We show that if M is a strongly minimal large homogeneous structure in a countable similarity type and the pregeometry of M is locally modular but not modular, then the pregeometry is affine over a division ring.

There is a distinction between locality and modularity. These two terms have often been used interchangeably in the target article and commentary. Using this distinction we argue in favor of a modularity. In addition we also argue that both PDP-type networks and box-and-arrow models have their own strengths and pitfalls.

We describe here a special subdirect decomposition of algebras with modular congruence lattice. Such a decomposition is based on the properties of the congruence lattices of algebras. We consider four properties of lattices: atomic, atomless, locally uniform and anti-uniform. In effect, we describe a star-decomposition of a given algebra with modular congruence lattice into two or three parts associated to these properties.

The claim that the human cognitive system tends to allocate resources to the processing of available inputs according to their expected relevance is at the basis of relevance theory. The main thesis of this chapter is that this allocation can be achieved without computing expected relevance. When an input meets the input condition of a given modular procedure, it gives this procedure some initial level of activation. Input-activated procedures are in competition for the energy resources that would allow them to (...) follow their full course. What determines which of the procedures in competition get sufficient resources to trigger their full operation is the dynamics of their activation. This dynamics depend both on the prior degree of mobilisation of a modular procedure and on the activation that propagates from other active modules. It is quite conceivable also that the mobilisation of some procedures has inhibitory effects on some others. The relevance-theoretic claim is that, at every instant, this dynamics of activation provides rough physiological indicators of expected relevance. The flow of energy in the system is locally regulated by these indicators. As a result, those input-procedure combinations that have the greatest expected relevance are the more likely ones to receive sufficient energy to follow their course. (shrink)

Structuralism can be defined as an important concept of using “units” as elements of form and space-giving, where the whole form is made not only up of a “texture”, a certain flexible grid, or an algorithm of shape-giving, but it depends also on the relationships created and how people use it. The hypothesis of this study is that “Modular Construction” can also have an aesthetically pleasing outlook and that modular housing can definitely have increasing importance in the future. Modular housing (...) due to its cheap, and manufactural way of construction can be replicable in various locations. Structuralism and Modular Construction have been at the core of the Modernist movement. Although postmodernism marked the decline of modular construction adopting methods known for a more close sense of place, vernacular, and locality, which differs from the ubiquitous model of modernism, it is the universality of space and habitation seen since the times of Hippodamus Grid, Japanese Tattami systems, and rational universal space in housing known in Europe that make grid and modular design an excellent system in obtaining, rationality, functionality, flexibility, and energy efficiency, including quick construction and possibility for replication. In this regard, modular construction has a future, especially in times of crisis and economic instability. (shrink)

The claim that the human cognitive system tends to allocate resources to the processing of available inputs according to their expected relevance is at the basis of relevance theory. The main thesis of this chapter is that this allocation can be achieved without computing expected relevance. When an input meets the input condition of a given modular procedure, it gives this procedure some initial level of activation. Input-activated procedures are in competition for the energy resources that would allow them to (...) follow their full course. What determines which of the procedures in competition get sufficient resources to trigger their full operation is the dynamics of their activation. This dynamics depend both on the prior degree of mobilisation of a modular procedure and on the activation that propagates from other active modules. It is quite conceivable also that the mobilisation of some procedures has inhibitory effects on some others. The relevance-theoretic claim is that, at every instant, this dynamics of activation provides rough physiological indicators of expected relevance. The flow of energy in the system is locally regulated by these indicators. As a result, those input-procedure combinations that have the greatest expected relevance are the more likely ones to receive sufficient energy to follow their course. (shrink)

Although disoriented young children reorient themselves in relation to the shape of the surrounding surface layout, cognitive accounts of this ability vary. The present paper tests three theories of reorientation: a snapshot theory based on visual image-matching computations, an adaptive combination theory proposing that diverse environmental cues to orientation are weighted according to their experienced reliability, and a modular theory centering on encapsulated computations of the shape of the extended surface layout. Seven experiments test these theories by manipulating four properties (...) of objects placed within a cylindrical space: their size, motion, dimensionality, and distance from the space’s borders. Their findings support the modular theory and suggest that disoriented search behavior centers on two processes: a reorientation process based on the geometry of the 3D surface layout, and a beacon-guidance process based on the local features of objects and surface markings. Ó 2010 Elsevier Inc. All rights reserved. (shrink)

The leading hypothesis concerning the “reuse” or “recycling” of neural circuits builds on the assumption that evolution might prefer the redeployment of established circuits over the development of new ones. What conception of cognitive architecture can survive the evidence for this hypothesis? In particular, what sorts of “modules” are compatible with this evidence? I argue that the only likely candidates will, in effect, be the columns which Vernon Mountcastle originally hypothesized some 60 years ago, and which form part of the (...) well-known columnar hypothesis in neuroscience—systems that cannot handle gross cognitive functions as distinct from strictly exiguous subfunctions. This is in stark contrast to the modules postulated by much of cognitive psychology, cognitive neuropsychology, and evolutionary psychology. And yet the fate of this revised notion is unclear. The main issue confronting it is the effect of the neural network context on local function. At some point the effects of context are so strong that the degree of specialization required for modularity is not able to be met. Still, despite indications from neuroimaging that peripheral and central systems deploy shared circuitry, some skills clearly do seem to display modularization and autonomy. This article: provides an in-depth analytical and historical review of the fortunes of modular thinking in cognitive science; offers a systematic calibration of brain regions in terms of degrees of functional specificity and robustness; and suggests another way of accounting for the partially encapsulated character of expertise and other highly practiced skills without having to resort to domain-specific modules. (shrink)

Three experiments investigated the modularity of harmonic expectations that are based on cultural schemata despite the availability of more predictive veridical information. Participants were presented with prime–target chord pairs and made an intonation judgment about each target. Schematic expectation was manipulated by the combination of prime and target, with some transitions being schematically more probable than others. Veridical information in the form of prime–target previews, local transition probabilities, or valid versus invalid previews was also provided. Processing was facilitated (...) when a schematically probable target chord followed the prime. Furthermore, this effect was independent of all manipulations of veridical expectation. A solution to L. B. Meyer's (1967b) query "On Rehearing Music" is suggested, in which schematic knowledge contributes to harmonic expectation in a modular manner regardless of whether any veridical knowledge exists. (shrink)

Currently, there is widespread skepticism that higher cognitive processes, given their apparent flexibility and globality, could be carried out by specialized computational devices, or modules. This skepticism is largely due to Fodor’s influential definition of modularity. From the rather flexible catalogue of possible modular features that Fodor originally proposed has emerged a widely held notion of modules as rigid, informationally encapsulated devices that accept highly local inputs and whose opera- tions are insensitive to context. It is a mistake, (...) however, to equate such features with computational devices in general and therefore to assume, as Fodor does, that higher cognitive processes must be non-computational. Of the many possible non-Fodorean architectures, one is explored here that offers possible solutions to computational problems faced by conventional modular systems: an ‘enzymatic’ architecture. Enzymes are computational devices that use lock-and-key template matching to iden- tify relevant information (substrates), which is then operated upon and returned to a common pool for possible processing by other devices. Highly specialized enzymes can operate together in a common pool of information that is not pre-sorted by information type. Moreover, enzymes can use molecular ‘tags’ to regulate the operations of other devices and to change how particular substrates are construed and operated upon, allowing for highly interactive, context-specific processing. This model shows how specialized, modular processing can occur in an open system, and suggests that skepti- cism about modularity may largely be due to failure to consider alternatives to the standard model. (shrink)

The main aim of this work is to relate integrability in QFT with a complete particle interpretation directly to the principle of causal localization, circumventing the standard method of finding sufficiently many conservation laws. Its precise conceptual-mathematical formulation as “modular localization” within the setting of local operator algebras also suggests novel ways of looking at general (non-integrable) QFTs which are not based on quantizing classical field theories.Conformal QFT, which is known to admit no particle interpretation, suggest the presence of (...) a “partial” integrability, referred to as “conformal integrability”. This manifests itself in a “braid-permutation” group structure which contains in particular informations about the anomalous dimensional spectrum. For chiral conformal models this reduces to the braid group as it is represented in Hecke- or Birman-Wenzl-algebras associated to chiral models.Another application of modular localization mentioned in this work is an alternative to the BRST formulation of gauge theories in terms of stringlike vectorpotentials within a Hilbert space setting. (shrink)

A large homogeneous model M is strongly minimal, if any definable subset is either bounded or has bounded complement. In this case is a pregeometry, where bcl denotes the bounded closure operation. In this paper, we show that if M is a large homogeneous strongly minimal structure and is non-trivial and locally modular, then M interprets a group. In addition, we give a description of such groups.

Currently, there is widespread skepticism that higher cognitive processes, given their apparent flexibility and globality, could be carried out by specialized computational devices, or modules. This skepticism is largely due to Fodor's influential definition of modularity. From the rather flexible catalogue of possible modular features that Fodor originally proposed has emerged a widely held notion of modules as rigid, informationally encapsulated devices that accept highly local inputs and whose operations are insensitive to context. It is a mistake, however, (...) to equate such features with computational devices in general and therefore to assume, as Fodor does, that higher cognitive processes must be non‐computational. Of the many possible non‐Fodorean architectures, one is explored here that offers possible solutions to computational problems faced by conventional modular systems: an ‘enzymatic’ architecture. Enzymes are computational devices that use lock‐and‐key template matching to identify relevant information (substrates), which is then operated upon and returned to a common pool for possible processing by other devices. Highly specialized enzymes can operate together in a common pool of information that is not pre‐sorted by information type. Moreover, enzymes can use molecular ‘tags’ to regulate the operations of other devices and to change how particular substrates are construed and operated upon, allowing for highly interactive, context‐specific processing. This model shows how specialized, modular processing can occur in an open system, and suggests that skepticism about modularity may largely be due to failure to consider alternatives to the standard model. (shrink)

We believe the dichotomy of processes introduced in the target article is highly speculative, because the dichotomy is shaped by the questionable assumption of modularity and the complementary assumption of locality. As a result, the author falls into a line of circular reasoning that biases his analysis of the experimental and neuropsychological data, and weakens the proposed model.

Evolutionary psychologists claim that the mind contains “hundreds or thousands” of “genetically specified” modules, which are evolutionary adaptations for their cognitive functions. We argue that, while the adult human mind/brain typically contains a degree of modularization, its “modules” are neither genetically specified nor evolutionary adaptations. Rather, they result from the brain’s developmental plasticity, which allows environmental task demands a large role in shaping the brain’s information-processing structures. The brain’s developmental plasticity is our fundamental psychological adaptation, and the “modules” that result (...) from it are adaptive responses to local conditions, not past evolutionary environments. If different individuals share common environ- ments, however, they may develop similar “modules,” and this process can mimic the development of genetically specified modules in the evolutionary psychologist’s sense. (shrink)

The invariance under interventions –account of causal explanation imposes a modularity constraint on causal systems: a local intervention on a part of the system should not change other causal relations in that system. This constraint has generated criticism against the account, since many ordinary causal systems seem to break this condition. This paper answers to this criticism by noting that explanatory models are always models of specific causal structures, not causal systems as a whole, and that models of (...) causal structures can have different modularity properties which determine what can and what cannot be explained with the model. (shrink)

Let D be a strongly minimal set in the language L, and $D' \supset D$ an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T' be the theory of the structure (D', D), where D interprets the predicate D. It is known that T' is ω-stable. We prove Theorem A. If D is not locally modular, then T' has Morley rank ω. We say that a strongly minimal set D is pseudoprojective (...) if it is nontrivial and there is a $k < \omega$ such that, for all a, b ∈ D and closed $X \subset D, a \in \mathrm{cl}(Xb) \Rightarrow$ there is a $Y \subset X$ with a ∈ cl(Yb) and |Y| ≤ k. Using Theorem A, we prove Theorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective. The following result of Hrushovski's (proved in $\S4$ ) plays a part in the proof of Theorem B. Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular. (shrink)

Farah argues that nonlocal models explain clinical data better. However, the locality assumption does not seem so implausible if different sorts of data are taken into account. In particular, priming experiments using evoked response potentials support modularity. I describe some ERP studies relevant to this issue.

The locality assumption (LA) seems rather awkward, especially when one considers centres of neuronal specialisation as defined by observed CNS activity. It is clear from electrophysiology that extra-striate functional compartmentalisation (modularity) is rather less well-defined than first thought; neuropsychological assessment attaching significance to varieties of preserved behaviour also reveals that some basic flaws must be inherent in current reasoning supporting LA.

When cognitive neuropsychologists make inferences about the functional architecture of the normal mind from selective cognitive impairments they generally assume that the effects of brain damage are local, that is, that the nondamaged components of the architecture continue to function as they did before the damage. This assumption follows from the view that the components of the functional architecture are modular, in the sense of being informationally encapsulated. In this target article it is argued that this “locality” assumption is (...) probably not correct in general. Inferences about the functional architecture can nevertheless be made from neuropsychological data with an alternative set of assumptions, according to which human information processing is graded, distributed, and interactive. These claims are supported by three examples of neuropsychological dissociations and a comparison of the inferences obtained from these impairments with and without the locality assumption. The three dissociations are: selective impairments in knowledge of living things, disengagment of visual attention, and overt face recognition. In all three cases, the neuropsychological phenomena lead to more plausible inferences about the normal functional architecture when the locality assumption is abandoned. Also discussed are the relations between the locality assumption in neuropsychology and broader issues, including Fodor's modularity hypothesis and the choice between top-down and bottom-up research approaches. (shrink)

Members of a human group are bound with one another by multiple flows of information. (Here we use “information” in a broad sense that includes not only the content of people’s knowledge, but also that of their beliefs, assumptions, fictions, rules, norms, skills, maps, images, and so on.) This information is materially realized in the mental representations of the people, and in their public productions, that is, their cognitively guided behaviors and the enduring material traces of these behaviors. Mentally represented (...) information is transmitted from individuals to individuals through public productions. Public representations such as speech, gestures, writing, or pictures are a special type of public productions, the function of which is to communicate a content. Public representations play a major role in information transmission. Much information, however, is communicated implicitly, that is, without being publicly represented. Information can also be transmitted without being properly speaking communicated, not even implicitly, as when one individual acquires a skill by observing and imitating the behavior of others. Most information transmitted among humans is about local and transient circumstances, and is not transmitted beyond these. Some information of more general relevance, however, is repeatedly transmitted, and propagates throughout the group. Talk of “culture” (whatever the preferred definition or theory of culture) is about this widely distributed information and about its material realizations inside people’s mind and in their common environment (see Sperber 1996). One can study cultural phenomena in two main ways. One can interpret them, that is, try and make their contents intelligible to people of another culture, or more intelligible to members of the culture in which these phenomena occurs, as do anthropologists and historians. One may also try and explain causally how these cultural phenomena emerge, stabilize and evolve.. (shrink)

When cognitive neuropsychologists make inferences about the functional architecture of the normal mind from selective cognitive impairments they generally assume that the effects of brain damage are local, that is, that the nondamaged components of the architecture continue to function as they did before the damage. This assumption follows from the view that the components of the functional architecture are modular, in the sense of being informationally encapsulated. In this target article it is argued that this “locality” assumption is (...) probably not correct in general. Inferences about the functional architecture can nevertheless be made from neuropsychological data with an alternative set of assumptions, according to which human information processing is graded, distributed, and interactive. These claims are supported by three examples of neuropsychological dissociations and a comparison of the inferences obtained from these impairments with and without the locality assumption. The three dissociations are: selective impairments in knowledge of living things, disengagment of visual attention, and overt face recognition. In all three cases, the neuropsychological phenomena lead to more plausible inferences about the normal functional architecture when the locality assumption is abandoned. Also discussed are the relations between the locality assumption in neuropsychology and broader issues, including Fodor's modularity hypothesis and the choice between top-down and bottom-up research approaches. (shrink)

The quantum theoretical concepts of modular momentum and dynamical non-locality, which were introduced four decades ago, have recently been used to explain single particle quantum interference phenomena. Although the non-local exchange of modular momentum associated with such phenomena cannot be directly observed, it has been suggested that effects induced by this exchange can be measured experimentally using weak measurements of pre- and post-selected ensembles of particles. This paper reports on such an optical experiment that yielded measured weak values that (...) were consistent with the theoretical prediction of an effect induced by a non-local exchange of modular momentum. (shrink)

In this paper we deal with the model theory of differentially closed fields of characteristic zero with finitely many commuting derivations. First we observe that the only known lower bound for the Lascar rank of types in differentially closed fields, announced in a paper of McGrail, is false. This gives us a new class of regular types which are orthogonal to fields. Then we classify the subgroups of the additive group of Lascar rank omega with differential-type 1 which are nonorthogonal (...) to fields. The last parts consist of an analaysis of the quotients of the heat variety. We show that the generic type of such a quotient is locally modular. Finally, we answer a question of Phylliss Cassidy about the existence of certain Jordan-Holder type series in the negative. (shrink)

Recently some philosophers have emphasized a potentially irreconcilable conceptual antagonism between the statistical characterization of natural selection and the standard scientific discussion of natural selection in terms of forces and causes. Other philosophers have developed an account of the causal character of selectionist statements represented in terms of counterfactuals. I examine the compatibility between such statisticalism and counterfactually based causal accounts of natural selection by distinguishing two distinct statisticalist claims: firstly the suggested impossibility for natural selection to be a cause (...) acting upon populations and secondly the conceptualization that all evolutionary causes occur at the level of interactions between individual organisms. I argue that deriving the latter from the former involves supplementary assumptions concerning precisely what causation is. I critically examine two of these assumptions purportedly preventing natural selection being regarded as a cause: the locality claim and the modularity claim. I conclude that justifying the strongest version of statisticalism—i.e. evolutionary causation only occurs at the level of individual interactions between organisms—would require further metaphysical arguments that are likely to be deemed highly problematic. Additionally, I argue that such a metaphysical position would be considered incongruous with both our scientific and ordinary use of the concepts of causality and explanation as employed within our everyday epistemological framework. (shrink)

A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be (...) closely related to modular pairs in the lattice of algebraically closed sets. (shrink)

We present an axiomatic approach for a class of finite, extensive form games of perfect information that makes use of notions like “rationality at a node” and “knowledge at a node.” We distinguish between the game theorist's and the players' own “theory of the game.” The latter is a theory that is sufficient for each player to infer a certain sequence of moves, whereas the former is intended as a justification of such a sequence of moves. While in general the (...) game theorist's theory of the game is not and need not be axiomatized, the players' theory must be an axiomatic one, since we model players as analogous to automatic theorem provers that play the game by inferring (or computing) a sequence of moves. We provide the players with an axiomatic theory sufficient to infer a solution for the game (in our case, the backwards induction equilibrium), and prove its consistency. We then inquire what happens when the theory of the game is augmented with information that a move outside the inferred solution has occurred. We show that a theory that is sufficient for the players to infer a solution and still remains consistent in the face of deviations must be modular. By this we mean that players have distributed knowledge of it. Finally, we show that whenever the theory of the game is group-knowledge (or common knowledge) among the players (i.e., it is the same at each node), a deviation from the solution gives rise to inconsistencies and therefore forces a revision of the theory at later nodes. On the contrary, whenever a theory of the game is modular, a deviation from equilibrium play does not induce a revision of the theory. (shrink)

Assume G is a superstable locally modular group. We describe for any countable model M of Th(G) the quotient group G(M) / Gm(M). Here Gm is the modular part of G. Also, under some additional assumptions we describe G(M) / Gm(M) relative to G⁻(M). We prove Vaught's Conjecture for Th(G) relative to Gm and a finite set provided that ℳ(G) = 1 and the ring of pseudoendomorphisms of G is finite.

The suggestion that cognition is holistic has become a prominent criticism of optimism about the prospects for cognitive science. This paper argues that the standard motivation for this holism, that of epistemological holism, does not justify this pessimism. An illustration is given of how the effects of epistemological holism on perception are compatible with the view that perceptual processes are highly modular. A suggestion for generalizing this idea to conceptual cognitive processing is made, and an account of the holists' failure (...) is offered. (shrink)

Müller misconstrues autonomy to mean strict locality of brain function, something quite different from the functional autonomy that linguists claim. Similarly, he misperceives the interaction of learned and innate components hypothesized in current generative models. Evidence from sign languages, Creole languages, and neurological studies of rare forms of aphasia also argues against his conclusions.

We give an exposition of Hrushovskiʼs New Strongly Minimal Set : A strongly minimal theory which is not locally modular but does not interpret an infinite field. We give an exposition of his construction.

We study the theory of lovely pairs of geometric structures, in particular o-minimal structures. We use the pairs to isolate a class of geometric structures called weakly locally modular which generalizes the class of linear structures in the settings of SU-rank one theories and o-minimal theories. For o-minimal theories, we use the Peterzil–Starchenko trichotomy theorem to characterize for a sufficiently general point, the local geometry around it in terms of the thorn U-rank of its type inside a lovely pair.

We study hyperdefinable groups, the most general kind of groups interpretable in a simple theory. After developing their basic theory, we prove the appropriate versions of Hrushovski's group quotient theorem and the Weil–Hrushovski group chunk theorem. We also study locally modular hyperdefinable groups and prove that they are bounded-by-Abelian-by-bounded. Finally, we analyze hyperdefinable groups in supersimple theories.

T is stable. We define the notion of meager regular type and prove that a meager regular type is locally modular. Assuming I < 2o and G is a definable abelian group with locally modular regular generics, we prove a counterpart of Saffe's conjecture. Using these results, for superstable T we prove the conjecture of vanishing multiplicities. Also, as a further application, in some additional cases we prove a conjecture regarding topological stability of pseudo-types over Q.

Let T be superstable. We say a type p is weakly minimal if R(p, L, ∞) = 1. Let $M \models T$ be uncountable and saturated, H = p(M). We say $D \subset H$ is locally modular if for all $X, Y \subset D$ with $X = \operatorname{acl}(X) \cap D, Y = \operatorname{acl}(Y) \cap D$ and $X \cap Y \neq \varnothing$ , dim(X ∪ Y) + dim(X ∩ Y) = dim(X) + dim(Y). Theorem 1. Let p ∈ S(A) be weakly (...) minimal and D the realizations of $\operatorname{stp}(a/A)$ for some a realizing p. Then D is locally modular or p has Morley rank 1. Theorem 2. Let H, G be definable over some finite A, weakly minimal, locally modular and nonorthogonal. Then for all $a \in H\backslash\operatorname{acl}(A), b \in G\operatorname{acl}(A)$ there are a' ∈ H, b' ∈ G such that $a' \in \operatorname{acl}(abb' A)\backslash\operatorname{acl}(aA)$ . Similarly when H and G are the realizations of complete types or strong types over A. (shrink)

We reconstruct the group action in the group configuration theorem. We apply it to show that in an ω-categorical theory a finitely based pseudolinear regular type is locally modular, and the geometry associated to a finitely based locally modular regular type is projective geometry over a finite field.

For a supersimple SU-rank 1 theory T we introduce the notion of a generic elementary pair of models of T . We show that the theory T* of all generic T-pairs is complete and supersimple. In the strongly minimal case, T* coincides with the theory of infinite dimensional pairs, which was used in 1184–1194) to study the geometric properties of T. In our SU-rank 1 setting, we use T* for the same purpose. In particular, we obtain a characterization of linearity (...) for SU-rank 1 structures by giving several equivalent conditions on T*, find a “weak” version of local modularity which is equivalent to linearity, show that linearity coincides with 1-basedness, and use the generic pairs to “recover” projective geometries over division rings from non-trivial linear SU-rank 1 structures. (shrink)

Finitely axiomatizable ℵ 1 categorical theories are locally modular.

We prove here:Theorem. LetT be a countable complete superstable non ω-stable theory with fewer than continuum many countable models. Then there is a definable groupG with locally modular regular generics, such thatG is not connected-by-finite and any type inG eq orthogonal to the generics has Morley rank.Corollary. LetT be a countable complete superstable theory in which no infinite group is definable. ThenT has either at most countably many, or exactly continuum many countable models, up to isomorphism.

In this paper we establish the following theorems. THEOREM A. Let T be a complete first-order theory which is uncountable. Then: (i) I(|T|, T) ≥ ℵ 0 . (ii) If T is not unidimensional, then for any λ ≥ |T|, I (λ, T) ≥ ℵ 0 . THEOREM B. Let T be superstable, not totally transcendental and nonmultidimensional. Let θ(x) be a formula of least R ∞ rank which does not have Morley rank, and let p be any stationary completion (...) of θ which also fails to have Morley rank. Then p is regular and locally modular. (shrink)

Let $M$ be a saturated model of a superstable theory and let $G = \operatorname{Aut}(M)$. We study subgroups $H$ of $G$ which contain $G_{(A)}, A$ the algebraic closure of a finite set, generalizing results of Lascar [L] as well as giving an alternative characterization of the simple superstable theories of [P]. We also make some observations about good, locally modular regular types $p$ in the context of $p$-simple types.

Suppose $D \subset M$ is a strongly minimal set definable in M with parameters from C. We say D is locally modular if for all $X, Y \subset D$ , with $X = \operatorname{acl}(X \cup C) \cap D, Y = \operatorname{acl}(Y \cup C) \cap D$ and $X \cap Y \neq \varnothing$ , dim(X ∪ Y) + dim(X ∩ Y) = dim(X) + dim(Y). We prove the following theorems. Theorem 1. Suppose M is stable and $D \subset M$ is strongly minimal. (...) If D is not locally modular then in M eq there is a definable pseudoplane. (For a discussion of M eq see [M, § A].) This is the main part of Theorem 1 of [Z2] and the trichotomy theorem of [Z3]. Theorem 2. Suppose M is stable and $D, D' \subset M$ are strongly minimal and nonorthogonal. Then D is locally modular if and only if D' is locally modular. (shrink)

We study the infinitely definable subgroups of the additive group in a separably closed field of finite positive imperfection degree. We give some constructions of families of such subgroups which confirm the diversity and the richness of this class of groups. We show in particular that there exists a locally modular minimal subgroup such that the division ring of its quasi-endomorphisms is not a fraction field of the ring of its definable endomorphisms, and that in contrast there exist 20 pairwise (...) orthogonal locally modular minimal subgroups whose induced structure is exactly that of an-vector space. We also show that there exist infinitely many pairwise orthogonal subgroups of infinite U-rank. Furthermore, these constructions are carried out in the additive group considered as a module over its ring of definable endomorphisms. (shrink)