Some Modifications of Hull Operators in Archimedean Lattice-Ordered Groups with Weak Unit

Abstract

${\textbf {W}}$ denotes the category, or class of algebras, in the title. A hull operator (ho) in ${\textbf {W}}$ is a function $\textbf{ho} {\textbf {W}}\overset{h}{\longrightarrow }\ {\textbf {W}}$ which can be called an essential closure operator. The family of these, denoted $\textbf{ho} {\textbf {W}}$, is a proper class and a complete lattice in the ordering as functions “pointwise", with the bottom ${{\,\textrm{Id}\,}}_{{\textbf {W}}}$ and top Conrad’s essential completion e. Other much studied hull operators are the divisible hull, maximum essential reflection, projectable hull, and Dedekind completion. This paper is the authors’ latest efforts to understand/create structure in $\textbf{ho} {\textbf {W}}$ through the nature of the interaction that an h might have with B, the bounded monocoreflection in ${\textbf {W}}$ (e.g., Bh=hB). We define and investigate three functions $\textbf{ho} {\textbf {W}}\longrightarrow \textbf{ho} {\textbf {W}}$ which stand in the relation

$$\begin{aligned} {{\,\textrm{Id}\,}}_{{\textbf {W}}} \le \overline{\alpha }(h) \le \overline{\lambda }(h) \le \overline{c}(h) \le h. \end{aligned}$$

General properties that an h might have, and particular choices of h, show various assignments of < and $=$ in this chain.