TY - JOUR
ID - 4264
TI - A pointfree version of remainder preservation
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Dube, Themba
AU - Naidoo, Inderasan
AD - Department of Mathematical Sciences, University of South Africa, P.O. Box 392, 0003 Unisa, South
Africa.
AD - Department of Mathematical Sciences, University of South
Africa, P.O. Box 392, 0003 Unisa, South Africa.
Y1 - 2013
PY - 2013
VL - 1
IS - 1
SP - 27
EP - 58
KW - frame
KW - remainder preservation
KW - Stone-v{Cech} compactification
KW - regular Lindel"{o}f coreflection
KW - realcompact coreflection
KW - proper map
KW - lax proper map
DO -
N2 - 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.
UR - https://cgasa.sbu.ac.ir/article_4264.html
L1 - https://cgasa.sbu.ac.ir/article_4264_91ce60eb77415d9197885588177906a7.pdf
ER -