Frame monomorphisms and a feature of the $l$-group of Baire functions on a topological space

Ball, Richard N., and Hager, Anthony W.. "Frame monomorphisms and a feature of the $l$-group of Baire functions on a topological space." Commentationes Mathematicae Universitatis Carolinae 54.2 (2013): 141-157. <http://eudml.org/doc/252540>.

@article{Ball2013,
abstract = {“The kernel functor” $W\xrightarrow\{\}\operatorname\{LFrm\}$ from the category $W$ of archimedean lattice-ordered groups with distinguished weak unit onto LFrm, of Lindelöf completely regular frames, preserves and reflects monics. In $W$, monics are one-to-one, but not necessarily so in LFrm. An embedding $\varphi \in W$ for which $k\varphi $ is one-to-one is termed kernel-injective, or KI; these are the topic of this paper. The situation is contrasted with kernel-surjective and -preserving (KS and KP). The $W$-objects every embedding of which is KI are characterized; this identifies the $\operatorname\{LFrm\}$-objects out of which every monic is one-to-one. The issue of when a $W$-map $G\xrightarrow\{\}\cdot $ is KI is reduced to when a related epicompletion of $G$ is KI. The poset $EC(G)$ of epicompletions of $G$ is reasonably well-understood; in particular, it has the functorial maximum denoted $\beta G$, and for $G=C(X)$, the Baire functions $B(X)\in EC(C(X))$. The main theorem is: $E\in EC(C(X))$ is KI iff $B(X)\overset\{*\}\{\le \}E\overset\{*\}\{\le \}\beta C(X)$ in the order of $EC(C(X))$. This further identifies in a concrete way many $\operatorname\{LFrm\}$-monics which are/are not one-to-one.},
author = {Ball, Richard N., Hager, Anthony W.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Baire functions; archimedean lattice-ordered group; Lindelöf frame; monomorphism; Baire functions; Archimedean lattice-ordered group; Lindelöf frame; monomorphism},
language = {eng},
number = {2},
pages = {141-157},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Frame monomorphisms and a feature of the $l$-group of Baire functions on a topological space},
url = {http://eudml.org/doc/252540},
volume = {54},
year = {2013},
}

TY - JOUR
AU - Ball, Richard N.
AU - Hager, Anthony W.
TI - Frame monomorphisms and a feature of the $l$-group of Baire functions on a topological space
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2013
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 54
IS - 2
SP - 141
EP - 157
AB - “The kernel functor” $W\xrightarrow{}\operatorname{LFrm}$ from the category $W$ of archimedean lattice-ordered groups with distinguished weak unit onto LFrm, of Lindelöf completely regular frames, preserves and reflects monics. In $W$, monics are one-to-one, but not necessarily so in LFrm. An embedding $\varphi \in W$ for which $k\varphi $ is one-to-one is termed kernel-injective, or KI; these are the topic of this paper. The situation is contrasted with kernel-surjective and -preserving (KS and KP). The $W$-objects every embedding of which is KI are characterized; this identifies the $\operatorname{LFrm}$-objects out of which every monic is one-to-one. The issue of when a $W$-map $G\xrightarrow{}\cdot $ is KI is reduced to when a related epicompletion of $G$ is KI. The poset $EC(G)$ of epicompletions of $G$ is reasonably well-understood; in particular, it has the functorial maximum denoted $\beta G$, and for $G=C(X)$, the Baire functions $B(X)\in EC(C(X))$. The main theorem is: $E\in EC(C(X))$ is KI iff $B(X)\overset{*}{\le }E\overset{*}{\le }\beta C(X)$ in the order of $EC(C(X))$. This further identifies in a concrete way many $\operatorname{LFrm}$-monics which are/are not one-to-one.
LA - eng
KW - Baire functions; archimedean lattice-ordered group; Lindelöf frame; monomorphism; Baire functions; Archimedean lattice-ordered group; Lindelöf frame; monomorphism
UR - http://eudml.org/doc/252540
ER -