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

Ball, Richard N. ; Hager, Anthony W.

Commentationes Mathematicae Universitatis Carolinae, Tome 54 (2013) no. 2, pp. 141-157

Résumé

“The kernel functor” $W\xrightarrow{k}\operatorname{LFrm}$ 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{\varphi }\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 $B(X)\overset{*}\leq E\overset{*}\leq \beta C(X)$ 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.

Détail

Classification : 06D22, 06F20, 18A20, 18A40, 26A21, 28A05, 54C30, 54C40, 54C50
Keywords: Baire functions; archimedean lattice-ordered group; Lindelöf frame; monomorphism