On the injectivity of Boolean algebras

summary:The functor taking global elements of Boolean algebras in the topos \text{\bold{Sh}\frak B} of sheaves on a complete Boolean algebra $\frak B$ is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in $\frak B$-valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts

Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory

In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters

Epimorphisms and maximal covers in categories of compact spaces

[EN] The category C is "projective complete"if each object has a projective cover (which is then a maximal cover). This property inherits from C to an epireflective full subcategory R provided the epimorphisms in R are also epi in C. When this condition fails, there still may be some maximal covers in R. The main point of this paper is illustration of this in compact Hausdorff spaces with a class of examples, each providing quite strange epimorphisms and maximal covers. These examples are then dualized to a category of algebras providing likewise strange monics and maximal essential extensions.Banaschewski, B.; Hager, A. (2013). Epimorphisms and maximal covers in categories of compact spaces. Applied General Topology. 14(1):41-52. doi:10.4995/agt.2013.1616.SWORD4152141H. Cook, Continua which admit only the identity mapping onto non-degenerate subcontinua, Fund. Math. 60 (1966) 214–249.R. Engelking, General Topology, Heldermann 1989.A. Gleason, Projective topological spaces, Ill. J. Math. 2 (1958), 482–489.L. Gillman and M. Jerison, Rings of Continuous Functions, Springer-Verlag 1976.HAGER, A. W. (1989). Minimal Covers of Topological Spaces. Annals of the New York Academy of Sciences, 552(1 Papers on Gen), 44-59. doi:10.1111/j.1749-6632.1989.tb22385.xHager, A. W., & Martinez, J. (1998). Singular Archimedean lattice-ordered groups. Algebra Universalis, 40(2), 119-147. doi:10.1007/s000120050086A. Hager and L. Robertson, Representing and ringifying a Riesz space, Symp. Math. XXI (1977), 411–431.H. Herrlich and G. Strecker, Category Theory, Allyn and Bacon 1973.Kennison, J. F. (1965). Reflective functors in general topology and elsewhere. Transactions of the American Mathematical Society, 118, 303-303. doi:10.1090/s0002-9947-1965-0174611-9Porter, J. R., & Woods, R. G. (1988). Extensions and Absolutes of Hausdorff Spaces. doi:10.1007/978-1-4612-3712-9V. Trnková, Non-constant continuous mappings of metric or compact Hausdorff spaces, Comm. Math. Univ. Carol. 13 (1972), 283–295

A new look at pointfree metrization theorems

summary:We present a unified treatment of pointfree metrization theorems based on an analysis of special properties of bases. It essentially covers all the facts concerning metrization from Engelking [1] which make pointfree sense. With one exception, where the generalization is shown to be false, all the theorems extend to the general pointfree context

Locally class-presentable and class-accessible categories

We generalize the concepts of locally presentable and accessible categories. Our framework includes such categories as small presheaves over large categories and ind-categories. This generalization is intended for applications in the abstract homotopy theory

On maximal immediate extensions of valued division algebras

We show an extension theorem for strictly contracting bilinear mappings into a spherically complete valued vector space and we apply this result to prove that every maximal valued division algebra having the same characteristic as its residue division algebra is spherically complete

Duration discrimination in the range of milliseconds and seconds in children with ADHD and their unaffected siblings

Background Detecting genetic factors involved in attention deficit hyperactivity disorder (ADHD) is complicated because of their small effect sizes and complex interactions. The endophenotype approach eases this by coming closer to the relevant genes. Different aspects of temporal information processing are known to be affected in ADHD. Thus, some of these aspects could represent candidate endophenotypes for ADHD. Method Fifty-four sib-pairs with at least one child with ADHD and 40 control children aged 6-18 years were recruited and asked to perform two duration discrimination tasks, one with a base duration of 50 ms on automatic timing and one with a base duration of 1000 ms on cognitively controlled timing. Results Whereas children with ADHD, but not their unaffected siblings, were impaired in discrimination of longer intervals, both groups were impaired in discriminating brief intervals. Furthermore, a significant within-family correlation was found for discrimination of brief intervals. Task performances of subjects of the control group correlated with individual levels of hyperactivity/impulsivity for discrimination of brief intervals, but not of longer intervals. Conclusions Cognitively controlled and also automatic processes of temporal information processing are impaired in children with ADHD. Discrimination of longer intervals appears as a typical ‘disease marker' whereas discrimination of brief intervals shows up as a ‘vulnerability marker'. Discrimination of brief intervals was found to be familial and linked to levels of hyperactivity/impulsivity. Taken together, discrimination of brief intervals represents a candidate endophenotype of ADH