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Start Over Creator Ball, Richard N.

1. Flow compactifications of nondiscrete monoids, idempotents and Hindman’s theorem

Creator:
Ball, Richard N. and Hagler, James N.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
flow, Stone-Čech compactification, and Hindman’s theorem
Language:
English
Description:
We describe the extension of the multiplication on a not-necessarily-discrete topological monoid to its flow compactification. We offer two applications. The first is a nondiscrete version of Hindman’s Theorem, and the second is a characterization of the projective minimal and elementary flows in terms of idempotents of the flow compactification of the monoid.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. Monomorphisms in spaces with Lindelöf filters

Creator:
Ball, Richard N. and Hager, Anthony W.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
compact Hausdorff space, Lindelöf set, and monomorphism
Language:
English
Description:
$\mathbf{SpFi}$ is the category of spaces with filters: an object is a pair $(X,\mathcal{F}) $, $X$ a compact Hausdorff space and $\mathcal{F}$ a filter of dense open subsets of $X$. A morphism $f\: (Y,\mathcal{G}) \rightarrow (X,\mathcal{F}) $ is a continuous function $f\: Y\rightarrow X$ for which $f^{-1}(F) \in \mathcal{G}$ whenever $F\in \mathcal{F}$. This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shall specifically address only one of these connections here in an appendix. Now we study the categorical monomorphisms in $\mathbf{SpFi}$. Of course, these monomorphisms need not be one-to-one. For general $\mathbf{SpFi}$ we derive a criterion for monicity which is rather inconclusive, but still permits some applications. For the category $\mathbf{LSpFi}$ of spaces with Lindelöf filters, meaning filters with a base of Lindelöf, or cozero, sets, the criterion becomes a real characterization with several foci ($C(X) $, Baire sets, etc.), and yielding a full description of the monofine coreflection and a classification of all the subobjects of a given $(X,\mathcal{F}) \in \mathbf{LSpFi}$. Considerable attempt is made to keep the discussion “topological,” i.e., within $\mathbf{SpFi}$, and to not get involved with, e.g., frames. On the other hand, we do not try to avoid Stone duality. An appendix discusses epimorphisms in archimedean $\ell $-groups with unit, roughly dual to monics in $\mathbf{LSpFi}$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public