Frame monomorphisms and a feature of the \(l\)-group of Baire functions on a topological space. (Q2856923)

The category whose objects are Archimedean \(\ell \)-groups with distinguished positive weak order unit and whose morphisms are \(\ell \)-group homomorphisms preserving the designated units is denoted by W. On the other hand, LFrm denotes the category of Lindelöf completely regular frames with frame maps. There is a functor \(k : \mathrm{W}\to \text{LFrm}\) which sends a W-object \(A\) to the Lindelöf frame \(kA\) of convex sub-\(\ell \)-groups (also called W-kernels) of \(A\), and a W-map \(\varphi \: A\to B\) to the frame homomorphism \(k\varphi \: kA\to kB\) which sends any \(I\in kA\) to the smallest W-kernel containing \(\varphi [I]\). It is a left adjoint whose right adjoint sends a Lindelöf frame \(L\) to \(C(L)=\{f\: \mathfrak {O}\mathbb {R}\to L\mid f\in \text{Frm}\}\). In this paper the authors call a morphism \(\varphi \in \text{W}\) kernel-injective (resp., kernel-surjective, kernel-preserving) if the morphism \(k\varphi \) is one-one (resp., onto, one-one and onto). The kernel-injective embeddings of W form the core of the paper. In the fourth section the authors undertake a thorough and very careful study of kernel-injective epicompletions. Baire functions are taken up in the last section in conjunction with kernel-injective epicompletion of the underlying \(\ell \)-group of the ring of real-valued continuous functions on a Tychonoff space. The main theorem (among many other very substantive results) appears in this section.