In [4] the authors studied the residue field of a model M of IΔ0 + Ω1 for the principal ideal generated by a prime p. One of the main results is that M/ has a unique extension of each finite degree. In this paper we are interested in understanding the structure of any quotient field of M, i.e. we will study the quotient M/I for I a maximal ideal of M. We prove that any quotient field of (...) M satisfies the property of having a unique extension of each finite degree. We will use some of Cherlin's ideas from [3], where he studies the ideal theory of non standard algebraic integers. (shrink)

We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere hyperdoctrine for which we describe a notion of quotient completion. That notion includes the exact completion on a category with weak finite limits as an instance as well as examples from type theory that fall apart from this.

The standard construction of quotient spaces in topology uses full separation and power sets. We show how to make this construction using only the predicative methods available in constructive type theory and constructive set theory.

Abstract.A keystone of the theory of noncommutative noetherian rings is the theorem that establishes a necessary and sufficient condition for a given ring to have a quotient ring. We trace the development of this theorem, and its applications, from its first version for noncommutative domains in the 1930s to Goldie’s theorems in the late 1950s.

Let ${\mathbb{B}}$ and ${\mathbb{C}}$ be Boolean algebras and ${e: \mathbb{B}\rightarrow \mathbb{C}}$ an embedding. We examine the hierarchy of ideals on ${\mathbb{C}}$ for which ${ \bar{e}: \mathbb{B}\rightarrow \mathbb{C} / \fancyscript{I}}$ is a regular (i.e. complete) embedding. As an application we deal with the interrelationship between ${\fancyscript{P}(\omega)/{{\rm fin}}}$ in the ground model and in its extension. If M is an extension of V containing a new subset of ω, then in M there is an almost disjoint refinement of the family ([ω]ω) V (...) . Moreover, there is, in M, exactly one ideal ${\fancyscript{I}}$ on ω such that ${(\fancyscript{P}(\omega)/{{\rm fin}})^V}$ is a dense subalgebra of ${(\fancyscript{P}(\omega)/\fancyscript{I})^M}$ if and only if M does not contain an independent (splitting) real. We show that for a generic extension V[G], the canonical embedding $$\fancyscript{P}^V(\omega){/}{\rm fin}\hookrightarrow \fancyscript{P}(\omega){/}(U(Os)(\mathbb{B}))^G$$ is a regular one, where ${U(Os)(\mathbb{B})}$ is the Urysohn closure of the zero-convergence structure on ${\mathbb{B}}$. (shrink)

Quine’s “New Foundations” (NF) was first presented in Quine [1937] and later on in Quine [1963]. Ernst Specker [1958, 1962], building upon a previous result of Ehrenfeucht and Mostowski [1956], showed that NF is consistent if and only if there is a model of the Theory of Negative (and positive) Types (TNT) with full extensionality that admits of a “shifting automorphism,” but the existence of a such a model remains an open problem.

We solve two problems from a work of Haskel and Pillay concerning maximal stable quotients of groups ∧-definable in NIP theories. The first result says that if G is a ∧-definable group in a distal theory, then Gst=G00 (where Gst is the smallest ∧-definable subgroup with G∕Gst stable, and G00 is the smallest ∧-definable subgroup of bounded index). In order to get it, we prove that distality is preserved under passing from T to the hyperimaginary expansion Theq. The second result (...) is an example of a group G definable in a nondistal NIP theory for which G=G00, but Gst is not an intersection of definable groups. Our example is a saturated extension of (R,+,[0,1]). Moreover, we make some observations on the question whether there is such an example which is a group of finite exponent. We also take the opportunity and give several characterizations of stability of hyperdefinable sets involving continuous logic. (shrink)

We study an extensive connection between quotient forcings of Borel subsets of Polish spaces modulo a σ-ideal and quotient forcings of subsets of countable sets modulo an ideal.

This monograph examines the meaning of religion.

A recursive algebra is a structure for which A is a recursive set of numbers and the Fi are uniformly recursive operations. We define an r.e. quotient algebra to be the quotient by an r.e. congruence .We say that is recursively stable among r.e. quotient algebras if, for each r.e. quotient algebra and each isomorphism from onto ′, the set {a,baA,bB and =[b]′} is r.e.We shall consider examples of recursive stability. Then, assuming that has a recursive (...) existential diagram, we show that the task of determining its recursive stability among r.e. quotient algebras can be reduced to a more routine consideration of syntactical conditions. To this result, we provide a counter-example which demonstrates the necessity of having a recursive existential diagram. This result and counter-example are on similar lines to ones obtained by Goncharov , for the recursive stability of recursive structures. (shrink)

We prove a converse to one of the theorems from [F], giving a description in terms of Turing complexity of sets which can be coded into recursive and r.e. quotient Boolean algebras.

For a NIP theory T, a sufficiently saturated model ${\mathfrak C}$ of T, and an invariant (over some small subset of ${\mathfrak C}$ ) global type p, we prove that there exists a finest relatively type-definable over a small set of parameters from ${\mathfrak C}$ equivalence relation on the set of realizations of p which has stable quotient. This is a counterpart for equivalence relations of the main result of [2] on the existence of maximal stable quotients of type-definable (...) groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in [3]. (shrink)

We show that the 'tail' of a doubly homogeneous chain of countable cofinality can be recognized in the quotient of its automorphism group by the subgroup consisting of those elements whose support is bounded above. This extends the authors' earlier result establishing this for the rationals and reals. We deduce that any group is isomorphic to the outer automorphism group of some simple lattice-ordered group.

J. Michael Dunn’s Theorem in 3-Valued Model Theory and Graham Priest’s Collapsing Lemma provide the means of constructing first-order, three-valued structures from classical models while preserving some control over the theories of the ensuing models. The present article introduces a general construction that we call a Dunn–Priest quotient, providing a more general means of constructing models for arbitrary many-valued, first-order logical systems from models of any second system. This technique not only counts Dunn’s and Priest’s techniques as special cases, (...) but also provides a generalized Collapsing Lemma for Priest’s more recent plurivalent semantics in general. We examine when and how much control may be exerted over the resulting theories in particular cases. Finally, we expand the utility of the construction by showing that taking Dunn–Priest quotients of a family of structures commutes with taking an ultraproduct of that family, increasing the versatility of the tool. (shrink)

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)

Open peer commentary on the article “A Cybernetic Approach to Contextual Teaching and Learning” by Philip Baron. Upshot: I present a new concept, the “In-sight Quotient,” which encourages learners to go deeper into the subconscious structure of learning. This enables them to manage the quality of their own individual learning experience. The lecturer becomes no longer a teacher but a “neuro-programmer.”.

In this paper, the following are proved:Theorem A. The quotient algebra ${\cal P} (\kappa )/I$ is complete if and only if the only non-trivial I -closed ideals extending I are of the form $I\lceil A$ for some $A\in I^+$ .Theorem B. If $\kappa$ is a stationary cardinal, then the quotient algebra ${\cal P} (\kappa )/ NS_\kappa$ is not complete.Corollary. (1) If $\kappa$ is a weak compact cardinal, then the quotient algebra ${\cal P} (\kappa )/NS_\kappa$ is not complete.(2) (...) If $\kappa$ bears $\kappa$ -saturated ideal, then the quotient algebra ${\cal P} (\kappa )/NS_\kappa$ is not complete.Theorem C. Assume that $\kappa$ is a strongly compact cardinal, I is a non-trivial normal $\kappa$ -complete ideal on $\kappa$ and B is an I -regular complete Boolean algebra. Then if ${\cal P} (\kappa )/I$ is complete, it is B -valid that for some $A\subseteq\check\kappa$ , ${\cal P} (\kappa )/({\bf J}\lceil A)$ is complete, where J is the ideal generated by $\check I$ in $V^B$ .Corollary. Let M be a transitive model of ZFC and in M , let $\kappa$ be a strongly compact cardinal and $\lambda$ a regular uncountable cardinal less than $\kappa$ . Then there exists a generic extension M [ G ] in which $\kappa =\lambda^+$ and $\kappa$ carries a non-trivial $\kappa$ -complete ideal I which is completive but not $\kappa^+$ -saturated. (shrink)

A study of the elementary theory of quotients of symmetric groups is carried out in a similar spirit to Shelah . Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S on an infinite cardinal μ are all of the form Sκ = the subgroup consisting of elements whose support has cardinality 20, cƒ 20 < κ, 0 < κ < 20, and κ = 0, we make a further analysis of the first order (...) theory of Sλ/Sκ. introducing many-sorted second order structures , all of whose sorts have cardinality at most 20, and in terms of which we can completely characterize the elementary theory of the groups Sλ/Sκ. (shrink)

We study non-homogeneity of quotients of Prikry and tree Prikry forcings with non-normal ultrafilters over some natural distributive forcing notions.

We investigate homomorphisms of degree structures with various relations, functions and constants. Our main emphasis is on pseudolattices, i.e., partially ordered sets with a join operation and relations simulating the meet operation. We show that there are no finite quotients of the pseudolattice of degrees or of the pseudolattice of degrees 0′, but that many finite distributive lattices are pseudolattice quotients of the pseudolattice of computably enumerable degrees.

This conceptual article aims to shed light on the significance of human spiritual dimension in the process of human resource development. It suggests spiritual intelligence as the missing link in the process of human development that should be identified and considered as an important factor for developed and morally qualified human resources. Moreover, this article also uncovered the growing interest of spiritual intelligence and its implications for HRD. The interest in spiritual intelligence in the body of knowledge has been established (...) in depth which assures that SI embedded in the fabric of the fields of organizational behaviour and psychology. The article emphasizes the vital influence of SI on bottom line of the organization, whereby, human resources can meaningfully contribute towards organization with improved performance. The conventional strategy of organizations in flourishing their employees is more on enhancement of individual’s knowledge, skills, abilities and emotions control. However, despite this immense interest in human capital, organizations tend to fail to have holistic approach of human resource development which eventually affects their performance. Further, the article indicates that without considering spiritual development of employees, the logic and emotions could not be the only ample source for human beings to perform utmost standard performances. Therefore, based on this, we conclude that the emerging notion of human spiritual quotient and its multi-dimensions must be included in HRD initiatives in order to have holistic mechanism. (shrink)

This paper addresses the problem of why the conditions under which standard proofs of the Dutch Book argument proceed should ever be met. In particular, the condition that there should be odds at which you would be willing to bet indifferently for or against are hardly plausible in practice, and relaxing it and applying Dutch book considerations gives only the theory of upper and lower probabilities. It is argued that there are nevertheless admittedly rather idealised circumstances in which the classic (...) form of the Dutch Book argument is valid. (shrink)

Explains why hope is so important and offers to teach seven key factors that unleash hope and improve life circumstances.

The quotient ℝ/G of the additive group of the reals modulo a countable subgroup G does not admit nontrivial Baire measurable automorphisms.

The Intelligence Quotient (IQ) tests and the corresponding psychometric explanations dominate both the scientific and popular views about human intelligence. Though the IQ tests have been in currency for long, there exists a gap in what they are believed to measure and what they do. While the IQ tests index the quality of cognitive functioning in selected domains of mental repertoire, the applied settings often inflate their predictive value leading to an interpretive gap. The present article contends that studying (...) the influence of motivational and affective processes on cognitive functioning would help to evolve a more psychologically comprehensive account of the IQ tests and bridge the interpretive gap. To conclude, the article suggests possible future research directions that could strengthen the predictive value of the IQ tests. (shrink)

Keywords: Recursively Enumerable; Computably Enumerable; Friedberg Splitting; Congruence Relation; Ideal; Quotient; Lattice.

The idea that intelligence is embedded not only in a single brain network, but instead in a complex, well-optimized system of complementary networks, has led to the development of whole brain network analysis. Using graph theory to analyze resting-state functional MRI data, we investigated the brain graph networks (or brain networks) of high intelligence quotient (HIQ) children. To this end, we computed the “hub disruption index κ”, an index sensitive to graph network modifications. We found significant topological differences in (...) the integration and segregation properties of brain networks in HIQ compared to standard IQ children, not only for the whole brain graph, but also for each hemispheric graph, and for the homotopic connectivity. Moreover, two profiles of HIQ children, homogenous and heterogeneous, based on the differences between the two main IQ subscales (verbal comprehension index (VCI) and perceptual reasoning index (PRI)), were compared. Brain networks changes were more pronounced in the heterogeneous HIQ than in the homogeneous HIQ subgroups. Finally, we found significant correlations between the graph networks’ changes and the full-scale IQ (FSIQ), as well as the subscales VCI and PRI. Specifically, the higher the FSIQ the greater was the brain organization modification in the whole brain, the left hemisphere, and the homotopic connectivity. These results shed new light on the relation between functional connectivity topology and high intelligence, as well as on different intelligence profiles. (shrink)

A decade or so ago, graphic depictions of female circumcision came to define the heart of a campaign presumably aimed at sensitizing the world about the tragic consequences of the practice. At the height of the campaign, it was easy to assume that the prospect for meaningful change was dim. Evolving knowledge about the practice illuminates the bottom-line of issues and demonstrates the centrality of empowerment as an elimination strategy. Interrogating an acclaimed initiative that has successfully helped bring about the (...) abandonment of female circumccision in several places, this article analyzes the definitive role of attending to the full belly for anti-circumcision purposes. (shrink)

In this note we shall describe the lattice of the congruences on a balanced Ockham algebra with the pseudocomplementation whose quotient algebras are boolean. This is an extension of the result obtained by Rodrigues and Silva who gave a description of the lattice of congruences on an Ockham algebra whose quotient algebras are boolean.

This study investigated the effect of Mahjong, which is a table game played by three or four players and involves intellectual activity, on the intelligence quotient of children. The participants were children between the age of 6 and 15 years, and their IQ was assessed immediately after enrolling in children's Mahjong classes and 1 year after the enrollment using the Wechsler Intelligence Scale for Children Fourth Edition. Twenty children were included in the analysis. Their mean age at the time (...) of the initial evaluation was 9 years and 6 months. In addition, we conducted a 1-year post-examination. The change in the IQ of this group was compared to that of a historical control group with a similar age range and test–retest interval. The mean overall full-scale IQ of the 20 children during the initial and post-1-year examinations was 106.05 and 113.75, respectively, and showed a statistically significant increase. Based on the subscale index, the verbal comprehension index and processing speed index scores both showed a statistically significant increase from 100.6 to 106.75 and from 108.05 to 119.05, respectively. The PSI of the children included in the analysis showed a statistically significant increase compared to the historical control group. This study suggests that children who participate in Mahjong classes during their childhood have increased PSI scores of WISC-IV. (shrink)