A pointfree version of remainder preservation | ||
| Categories and General Algebraic Structures with Applications | ||
| مقاله 4، دوره 1، شماره 1، اسفند 2013، صفحه 27-58 اصل مقاله (692.71 K) | ||
| نوع مقاله: Research Paper | ||
| نویسندگان | ||
| Themba Dube؛ Inderasan Naidoo | ||
| Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Unisa, South Africa. | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| frame؛ remainder preservation؛ Stone-v{Cech} compactification؛ regular Lindel"{o}f coreflection؛ realcompact coreflection؛ proper map؛ lax proper map | ||
| مراجع | ||
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