Applied Categorical Structures – Official Blog

by Kasch, Friedrich

Extending the notion of von Neumann regular elements in a ring R, a homomorphism f:A→M between R-modules is said to be regular if there exists some g:M → A such that fgf = f. In this paper we report about the use of this term in module theory.

DOI: 10.1007/s10485-007-9068-5
Online Date: 3/29/2007
Print publication date: 4/1/2008
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by Chen, Guiyun; Shi, Wujie

It is an interesting topic to determine the structure of a finite group which has a given number of elements of maximal order. In this article, the author classified finite groups with 30 elements of maximal order.

DOI: 10.1007/s10485-007-9067-6
Online Date: 3/29/2007
Print publication date: 4/1/2008
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by Gerlo, A.

In a topological construct $\mathcal{X}$ endowed with a proper $(\mathcal{E}, \mathcal{M})$-factorization system and a concrete functor $\Lambda:\mathcal{X}\rightarrow \mathbf{Prap}$, we study $\mathcal{F}_{\Lambda}$-compactness and $\mathcal{F}_{\Lambda}$-Hausdorff separation, where $\mathcal{F}_{\Lambda}$ is a class of “closed morphisms” in the sense of Clementino et al. (A functional approach to general topology. In: Categorical Foundations. Encyclopedia of Mathematics and Its Applications, vol. 97, pp. 103–163. Cambridge University Press, Cambridge, 2004), determined by Λ. In particular, we point out under which conditions on Λ, the notion of $\mathcal{F}_{\Lambda}$-compactness of an object $\underline{X}$ of $\mathcal{X}$ coincides with 0-compactness of the image $\Lambda(\underline{X})$ in . Our results will be illustrated by some examples: except for some well-known ones, like b-compactness of a topological space, we also capture some compactness notions that were not considered before in the literature. In particular, we obtain a generalization of b-compactness to the setting of approach spaces. This notion is shown to play an important role in the study of uniformizability.

DOI: 10.1007/s10485-007-9066-7
Online Date: 3/29/2007
Print publication date: 8/1/2008
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by Ball, Richard N.; Hager, Anthony W.; Walters-Wayland, Joanne

We characterize monomorphisms in , the category of regular Lindelöf locales. Though somewhat complicated, the characterization is intrinsic in the sense that it refers only to the properties of the morphism itself, rather than to properties of some lifting of it to a distant category.

DOI: 10.1007/s10485-007-9070-y
Online Date: 3/29/2007
Print publication date: 4/1/2007
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by Ma, Jingjing; Redfield, R. H.

Most results on the structure of lattice-ordered fields require that the field have a positive multiplicative identity. We construct a functor from the category of lattice-ordered fields with a vector space basis of d-elements to the full subcategory of such fields with positive multiplicative identities. This functor is a left adjoint to the forgetful functor and, in many cases, allows us to write all compatible lattice orders in terms of orders with positive multiplicative identities. We also use these results to characterize algebraically those extensions of totally ordered fields that have vℓ-bases of d-elements.

DOI: 10.1007/s10485-007-9063-x
Online Date: 3/6/2007
Print publication date: 4/1/2007
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