eventually constant complexity growth

2016 ◽  
Vol 37 (7) ◽  
pp. 2099-2130
Author(s):  
MICHAEL DAMRON ◽  
JON FICKENSCHER
Keyword(s):  
Symbolic Dynamics ◽  
Explicit Description ◽  
Probability Measures ◽  
Bounded Number ◽  
Ergodic Measures ◽  
The One ◽  
Constant Growth ◽  
Block Growth ◽  
Linear Block

In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and Kra. In this paper, we show that under the stronger assumption of eventually constant growth, an improved bound exists. To this end, we introduce special Rauzy graphs. Variants of the well-known Rauzy graphs from symbolic dynamics, these graphs provide an explicit description of how a Rauzy graph for words of length $n$ relates to the one for words of length $n+1$ for each $n=1,2,3,\ldots \,$.

scholarly journals On automorphism groups of low complexity subshifts

2015 ◽  
Vol 36 (1) ◽  
pp. 64-95 ◽  
Author(s):  
SEBASTIÁN DONOSO ◽  
FABIEN DURAND ◽  
ALEJANDRO MAASS ◽  
SAMUEL PETITE
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Polynomial Complexity ◽  
Automorphism Groups ◽  
Low Complexity ◽  
Topological Dynamics ◽  
Main Technique ◽  
Shift Map ◽  
The One ◽  
A Finite Group

In this article, we study the automorphism group$\text{Aut}(X,{\it\sigma})$of subshifts$(X,{\it\sigma})$of low word complexity. In particular, we prove that$\text{Aut}(X,{\it\sigma})$is virtually$\mathbb{Z}$for aperiodic minimal subshifts and certain transitive subshifts with non-superlinear complexity. More precisely, the quotient of this group relative to the one generated by the shift map is a finite group. In addition, we show that any finite group can be obtained in this way. The class considered includes minimal subshifts induced by substitutions, linearly recurrent subshifts and even some subshifts which simultaneously exhibit non-superlinear and superpolynomial complexity along different subsequences. The main technique in this article relies on the study of classical relations among points used in topological dynamics, in particular, asymptotic pairs. Various examples that illustrate the technique developed in this article are provided. In particular, we prove that the group of automorphisms of a$d$-step nilsystem is nilpotent of order$d$and from there we produce minimal subshifts of arbitrarily large polynomial complexity whose automorphism groups are also virtually$\mathbb{Z}$.

scholarly journals Birational automorphism groups and differential equations

1990 ◽  
Vol 119 ◽  
pp. 1-80 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Critical Point ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Rigorous Proof ◽  
Differential Galois Theory ◽  
General Attitude ◽  
Transcendental Functions ◽  
Birational Automorphism

Painlevé studied the differential equations y″ = R(y′ y, x) without moving critical point, where R is a rational function of y′ y, x. Most of them are integrated by the so far known functions. There are 6 equations called Painlevé’s equations which seem to be irreducible or seem to define new transcendental functions. The simplest one among them is y″ = 6y2 + x. Painlevé declared on Comptes Rendus in 1902-03 that y″ = 6y2 + x is irreducible. It seems that R. Liouville pointed out an error in his argument. In fact there are discussions on this subject between Painlevé and Liouville on Comptes Rendus in 1902-03. In 1915 J. Drach published a new proof of the irreducibility of the differential equation y″ = 6y2 + x. The both proofs depend on the differential Galois theory developed by Drach. But the differential Galois theory of Drach contains errors and gaps and it is not easy to understand their proofs. One of our contemporaries writes in his book: the differential equation y″ = 6y2 + x seems to be irreducible dans un sens que on ne peut pas songer à préciser. This opinion illustrates well the general attitude of the nowadays mathematicians toward the irreducibility of the differential equation y″ = 6y2 + x. Therefore the irreducibility of the differential equation y″ = 6y2 + x remains to be proved. We consider that to give a rigorous proof of the irreducibility of the differential equation y″ = 6y2 + x is one of the most important problem in the theory of differential equations.

The Academic Vocation

1992 ◽  
Author(s):  
Mark R. Schwehn
Keyword(s):  
Research University ◽  
The United States ◽  
Faculty Members ◽  
University Education ◽  
Knowledge And Skills ◽  
College And University ◽  
Teaching History ◽  
The One ◽  
The University ◽  
Number Of Classes

In this chapter, I shall try to advance our thinking about college and university education in the United States through a critical study of contemporary conceptions of the academic vocation. Current reflection upon the state of higher learning in America makes this task at once more urgent and more difficult than it has ever been since the rise of the modern research university. Consider, for example, former Harvard President Derek Bok’s 1986–87 report to the Harvard Board of Overseers. On the one hand, Bok repeatedly insists that universities are obliged to help students learn how to lead ethical, fulfilling lives. On the other hand, he admits that faculty are ill-equipped to help the university discharge this obligation. “Professors,” Bok writes, “. . . are trained to transmit knowledge and skills within their chosen discipline, not to help students become more mature, morally perceptive human beings.” Notice Bok’s assumptions. Teaching history or chemistry or mathematics or literature has little or nothing to do with forming students’ characters. Faculty members must therefore be exhorted, cajoled, or otherwise maneuvered to undertake this latter endeavor in addition to teaching their chosen disciplines. The pursuit of knowledge and the cultivation of virtue are, for Bok at least, utterly discrete activities. To complicate matters still further, the Harvard faculty, together with most faculty members at other modern research universities, would very probably resist the notion that their principal vocational obligation is, as Bok suggested, to transmit the knowledge and skills of their disciplines. They believe that their calling primarily involves making or advancing knowledge, not transmitting it. How else could we explain the familiar academic lament “Because this is a terribly busy semester for me, I do not have any time to do my own work”? Among all occupational groups other than the professoriate, such a complaint, voiced under conditions of intensive labor, is inconceivable. Among university faculty members, it is expected. Never mind the number of classes taught, courses prepared, papers graded, and committees convened.

Commutative regular rings and Boolean-valued fields

1984 ◽  
Vol 49 (1) ◽  
pp. 281-297 ◽  
Author(s):  
Kay Smith
Keyword(s):  
Set Theory ◽  
Regular Ring ◽  
Galois Theory ◽  
Algebraic Closure ◽  
Original Proof ◽  
Valued Fields ◽  
Boolean Space ◽  
Equality Relation ◽  
The One ◽  
Regular Rings

In this paper we present an equivalence between the category of commutative regular rings and the category of Boolean-valued fields, i.e., Boolean-valued sets for which the field axioms are true. The author used this equivalence in [12] to develop a Galois theory for commutative regular rings. Here we apply the equivalence to give an alternative construction of an algebraic closure for any commutative regular ring (the original proof is due to Carson [2]).Boolean-valued sets were developed in 1965 by Scott and Solovay [10] to simplify independence proofs in set theory. They later were applied by Takeuti [13] to obtain results on Hilbert and Banach spaces. Ellentuck [3] and Weispfenning [14] considered Boolean-valued rings which consisted of rings and associated Boolean-valued relations. (Lemma 4.2 shows that their equality relation is the same as the one used in this paper.) To the author's knowledge, the present work is the first to employ the Boolean-valued sets of Scott and Solovay to obtain results in algebra.The idea that commutative regular rings can be studied by examining the properties of related fields is not new. For several years algebraists and logicians have investigated commutative regular rings by representing a commutative regular ring as a subdirect product of fields or as the ring of global sections of a sheaf of fields over a Boolean space (see, for example, [9] and [8]). These representations depend, as does the work presented here, on the fact that the set of central idempotents of any ring with identity forms a Boolean algebra. The advantage of the Boolean-valued set approach is that the axioms of classical logic and set theory are true in the Boolean universe. Therefore, if the axioms for a field are true for a Boolean-valued set, then other properties of the set can be deduced immediately from field theory.

Restricted orbit equivalence for ergodic ${\Bbb Z}^{d}$ actions I

1997 ◽  
Vol 17 (5) ◽  
pp. 1083-1129 ◽  
Author(s):  
JANET WHALEN KAMMEYER ◽  
DANIEL J. RUDOLPH
Keyword(s):  
Equivalence Relation ◽  
Dimensional Case ◽  
Orbit Equivalence ◽  
One Dimensional ◽  
Ergodic Transformations ◽  
Higher Dimensional ◽  
The One ◽  
Equivalence Invariant ◽  
Classical Entropy

In [R1] a notion of restricted orbit equivalence for ergodic transformations was developed. Here we modify that structure in order to generalize it to actions of higher-dimensional groups, in particular ${\Bbb Z}^d$-actions. The concept of a ‘size’ is developed first from an axiomatized notion of the size of a permutation of a finite block in ${\Bbb Z}^d$. This is extended to orbit equivalences which are cohomologous to the identity and, via the natural completion, to a notion of restricted orbit equivalence. This is shown to be an equivalence relation. Associated to each size is an entropy which is an equivalence invariant. As in the one-dimensional case this entropy is either the classical entropy or is zero. Several examples are discussed.

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