We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an F_σ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over ∅. As an easy conclusion of our main theorem, we get the main result of [K. Krupiński, A. Pillay and T. (...) Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math.228 (2018) 863–932] which says that for any strong type defined on a single complete type over ∅, smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the paper mentioned above about bounded quotients of type-definable subgroups of definable groups. (shrink)

We study the groups Gal L and Gal KP, and the associated equivalence relations EL and EKP, attached to a first order theory T. An example is given where EL≠ EKP. It is proved that EKP is the composition of EL and the closure of EL. Other examples are given showing this is best possible.

We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p‐valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in, we establish a relative Galois correspondence for relatively definable subgroups (...) of the group of differential order automorphisms. (shrink)

In [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. 19 865–877], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that [Formula: see text] has the canonical base property in a strong form; “internality to” being replaced by “algebraicity in”. In the current paper, we give a reasonably robust definition of the “strong canonical base property” in a rather more general finite rank context than (...) [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. 19 865–877], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that [Formula: see text]-based groups are rigid. (shrink)

In this paper, I will give a new characterisation of the spaces of complete theories of pseudofinite fields and of algebraically closed fields with a generic automorphism (ACFA) in terms of the Vietoris topology on absolute Galois groups of prime fields.

Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is (...) not injective. (shrink)

In his testamentary letter to Auguste Chevalier, Évariste Galois states that, in modern terminology, the smallest simple group has order 60. No proof of this statement survives in his papers, and it has been suggested that a proof would have been impossible using the methods available at the time. We argue that this assertion is unduly pessimistic. Moreover, one fragmentary document, dismissed as a triviality and misunderstood, looks suspiciously like cryptic notes related to this result. We give an (...) elementary proof of Galois’s statement, explain why it is likely that he would have been aware of the methods involved, and discuss the potential relevance of the fragment. (shrink)

In this paper we investigate the properties of automorphism groups of countable short recursively saturated models of arithmetic. In particular, we show that Kaye's Theorem concerning the closed normal subgroups of automorphism groups of countable recursively saturated models of arithmetic applies to automorphism groups of countable short recursively saturated models as well. That is, the closed normal subgroups of the automorphism group of a countable short recursively saturated model of PA are exactly the stabilizers of the invariant cuts of (...) the model which are closed under exponentiation. This Galois correspondence is used to show that there are countable short recursively saturated models of arithmetic whose automorphism groups are not isomorphic as topological groups. Moreover, we show that the automorphism groups of countable short arithmetically saturated models of PA are not topologically isomorphic to the automorphism groups of countable short recursively saturated models of PA which are not short arithmetically saturated. (shrink)

We carry out some of Galois' work in the setting of an arbitrary first-order theory T. We replace the ambient algebraically closed field by a large model M of T, replace fields by definably closed subsets of M, assume that T codes finite sets, and obtain the fundamental duality of Galois theory matching subgroups of the Galois group of L over F with intermediate extensions F ≤ K ≤ L. This exposition of a special case of (...) [10] has the advantage of requiring almost no background beyond familiarity with fields, polynomials, first-order formulae, and automorphisms. (shrink)

We achieve several results. First, we develop a variant of the theory of absolute Galois groups in the context of many sorted structures. Second, we provide a method for coding absolute Galois groups of structures, so they can be interpreted in some monster model with an additional predicate. Third, we prove the “Weak Independence Theorem” for pseudo-algebraically closed (PAC) substructures of an ambient structure with no finite cover property (nfcp) and the property [Formula: see text]. Fourth, we describe (...) Kim-dividing in these PAC substructures and show several results related to the SOPnhierarchy. Fifth, we characterize the algebraic closure in PAC structures. (shrink)

This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard—Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal extension of iterative differential fields, and then prove that these extensions have good Galois theory and that a G-primitive element theorem holds. In (...) addition, making use of the basic theory of arc spaces of algebraic groups, we define iterative logarithmic equations, finally proving that our strongly normal extensions are Galois extensions for these equations. (shrink)

We analyze the notions of indiscernibility and indeterminacy in the light of the Galois theory of field extensions and the generalization to \(K\) -algebras proposed by Grothendieck. Grothendieck’s reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a Galois–Grothendieck duality between \(G\) -spaces and the minimal observable algebras that discern (or separate) their points. According to the natural epistemic interpretation of the original Galois theory, the possible \(K\) -indiscernibilities (...) between the roots of a polynomial \(p(x)\in K[x]\) result from the limitations of the field \(K\) . We discuss the relation between this epistemic interpretation of the Galois–Grothendieck duality and Leibniz’s principle of the identity of indiscernibles. We then use the conceptual framework provided by Klein’s Erlangen program to propose an alternative ontologic interpretation of this duality. The Galoisian symmetries are now interpreted in terms of the automorphisms of the symmetric geometric figures that can be placed in a background Klein geometry. According to this interpretation, the Galois–Grothendieck duality encodes the compatibility condition between geometric figures endowed with groups of automorphisms and the ‘observables’ that can be consistently evaluated at such figures. In this conceptual framework, the Galoisian symmetries do not encode the epistemic indiscernibility between individuals, but rather the intrinsic indeterminacy in the pointwise localization of the figures with respect to the background Klein geometry. (shrink)

The objective of this article is to characterise elimination of finite generalised imaginaries as defined in [9] in terms of group cohomology. As an application, I consider series of Zariski geometries constructed [10, 23, 24] by Hrushovski and Zilber and indicate how their nondefinability in algebraically closed fields is connected to eliminability of certain generalised imaginaries.

Profinite groups are Galois groups. The effective study of infinite Galois groups was initiated by Metakides and Nerode [8] and further developed by LaRoche [5]. In this paper we study profinite groups without considering Galois extensions of fields. The Artin method of representing a finite group as a Galois group has been generalized by Waterhouse [14] to profinite groups. Thus, there is no loss of relevance in our approach.The fundamental notions of a co-r.e. profinite (...) group, recursively profinite group, and the degree of a co-r.e. profinite group are defined in §1. In this section we prove that every co-r.e. profinite group can be effectively represented as an inverse limit of finite groups. The degree invariant is shown to behave very well with respect to open subgroups and quotients. The work done in this section is basic to the rest of the paper.The commutator subgroup, the Frattini subgroup, thep-Sylow subgroups, and the center of a profinite group are essential in the study of profinite groups. It is only natural to ask if these subgroups are effective. The following question exemplifies our approach to this problem: Is the center a co-r.e. profinite group? Theorem 2 provides a general method for answering this type of question negatively. Examples 3,4 and 5 are all applications of this theorem. (shrink)

We analyze the notion of indiscernibility in the light of the Galois theory of field extensions and the generalization to K-algebras proposed by Grothendieck. Grothendieck's reformulation of Galois theory permits to recast the Galois correspondence between symmetry groups and invariants as a duality between G-spaces and the minimal observable algebras that separate theirs points. In order to address the Galoisian notion of indiscernibility, we propose what we call an epistemic reading of the Galois-Grothendieck theory. According to (...) this viewpoint, the Galoisian notion of indiscernibility results from the limitations of the `resolving power' of the observable algebras used to discern the corresponding `coarse-grained' states. The resulting Galois-Grothendieck duality is rephrased in the form of what we call a Galois indiscernibility principle. According to this principle, there exists an inverse correlation between the coarsegrainedness of the states and the size of the minimal observable algebra that discern these states. (shrink)

In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory T is G-compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note. -/- For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such types (...) have a certain model theoretic property that we call divisible amalgamation. -/- The main result of this paper is that if c is a finite tuple algebraic over a tuple a, the Lascar group of stp(ac) is abelian, and the underlying theory is G-compact, then the Lascar groups of stp(ac) and of stp(a) are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup. -/- Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of stp(ac) is abelian, nor the assumption of c being finite can be removed. (shrink)

We discuss some new properties of the natural Galois connection among set relation algebras, permutation groups, and first order logic. In particular, we exhibit infinitely many permutational relation algebras without a Galois closed representation, and we also show that every relation algebra on a set with at most six elements is Galois closed and essentially unique. Thus, we obtain the surprising result that on such sets, logic with three variables is as powerful in expression as full first (...) order logic. (shrink)

I examine the way a relevant conceptual novelty in mathematics, that is, the notion of group, has been constructed in order to show the kinds of heuristic reasoning that enabled its manufacturing. To this end, I examine salient aspects of the works of Lagrange, Cauchy, Galois and Cayley. In more detail, I examine the seminal idea resulting from Lagrange’s heuristics and how Cauchy, Galois and Cayley develop it. This analysis shows us how new mathematical entities are generated, (...) and also how what counts as a solution to a problem is shaped and changed. Finally, I argue that this case study shows us that we have to study inferential micro-structures, that is, the ways similarities and regularities are sought, in order to understand how theoretical novelty is constructed and heuristic reasoning is put forward. (shrink)

We study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

We would like to open some questions here about the institutional and cultural conditions of anything that might be called cultural studies or multiculturalism. By introducing cultural studies and multiculturalism many intellectuals aim at a more democratic culture. We share this aim. In this essay, however, we would like to argue that the projects of cultural studies and multiculturalism require: a more international model of cultural studies than the dominant Anglo-American versions; renewed attention to the institutional environments of cultural studies; (...) and a questioning of the relations between multiculturalism and identity politics. We seek less to “fix” these problems than to provide a critical analysis of the languages, the methods of criticism, and the assumptions about identity, culture, and politics that present the problems to us. Because the thickets entangling what our group calls cultural studies are so deeply rooted in Western academia, which to a large degree constitutes our own group, the counterexample of cultural criticism in other contexts can be more than usually instructive. We begin by considering the position of cultural studies in China, since our group includes a number of Chinese intellectuals, on whose experience the following section is largely based. The Chicago Cultural Studies Group began meeting in June 1990. It includes Lauren Berlant, David Bunn, Vinay Dharwadker, Norma Field, Dilip Gaonkar, Marilyn Ivy, Benjamin Lee, Leo Ou-fan Lee, Xinmin Liu, Mathew Roberts, Sharon Stephens, Katie Trumpener, Greg Urban, Michael Warner, Jianyang Zha, and Jueliang Zhou. (shrink)

Malm and colleagues (2008) consider (and reject) five arguments putatively justifying the idea that healthcare workers (HCWs) have a duty to treat (DTT) during a pandemic. We do not have sufficient...

Militante et réflexive, l’histoire des femmes a aussi besoin d’exprimer ses doutes et ses inconforts. Rattrapée par l’actualité, la violence et nombre d’événements cruels et tragiques, par des incertitudes majeures vécues par l’ensemble du monde intellectuel, elle pose aujourd’hui en cet article collectif des interrogations et des inquiétudes. Ce travail à plusieurs (les participantes du Groupe d’histoire des femmes du CRH) cherche à mettre à plat ce qui actuellement se dérobe à notre connaissance, en partant de notions-clés souvent utilisées, comme (...) consentement, sexualité, écriture, patriarcat, etc. Chacune de ces notions est discutée à la lumière des brouillages actuels, des temporalités nouvelles, de l’avènement triomphant du corps marchandise, de la pauvreté grandissante, etc. (shrink)

En 1983, un groupe de femmes japonaises francophiles, très intéressées par le féminisme français et les changements rapides de la vie des femmes en France depuis les années 1970, créent la Société Franco-japonaise des Études sur les Femmes. Le but est de promouvoir une meilleure compréhension entre les deux cultures et de développer des liens avec les Françaises sur un grand nombre de questions féminines. Le projet se révéla vite être un succès. Parmi les études entreprises par la Société, l’analyse (...) comparée de l’histoire des Françaises et des Japonaises, domaine très peu exploré alors, a suscité un vif intérêt chez les participantes. Cet article revient sur vingt années de réflexion, de recherche, et d’échange par un groupe d’historiennes non-professionnelles sur les relations entre histoire et expériences de vie personnelles. Il nous montre comment l’étude de l’histoire des femmes dans un contexte international peut être fructueuse et bénéfique, et combien une libre discussion collective, sans tabous, sur ce sujet peut éveiller ou approfondir la conscience féministe. (shrink)

We concur with Baum and colleagues (2009) on the importance of pandemic planners taking explicit steps to employ public engagement methodologies. Thus far, as Baum and colleagues note, there have b...

Originally published between 1925 and 1931 the volumes in this mini-set cover education, from early years to further education, as well as the role parents and society have in the moral education of children and young people. Among others the authors included are: C E M Joad, W J K Diplock, M Chaning Pearce and J F Roxburgh.

The focus of the titles in this mini-set, originally published between 1926 and 1928 is on the challenges of feeding an increasing global population, technological changes in the home, the evolutionary role of clothes and the dangers and benefits of alcohol.

Originally published between 1924 and 1929 discuss the future of psychology and the break-throughs it could provide for modern medicine. Authors included are E N Bennett, H F Carlill,D F Fraser Harris, M Jaeger and C J Patten.

_Routledge Library Editions: Psychoanalysis_ brings together as one set, or individual volumes, a series of 8 previously out-of-print titles, originally published between 1923 and 1993. Written by international authors from a variety of backgrounds, this set looks at psychoanalysis in a number of different areas including, culture, religion, sociology, postmodernism, literary criticism and others.

This set collects together a vital selection of works on Existentialism, including the key Introduction to the New Existentialism by Colin Wilson. Some of the titles were early works written as this new philosophy spread into the English language, while others are more recent examinations.

Originally published between 1982 and 1991 the 3 volumes in this set Reflect the diversity in Hegelianism and every branch of philosophy which he contributed to. Examine Hegel's work in relation to Marx and Wittgenstein Discuss Hegel's social theory Examine British Hegelian thinking and the lines of its development Offer an interpretation of Hegelian theory that is relevant for the understanding of modern republican constitutions.

Reissuing works originally published between 1949 and ‘79, this set presents a rich selection of renowned scholarship across the subject, touching also on ethics, religion, and psychology and other behavioural science. Classic previously out-of-print works are brought back into print here in this set of important discourse and theory.

This 20 volume Routledge Revivals collection brings together a selection of groundbreaking Philosophy titles, from the rich and diverse Routledge backlist. With titles published between 1933 and 1991, this is a truly wide-ranging selection, encompassing works by distinguished authors such as: Simone Weil, Hilary Putnam, Franz Brentano, Anthony Kenny, Karl Jaspers and Israel Scheffler. Dealing with everything from the notion of freewill, to concepts of time and space, to theories of morality, this set offers a collection of the best of (...) Routledge publishing in the field of Philosophy from across the Twentieth Century. Please note that all titles have been previously available for sale individually through the Routledge Revivals programme. (shrink)

The thirteen titles in this mini-set include works by some of the most well-known scientists and medical professionals of the twentieth century: Daedalus by J B S Haldane, Eos, or the Wider Aspects of Cosmogony by J H Jeans, Archimedes, or the Future of Physics by L L Whyte and the The Conquest of Cancer by H W S Wright to name but a few. Ground breaking in their day, some of the works remain controversial nearly 100 years after their (...) original publication. (shrink)

Originally published between 1926 and 1931 the titles in this mini-set discuss the role of women in both domestic and professional areas, contraception, sexual relationships and the changing nature of the family within society.

We study the degree of elimination of imaginaries needed for the three main applications: to have canonical bases for types over models, to define strong types as types over algebraically closed sets and to have a Galois correspondence between definably closed sets B such that A ⊆ B ⊆ acl and closed subgroups of the Galois group Aut/A). We also characterize when the topology of the Galois group is the quotient topology.

Zemyarska MS. Is it ethical to provide IVF add-ons when there is no evidence of a benefit if the patient requests it? J Med Ethics 2019;45:346–50. doi: 10.1136/medethics-2018-104983. The Acknowledgements section of ….

Biology is seen not merely as a privileged oppressor of women but as a co-victim of masculinist social assumptions. We see feminist critique as one of the normative controls that any scientist must perform whenever analyzing data, and we seek to demonstrate what has happened when this control has not been utilized. Narratives of fertilization and sex determination traditionally have been modeled on the cultural patterns of male/female interaction, leading to gender associations being placed on cells and their components. We (...) also find that when gender biases are controlled, new perceptions of these intracellular and extracellular relationships emerge. (shrink)