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Compactifications, C(X) and ring epimorphisms

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W.D. Burgess and R. Raphael

Given a topological space $X$, $K(X)$ denotes the upper semi-lattice of its
(Hausdorff) compactifications. Recent studies have asked when, for $\alpha X \in
K(X)$, the restriction homomorphism $\rho : C(\alpha X) \to C(X)$ is an
epimorphism in the category of commutative rings. This article continues this
study by examining the sub-semilattice, $K_{epi}(X)$, of those compactifications
where $\rho$ is an epimorphism along with two of its subsets, and its complement
$K_{nepi}(X)$. The role of $K_z(X)\subseteq K(X)$ of those $\alpha X$ where $X$
is $z$-embedded in $\alpha X$, is also examined. The cases where $X$ is a
$P$-space and, more particularly, where $X$ is discrete, receive special
attention.

Keywords:
epimorphism, ring of continuous functions, category of rings, compactifications

2000 MSC:
18A20, 54C45, 54B40

*Theory and Applications of Categories,*
Vol. 16, 2006,
No. 21, pp 558-584.

http://www.tac.mta.ca/tac/volumes/16/21/16-21.dvi

http://www.tac.mta.ca/tac/volumes/16/21/16-21.ps

http://www.tac.mta.ca/tac/volumes/16/21/16-21.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/21/16-21.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/21/16-21.ps

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