A pointfree version of remainder preservation
| Autor: | Themba Dube, Inderasan Naidoo |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2013 |
| Předmět: |
Frame
remainder preservation Stone-v{Cech} compactification realcompact coreflection proper map lax proper map Mathematics QA1-939 |
| Zdroj: | Categories and General Algebraic Structures with Applications, Vol 1, Iss 1, Pp 27-58 (2013) |
| Druh dokumentu: | article |
| ISSN: | 2345-5853 2345-5861 |
| Popis: | Recall that a continuous function $fcolon Xto Y$ between Tychonoff spaces is proper if and only if the Stone extension $f^{beta}colon beta Xtobeta Y$ takes remainder to remainder, in the sense that $f^{beta}[beta X-X]subseteq beta Y-Y$. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism $hcolon Lto M$ to be $beta$-proper, $lambda$-proper or $upsilon$-proper in case the lifted homomorphism $h^{beta}colonbeta Ltobeta M$, $h^{lambda}colonlambda Ltolambda M$ or $h^{upsilon}colonupsilon Ltoupsilon M$ takes remainder to remainder. These turn out to be weaker forms of properness. Indeed, every proper homomorphism is $beta$-proper, every $beta$-proper homomorphism is $lambda$-proper, and $lambda$-properness is equivalent to $upsilon$-properness. A characterization of $beta$-proper maps in terms of pointfree rings of continuous functions is that they are precisely those whose induced ring homomorphisms contract free maximal ideals to free prime ideals. |
| Databáze: | Directory of Open Access Journals |
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