Let G be a group definable in an o-minimal structure M. In this paper we show: Theorem. If G is a two-dimensional definably connected nonabelian group, then G is centerless and G is isomorphic to R+R*>0, for some real closed field R. Theorem. If G is a three-dimensional nonsolvable, centerless, definably connected group, then either G SO3 or G PSL2, for some real closed field R.

We study structures equipped with generic predicates and/or automorphisms, and show that in many cases we obtain simple theories. We also show that a bounded PAC field is simple. 1998 Published by Elsevier Science B.V. All rights reserved.

The aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$, with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$. We call such groups definable f-generic groups.So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$, and (...) the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o-minimal groups in the p-adic context.In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$, so as to make sense of the notions of f-generic type, etc. In this paper we will show that every definable f-generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$. This is analogous to the o-minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f-generic types coincide with almost periodic types in the p-adic case. (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.

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.

We prove the Main Gap for the class of a -models (sufficiently saturated models) of an arbitrary stable 1-based theory T . We (i) prove a strong structure theorem for a -models, assuming NDOP, and (ii) roughly compute the number of a -models of T in any given cardinality. The analysis uses heavily group existence theorems in 1-based theories.

We show that the structure (C,+,·) has no proper non locally modular reducts which contain +. In other words, if $X \subset \mathbf{C}^n$ is constructible and not definable in the module structure (C,+,λ a ) a ∈ C (where λ a denotes multiplication by a) then multiplication is definable in (C,+,X).

We explore the existence and the size of infinite models of categorical theories having cardinality less than the size of the associated Tarski-Lindenbaum algebra. Restricting to totally transcendental, categorical theories we show that “Every tiny model is countable” is independent of ZFC. IfT is trivial there is at most one tiny model, which must be the algebraic closure of the empty set. We give a new proof that there are no tiny models ifT is not totally transcendental and is non-trivial.

We study perfectly trivial theories, 1-based theories, stable varieties, and their mutual interaction. We give a structure theorem for the models of a complete perfectly trivial stable theory without DOP: any model is the algebraic closure of a nonforking regular tree of elements. We also give a structure theorem for stable varieties, all of whose completions have NDOP. Such a variety is a varietal product of an affine variety and a combinatorial variety of an especially simple form.

We introduce and study the notions of a PAC-substructure of a stable structure, and a bounded substructure of an arbitrary substructure, generalizing [10]. We give precise definitions and equivalences, saying what it means for properties such as PAC to be first order, study some examples (such as differentially closed fields) in detail, relate the material to generic automorphisms, and generalize a "descent theorem" for pseudo-algebraically closed fields to the stable context. We also point out that the elementary invariants of pseudo-algebraically (...) closed fields from [6] are also valid for pseudo-differentially closed fields. (shrink)

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

Given a lovely pair P ≺ M of models of a simple theory T, we study the structure whose universe is P and whose relations are the traces on P of definable (in ℒ with parameters from M) sets in M. We give a necessary and sufficient condition on T (which we call weak lowness) for this structure to have quantifier-elimination. We give an example of a non-weakly-low simple theory.

In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the forma/Ewhereais a possibly (...) infinite tuple andEa type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2].Throughout this paper we will work in a large, saturated modelMof a complete theoryT. All types, sets and sequences will have size smaller than the size ofM. We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9]. (shrink)

For G a group definable in some structure M, we define notions of “definable” compactification of G and “definable” action of G on a compact space X , where the latter is under a definability of types assumption on M. We describe the universal definable compactification of G as View the MathML source and the universal definable G-ambit as the type space SG. We also point out the existence and uniqueness of “universal minimal definable G-flows”, and discuss issues of amenability (...) and extreme amenability in this definable category, with a characterization of the latter. For the sake of completeness we also describe the universal compactification and universal G-ambit in model-theoretic terms, when G is a topological group. (shrink)

We develop a basic theory of rosy groups and we study groups of small Uþ-rank satisfying NIP and having finitely satisfiable generics: Uþ-rank 1 implies that the group is abelian-by-finite, Uþ-rank 2 implies that the group is solvable-by-finite, Uþ-rank 2, and not being nilpotent-by-finite implies the existence of an interpretable algebraically closed field.

Adequate measurement is an essential component of the assessment of mental health disorders and symptoms such as depression and anxiety. The present study investigated sex-specific differences in the Patient Health Questionnaire-9 and Generalized Anxiety Disorder-7. This comprehensive cross-sectional design study pursued four objectives: measurement invariance of PHQ-9 and GAD-7 between male and female; depression and anxiety prevalence differences; cross-sex differences in the relationship between depression and anxiety; and a comparison of symptom heterogeneity. A sample of 1966 students from South Africa (...) completed the PHQ-9 and the GAD-7. Data analyses for measurement invariance, latent class analysis, inter-variable correlations and group comparisons were conducted in Mplus. The two-dimensional PHQ-9 achieved scalar invariance, while the GAD-7 yielded metric invariance. The somatic and non-somatic latent dimensions of depression were compared and showed no significant difference between male and female groups. The positive relationship between depression and anxiety was also not significantly different between the two groups. While the PHQ-9 symptoms formed three classes in the male group, and four classes in the female group, the GAD-7 had the same number of classes and a similar pattern between the two groups. These findings hold implications for the measurement, assessment and understanding of symptom manifestation and distribution, as well as the treatment of depression and anxiety in South Africa. (shrink)

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.

This article assesses the shortcomings and the disconnectedness of the current academic theological education in South Africa. It offers a brief history to provide a guiding principle for academic theology. It then proceeds to show the current disconnect and challenges between academic theology and the church, with its primary focus on academic theology. Drawing on original research and reflection on these responses, commodification, euro-centricity and rankings are seen as three traps of modern academics. These three areas have distorted the true (...) content of theological reflection. This article thus clearly highlights the current problem and motivates the need for academic theology and the local church to reconnect with each other. With this article focussing on academic institutions, it calls for the academy to change not in nature but in content, and to draw its content for the local context. (shrink)

A theory T is called almost [MATHEMATICAL SCRIPT CAPITAL N]0-categorical if for any pure types p1,…,pn there are only finitely many pure types which extend p1 ∪…∪pn. It is shown that if T is an almost [MATHEMATICAL SCRIPT CAPITAL N]0-categorical theory with I = 3, then a dense linear ordering is interpretable in T.

We introduce the notion of a lovely pair of models of a simple theory T, generalizing Poizat's “belles paires” of models of a stable theory and the third author's “generic pairs” of models of an SU-rank 1 theory. We characterize when a saturated model of the theory TP of lovely pairs is a lovely pair , finding an analog of the nonfinite cover property for simple theories. We show that, under these hypotheses, TP is also simple, and we study forking (...) and canonical bases in TP. We also prove that assuming only that T is low, the existentially universal models of the universal part of a natural expansion TP+ of TP, are lovely pairs, and “simple Robinson universal domains”. (shrink)

A lay movement known as the Poor of Christ, incorrectly referred to as the Waldensians, started in the Roman Church in 1176 and rapidly spread through Europe despite severe persecution by the church. Through their values and methods, they impacted on the communities where they were present. This article aims to show their role and contributions indicating its impact on the 16th century Reformation and relevance for the church today. It does this by examining selected themes from the Poor of (...) Christ: their life-transforming faith, the strengthening of laity, the equal role of women in ministry, the importance of teaching and living the faith, a focused rather than fenced theology and loyalty to the one Church of Christ. The article also draws reference to the COVID-19 pandemic and explores what we can learn from the Poor of Christ in addressing this situation.Contribution: This article puts the spotlight on an overlooked lay movement that contributed significantly in preparing the climate for the 16th Century Reformation. Their example shows that the real strength of church is through members empowered to holistically live their faith. The article shows the current relevance of their values and methods. (shrink)

We prove (Proposition 2.1) that if $\mu$ is a generically stable measure in an NIP (no independence property) theory, and $\mu(\phi(x,b))=0$ for all $b$ , then for some $n$ , $\mu^{(n)}(\exists y(\phi(x_{1},y)\wedge \cdots \wedge\phi(x_{n},y)))=0$ . As a consequence we show (Proposition 3.2) that if $G$ is a definable group with fsg (finitely satisfiable generics) in an NIP theory, and $X$ is a definable subset of $G$ , then $X$ is generic if and only if every translate of $X$ does not (...) fork over $\emptyset$ , precisely as in stable groups, answering positively an earlier problem posed by the first two authors. (shrink)

In this paper we construct a non-CM-trivial stable theory in which no infinite field is interpretable. In fact our theory will also be trivial and ω-stable, but of infinite Morley rank. A long term aim would be to find a nonCM-trivial theory which has finite Morley rank (or is even strongly minimal) and does not interpret a field. The construction in this paper is direct, and is a “3-dimensional” version of the free pseudoplane. In a sense we are cheating: the (...) original point of the notion ofCM-triviality was to describe the geometry of a strongly minimal set, or even of a regular type. In our example, non-CM-triviality will come from the behaviour of three orthogonal regular types.A stable theory is said to beCM-trivial if wheneverA⊆Band acl(Ac) ∩ acl(B) = acl(A) inTeq, then Cb(stp(c/A)) ⊆ Cb(stp(c/B)). ( An infinite stable field will not beCM-trivial.) The notion is due to Hrushovski [3], where he gave several equivalent definitions, as well as showing that his new strongly minimal sets constructed “ab ovo” wereCM-trivial. The notion was studied further in [6] where it was shown thatCM-trivial groups of finite Morley rank are nilpotent-by-finite. These results were generalized in various ways to the superstable case in [8]. (shrink)

It is known Hrushovski and Pillay that a group G definable in the field \ of p-adic numbers is definably locally isomorphic to the group \\) of p-adic points of a algebraic group H over \. We observe here that if H is commutative then G is commutative-by-finite. This shows in particular that any one-dimensional group G definable in \ is commutative-by-finite. This result extends to groups definable in p-adically closed fields. We prove our results in the more general (...) context of geometric structures. (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)

The ongoing digital transformation (DT) in our world has not only brought change to secular systems but also to how things are done in the mission and ministry of the Christian faith. Although before the coronavirus disease 2019 (COVID-19) pandemic, some churches were interacting with DT, the post COVID-19 experience has shown that many more Christian organisations, especially, the African Neo-Pentecostal Churches (ANPC), have carved their niche in the digital space. With South Africa and Nigeria in view, this article investigated (...) how the ANPC interacts with DT to improve missional work and the possible implications of doing so in the last 15 years. In the process, three strategies were applied. Firstly, a literature review and some online investigations were undertaken to articulate how the ANPC have engaged with DT in the past. Secondly, as a primary source, a quantitative empirical survey was conducted through data collection from 109 Neo-Pentecostal leaders to assess the current development in the use of digital technology (DTECH) by the ANPC. Thirdly, the article applied Albert Humphrey’s Strengths, Weaknesses, Opportunities and Threats (SWOT) framework to analyse the implications of the interaction. The outcome showed that the strengths and opportunities of digital interaction such as the wider and easier reach of audience, online training, online giving, improved sound and light technology and advertisements among others are helpful in missional assignments. Nevertheless, some observed Weaknesses and Threats need attention. Most importantly, the lesson from the findings is that ‘techno church’ practices come with gains; however, interaction with DT is not sufficient for missional services.Contribution: Aligning with HTS Teologiese Studies/Theological Studies’s focus and scope, this article contributed to an interdisciplinary religious aspect of research as it brought forward the interactions of African Pentecostalism, Historical Theology and Information Technology. (shrink)

We define the notion has the NIP (not the independence property) in A, where A is a subset of a model, and give some equivalences by translating results from function theory. We also discuss the number of coheirs when A is not necessarily countable, and revisit the notion “ has the NOP (not the order property) in a model M”.

This study investigated the effects of music therapy and pictorial illustration on the attention span of children with mild intellectual difficulties. A pre-test, post-test and control group quasi-experimental research design was used with a sample of children diagnosed with mild intellectual disability from three special schools in Ibadan, Nigeria. Fifty children were randomly selected and assigned to one of three groups: music therapy, pictorial illustration, or control. Twenty-four sessions of music therapy and pictorial illustration classes were held with the experimental (...) group only. The Moss Attention Rating Scale was used before and after the intervention to collect data on participants’ attention span. Analysis of Covariance indicated that there was a significant statistical difference between pre-test and post-test results of the two groups. The estimated marginal means of post-attention span by treatment indicated that pictorial illustration had the highest post-attention span score, followed by music therapy, while post-attention span score for the control group was the least. Based on the findings, it is recommended that teachers, caregivers, and parents of children with mild intellectual disability adopt pictorial illustration and music therapy as teaching strategies to enhance their attention span. (shrink)

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 aboutCM-triviality, asking in particular whether a structure of finite rank, which is “coordinatized” byCM-trivial types of rank 1, is itselfCM-trivial. (Actually Wagner (...) worked in a slightly more general context, adapting the definitions to a certain “local” framework, in which algebraic closure is replaced byP-closure, forPsome 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 andCM-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. (shrink)

This article traces the historical impact of the church in transforming, developing and changing society. It looks at how the church in selected periods in history, mainly in the reformation era, worked towards the transformation of society and communities. Extracting from these the author attempts to show how this can be applied to the church in the South African context. Further, the author explores the changing dynamics and characteristics of being church today and establishes that it is not so much (...) about what the church believes, but in what it does that matters most in this day and age. The author, with the latter thought in mind, expresses that community transformation has to be an integral part of the ministry of the church today. (shrink)

The Accra Confession was formulated and adopted by the World Alliance of Reformed Churches in Accra, Ghana, in 2004. This article traces the historical development of the Accra Confession and its continued impact on the present. It aims to show the confession as a Reformed response to 'empire'. The article explores the content, debates and challenges the Accra Confession offers to the notion of empire.

We give a characterisation of forking in the context of simple theories in terms of the fundamental order.

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)

We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures (...) in the special cases when G is 1-dimensional, and when G is definably simple. (shrink)

As clinicians, this case1 raises both personal and professional challenges. A key issue is who carries legal parental responsibility for the difficult decisions that may be required around life-sustaining care in baby T. Medicolegally, we understand that the surrogate mother holds legal parental responsibility for baby T until this can be transferred to the intended parents.2 But this process can take many months to complete, after the birth of baby. As M is now critically ill and unable to engage in (...) any discussion around the care of her baby, who becomes the legal guardian of the baby, for complex decision making that involves either reorientation of care away from intensive care with the inevitable consequence of death in this extremely premature baby, or continuing this life-sustaining treatment with a high likelihood for major neonatal morbidity and longer term disability? Does it pass onto one of the IPs as the genetic father, or does this responsibility fall on the neonatal consultant managing the baby in intensive care? And if an antenatal surrogacy agreement had existed, is it legally binding for inclusion of the IPs perspectives in the decision-making, especially if our medical decision making turns out to be at discord with the beliefs and wishes of the IPs for baby T? Box 1 ### Current surrogacy proceedings in the UK.2 6–8 The Law Commission UK5 is currently reviewing the decision on whether legal parenthood to intended parents can be effected immediately at the time of birth of the baby. Parental orders. (shrink)