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\begin{document}
\maketitle

\begin{abstract}We use a landmark result in the theory of Riesz spaces --- Freudenthal's 1936 Spectral Theorem --- to canonically represent any Archimedean  lattice-ordered  group $G$ with a strong unit  as a (non-separating) lattice-group of real valued continuous functions on  an appropriate $G$-indexed  zero-di\-men\-sional compactification $w_{G}Z_{G}$ of  its space  $Z_{G}$ of \emph{minimal} prime ideals. The two further ingredients needed to establish  this representation are the    Yosida representation of $G$ on its space $X_{G}$ of \emph{maximal} ideals, and the well-known continuous surjection of $Z_{G}$ onto $X_{G}$. We then establish our main result by showing that the inclusion-minimal extension of this representation of $G$ that separates the points of $Z_{G}$ --- namely, the sublattice subgroup of $\C{(Z_{G})}$ generated by the image of $G$ along with all  characteristic functions of clopen (closed and open) subsets of $Z_{G}$ which are determined by elements of $G$ --- is precisely the classical projectable hull of $G$. Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and  provides  the most direct and explicit construction of projectable hulls to date. Our  techniques do require the presence of a strong unit.
\end{abstract}

\section{Introduction}\label{s:intro}

In 1936, Freudenthal proved his well-known Spectral Theorem  \cite{Freudenthal1936} for Riesz spaces (real linear spaces with a compatible lattice order) with motivations coming from the theory of integration. (See \cite[40.2]{LuxZaan71} for a handbook treatment.)

 In its basic version, the theorem asserts that any element of a Riesz space $R$ with a strong unit $u$ and the principal projection property may be uniformly approximated,  in the norm that $u$ induces on $R$, by abstract characteristic functions --- ``components of  the unit  $u$''.  See Subsection \ref{ss:pol} for more details. Freudenthal's theorem led to a considerable amount of research on Riesz spaces and their generalisations, the lattice-ordered Abelian groups that concern us here, and which we  call \emph{$\ell$-groups} for short. (For background we refer to \cite{LuxZaan71, Darnel, Glass}.) One main line of research concentrated on extending one given structure $G$ to a minimal completion  that enjoys the principal projection property, where Freudenthal's theorem therefore applies. Such an  extension  is called the \emph{projectable hull} of $G$; please see Subsection \ref{ss:esspol} for details.

In 1973, Conrad \cite{Conrad73} proved the existence and  uniqueness of  projectable hulls of (a class of lattice-groups more general than) Archimedean $\ell$-groups, using his previous construction in \cite{Conrad71} of the \emph{essential closure}  of such an $\ell$-group --- the largest extension of the structure that is \emph{essential}, in the sense recalled in Subsection \ref{ss:esspol}. At about the same time, Chambless \cite{Chambless72} exhibited a different construction of the projectable hull based on direct limits; cf.\ also Bleier's construction in \cite{Bleier74}. Here we present a new construction of the projectable hull of an Archimedean $\ell$-group equipped with a strong order unit $u$ --- an element whose multiples eventually dominate any other element in the $\ell$-group --- that does not use direct limits, nor essential closures. Our construction exposes instead the intimate connection between projectable hulls and   zero-dimensional compactifications of  spectral spaces of minimal prime ideals. Closing the circle of ideas beginning with Freudenthal, to establish this connection we will need to apply his Spectral Theorem at a key step of the construction. We now recall some standard notions, and introduce notations that will remain in force throughout the paper.

Throughout,  all lattice-ordered groups are Abelian, and referred to simply as $\ell$-groups for short. We write $\U$ for the category whose typical object is a pair $(G,u)$, where $G$ is an  $\ell$-group that is \emph{Archimedean} --- whenever $0\leq ng\leq h$ for $h,g \in G$ and all integers $n\geq 1$, then $g=0$ ---
equipped with a distinguished (\emph{strong order}) \emph{unit} $u\in G$ --- an element $u\geq 0$   such that for all $g\in G$ there is an integer $n\geq 1$ such that $nu\geq g$. As morphisms, we take the   lattice-group homomorphisms (\emph{$\ell$-homomorphisms}) that are \emph{unital}, i.e.\ preserve the distinguished units.  It will transpire that our  techniques do require the existence of a strong unit, as opposed to the existence of a weak unit. Recall that a \emph{weak \textup{(}order\textup{)} unit} of $G$ is an element $w\in G^{+}$ such that for each $g \in G$,  $w\wedge |g|=0$ implies  $g=0$. Here,  $|g|:=(g\vee 0) + (-g\vee 0)$ is the \emph{absolute value} of $g$.

By an \emph{ideal} in an $\ell$-group we mean, as usual, a sublattice subgroup $I$ of $G$ that is \emph{order-convex}: whenever $a,c\in I$, $b\in G$, and $a\leq b\leq c$, then $b\in G$. Ideals are exactly the kernels of (unital) $\ell$-homomorphisms, i.e.\ the morphisms in the unrestricted category of abelian $\ell$-groups, and the usual homomorphism theorems hold. An ideal $\p$ of $G$ is \emph{prime} if, and only if,  it is proper ($\p\neq G$) and the quotient $\ell$-group $G/\p$ is totally ordered. A prime ideal is \emph{maximal}  if it is   inclusion-maximal --- equivalently, if $G/\p$ is non-trivial and \emph{simple}, i.e.\ it has no non-trivial proper ideals. Ideals that are inclusion-maximal are automatically prime. A prime ideal is  \emph{minimal} if it is inclusion-minimal. For any unital $\ell$-group  $(G,u)$, we denote by $\MaxS{G}$ the collection of its maximal (prime) ideals, and by $\MinS{G}$ the collection  of its minimal prime ideals.
We  topologize both $\MaxS{G}$ and $\MinS{G}$  using the \emph{spectral}, or \emph{Zariski} topology. The closed sets for this topology are given by  subsets of the form

\begin{align*}
\VV_{M}{(A)}:=\{\mathfrak{m}\in \MaxS{G} \,\mid\, \mathfrak{m}\supseteq A\}
\end{align*}

and

\begin{align*}
\VV_{m}{(A)}:=\{\mathfrak{p}\in \MinS{G} \,\mid\, \mathfrak{p}\supseteq A\},
\end{align*}

as $A$ ranges over arbitrary subsets of $G$. The resulting topological spaces  are called the \emph{maximal} and \emph{minimal prime spectrum} of $G$, respectively. The topology
on $\MaxS{G}$ is also called the \emph{hull-kernel} topology, because  it agrees with the classical  hull-kernel topology for rings of continuous functions \cite{GillmanJerison76}, \textit{mutatis mutandis}. Accordingly, we  call $\VV_M{(A)}$ (or $\VV_m{(A)}$)  the \emph{zero set} of $A$ (on the appropriate space), and its complement --- denoted by $\Ss_{M}{(A)}$ (or $\Ss_{m}{(A)}$) --- the \emph{support} of $A$. It is known \cite[10.2.1]{BKW} that the collections
\begin{equation*}
 \{\VV_{M}(\{g\})\}_{g\in G}\qquad\text{and}\qquad\{\VV_{m}(\{g\})\}_{g\in G}
\end{equation*}
form a closed base for $\MaxS{G}$ and $\MinS{G}$, respectively. Throughout we write $\VV_{M}{(g)}$ in place of $\VV_{M}{(\{g\})}$, and similarly for $\VV_{m}$, $\Ss_{M}$, and $\Ss_{m}$.

The space $\MaxS{G}$ is a Hausdorff space that is compact precisely because of the assumption that $G$ has a (strong) unit $u$; see \cite[10.2.5]{BKW}. The space $\MinS{G}$ is a Hausdorff zero-dimensional space that need not be compact  \cite[10.2.1]{BKW}. Whether it is or not has nothing to do with the existence of a strong unit, but rather with  complementation properties of the lattice $G^{+}:=\{g\in G \mid g\geq 0\}$, the \emph{positive cone} of $G$; see Section \ref{s:repG}.

\begin{notation}For the rest of this paper, we let $(G,u)$ denote a $\U$-object, and set

\begin{align*}
 X_{G}&:=\MaxS{G},\\
 Z_{G}&:=\MinS{G}.
\end{align*}

\end{notation}

If $X$ is any topological space, always at least Tychonoff, we write $\C{(X)}$ for the $\ell$-group of continuous functions $X\to\R$ under pointwise operations. If $X$ is compact, the function $1_{X}$ constantly equal to $1$ over $X$ is a strong unit of $\C{(X)}$ by the Extreme Value Theorem. We  always tacitly consider $\C{(X)}$ endowed with the distinguished unit $1_{X}$, and hence as a $\U$-object when $X$ is compact Hausdorff. The classical \emph{Yosida representation} \cite{Yosida41} of $(G,u)$ yields a canonical unital lattice-group embedding  $\,\widehat{\cdot}\colon G \hookrightarrow \C{(X_G)}$; details are recalled in Subsection \ref{ss:Yosida}.

It is well known that $Z_{G}$ is canonically thrown onto $X_G$, as follows.  Given $\a \in Z_{G}$, a standard argument \cite[27.4]{LuxZaan71} shows that, by virtue of the presence of the (strong) unit $u$,   there exists at least one $\m_{\a} \in X_G$ such that $\a \seq \m_{\a}$. Since the prime ideals of $G$ form a \emph{root system} under set-theoretic inclusion \cite[2.4.3]{BKW} --- that is, the set of prime ideals containing any given prime ideal is linearly ordered --- such an $\m_{\a}$ must be unique; in other words, the set $\up{\a}\cap X_G$ is a singleton, where $\up{\a}:=\{\b\seq G \mid \a \seq \b, \b \text{ a prime ideal}\}$. Hence there is a function

\begin{align}\label{eq:lambda}
\lambda \colon Z_{G}\twoheadrightarrow X_{G}
\end{align}

defined by

\begin{align}\label{eq:action}
\a \in Z_{G}\overset{\lambda}{\longmapsto} \m_{\a} \in X_G.
\end{align}

By \cite[10.2.5]{BKW}, the map $\lambda$ is continuous, and is a surjection by the standard fact that each prime ideal contains a minimal prime ideal \cite[2.4.5]{BKW}.

\begin{remark}The strong unit is crucial here. There exist non-trivial Archimedean  (and even Dedekind-complete) $\ell$-groups with a weak unit and no maximal ideal at all; see \cite[27.8]{LuxZaan71}.
Thus, while the  Yosida representation has an important extension to $\ell$-groups with a weak unit \cite{Yosida42, HagerRobertson77}, the existence of the map (\ref{eq:lambda}), a key ingredient to our construction, does require a strong unit. A generalisation of our results to the case of  weak  units would surely be of interest, but it would require substantial modifications. \end{remark}

Composition of the map $\lambda$ with the Yosida representation of $G$ embeds $G$ as a unital sublattice-subgroup (\emph{$\ell$-subgroup}) into
$\C{(Z_{G})}$: one sends $g\in G$ to $\widehat{g}\circ \lambda \colon Z_{G}\to\R$. The assignment is injective because $\lambda$ is surjective.
In  Section \ref{s:repG}  this observation is considerably strengthened. It turns out that $G$ determines a specific zero-dimensional compactification of its minimal spectrum which we denote $w_{G}Z_{G}$.  The clopen subsets of $w_{G}{Z_{G}}$, we will see, are (finitely)
 generated from the clopen subsets of $Z_{G}$ of the form $\VV_m(g)$. (Please see   Section \ref{s:repG} for details on this compactification.) We prove
in Theorem \ref{t:min-rep} that $G$ embeds as a unital $\ell$-subgroup of $\C{(w_{G}{Z_{G}})}$.    We will see that this stronger embedding of $G$ is in fact granted by Freudenthal's Spectral Theorem. Now, by the  Yosida theory (see again Subsection \ref{ss:Yosida}), the image of $G$ in $\C{(w_{G}{Z_{G}})}$  does not separate
the points of the base space, unless $X_G$ and $w_{G}{Z_{G}}$ are homeomorphic. We can however consider a minimal extension of the image of $G$ inside $\C{(w_{G}{Z_{G}})}$ which separates the points. Indeed, since $w_{G}{Z_{G}}$ is zero-dimensional, there is a canonical such extension: we must adjoin to
the image of $G$ all characteristic functions of clopen  subsets of $w_{G}{Z_{G}}$. We thereby obtain a unital embedding

\begin{align}\label{eq:phintro}
\pi_{G} \colon G \hookrightarrow \PH{(G)},
\end{align}

where $\PH{(G)}$ denotes the unital $\ell$-subgroup of $\C{(w_{G}Z_{G})}$ generated by the  representation  of $G$ into  $\C{(w_{G}Z_{G})}$, together with all \emph{characteristic functions} $w_{G}Z_{G}\to \R$ --- the continuous maps with range contained in $\{0,1\}$. We now have the homeomorphism $\MaxS{\PH{(G)}}\cong w_G{Z_{G}}$. In Theorem \ref{t:decomposition} we show that the elements of $\PH{(G)}$ may be characterised amongst elements of $\C{(w_{G}{Z_{G}})}$ as those functions with the property that, for an appropriate finite partition of $w_{G}Z_{G}$ into clopens, they agree with the image of some element of $G$ locally at each clopen. Building on this
we finally show in Theorem \ref{t:ph} that \eqref{eq:phintro} is the projectable hull of $(G,u)$, thus obtaining our main result. Summarising, we prove the existence of the projectable hull  of any  $\U$-object $(G,u)$ by exhibiting it as a natural substructure of $\C{(w_{G}Z_{G})}$, namely, $\PH{(G)}$.

\smallskip
Several intermediate  results in this paper admit a fuller development  of considerable potential  interest. We focus here on the proof of our main Theorem \ref{t:ph}, and postpone further results to future work.

\section{Preliminaries}\label{s:back}

\subsection{Polars and projection properties}\label{ss:pol}For the standard notions that we recall in this subsection, see \cite[Ch.\ 3, 6, and 11]{BKW}, together with \cite[Ch.\ 4, \S 24]{LuxZaan71}.

Given any $\ell$-group $A$, the elements $x,y \in A$ are \emph{orthogonal}, written $\orth{x}{y}$, if
$|x|\wedge |y|=0$. For $T \seq A$, we set

\begin{align*}
T^\perp:=\{x \in A\mid \orth{x}{y} \text{ for all } y \in T\};
\end{align*}

we  write $T^\pperp$ instead of $(T^\perp)^\perp$, and $x^\perp$ instead of $\{x\}^\perp$ for $x \in A$.
A subset $S\seq A$ is a \emph{polar} if it satisfies $S=S^\pperp$, or equivalently, if there exists $T\subseteq A$ such that $S=T^\perp$. When necessary, we will denote polars computed in $A$ by $T^{\perp_{A}}$. We write
$\Pol{A}$ to denote  the set of polars of $A$. Under the inclusion order,  $\Pol{A}$ is a complete distributive lattice with  $A=0^\perp$ as maximum,  $\{0\}=A^\perp=0^\pperp$ as minimum, meets given by intersections, and joins given by $\bigvee S_i:=(\bigcup S_i)^\pperp$. It can be shown that $\Pol{A}$ is a complete Boolean algebra, with complementation given by the map $S\in\Pol{A}\mapsto S^\perp\in\Pol{A}$. In particular, for any subset $T\seq A$ we have $T^{\pperp\perp}=T^\perp$.

If $x\in A$, the set $x^\pperp$ is called the \emph{principal polar} generated by $x$. Then $x^\pperp\in\Pol{A}$, and $x^\pperp=\bigcap_{x \in S\in\Pol{A}} S$, that is, $x^\pperp$ is the inclusion-smallest polar containing $\{x\}$. We write $\pPol{A}$ to denote the set of principal polars of $A$; it is a sublattice of $\Pol{A}$, because of the identities

\begin{align}
(x\wedge y)^\pperp&=x^\pperp\cap y^\pperp\label{e:ppolmeet}\\
(x\vee y)^\pperp&=x^\pperp\vee y^\pperp,\label{e:ppoljoin}
\end{align}

which hold for each $x,y\in A^+$. Further, the minimum $0^\pperp$ of $\Pol{A}$ lies in $\pPol{A}$. However, the maximum $A=0^\perp$ of $\Pol{A}$ need not be a principal polar: in fact, this happens precisely when $A$ has a weak unit $w$, and in that case $A=w^\pperp$.  Even when $A$ has a weak unit, $\pPol{A}$ may fail to be a Boolean subalgebra of $\Pol{A}$, because the complement of a principal polar need not be principal.

An ideal $I\subseteq A$ is  \emph{closed}, or is a \emph{band}, if for each $S\subseteq I$ such that $\bigvee S$ exists in $A$, we have $\bigvee S \in I$. It can be shown that each polar is a band; for the converse, we have the important

\begin{lemma}\label{l:pol=band}An $\ell$-group $A$ is such that its polars coincide with its bands if, and only if, $A$ is Archimedean.
\end{lemma}
\begin{proof}\cite[11.1.10]{BKW}.
\end{proof}

A band $I \seq A$ is a \emph{projection band} if there is a product splitting $A \cong I \card I^\perp$.

\begin{definition}[{Cf.\ \cite[24.8]{LuxZaan71}}]\label{d:projprop}An $\ell$-group $A$ is said to have the \emph{principal projection property}, or to be \emph{projectable}, if each principal band of $A$ is a projection band. Further, $A$ is said to have the \emph{projection property}, or to be \emph{strongly projectable}, if each band of $G$ is a projection band.
\end{definition}

We recall here a standard fact:

\begin{lemma}\label{l:pppimpliesarch}An $\ell$-group with the principal projection property must be Archimedean.
\end{lemma}
\begin{proof}The (easy) proof for vector lattices given in \cite[24.9]{LuxZaan71} works for $\ell$-groups without changes.
\end{proof}

\begin{remark}\label{r:terminology}
Projection properties are a  classical topic in the theory of vector lattices, see \cite[Ch.\ 4]{LuxZaan71}. In the literature on $\ell$-groups, it is standard to call $A$ \emph{projectable} when each of its principal  polars is a cardinal summand (i.e.\ a factor of a product splitting) of $A$, and \emph{strongly projectable} when the same holds for all polars. Thus, we see from Lemmas \ref{l:pppimpliesarch} and \ref{l:pol=band} that an $\ell$-group $A$ has the  principal projection property if, and only if, it is  projectable in the present sense; and that it has the  projection property if, and only if, it is strongly projectable in the present sense. Cf.\ also \cite[7.5]{BKW}. This explains the alternative terminologies in Definition \ref{d:projprop}.  In the rest of this paper we shall use the terminology \emph{projectable}.
\end{remark}

 A \emph{component of  the unit $u$} is an element $\chi \in G$ such that $\chi\vee (u-\chi)=u$ and $\chi\wedge (u-\chi)=0$. It is well known that this  entails  the existence of a product splitting $G\cong \chi^{\perp}\times\chi^{\perp\perp}$. Conversely, if $G\cong A\times B$ in $\U$, then there is a unique  $\chi \in G$ --- namely, the image in $G$ of the unit of $B$ under the unital isomorphism $G\cong A\times B$ ---  that is a component of the unit $u$ such that $A\cong \chi^{\perp}$ and $B\cong \chi^{\perp\perp}$. We use these elementary facts  without further justification throughout.

Finally, we recall the version of Freudenthal's Spectral Theorem that we will use.
\begin{theorem}\label{t:freudenthal} Let $R$ be a Riesz space that is projectable and has a unit $u$. For $v \in R$, set $||v||_{u}:=\inf{\{\lambda\in\R\mid \lambda \geq 0 \text{ and } \lambda u \geq |v|\}}$. Then $||v||_{u}$ is a norm on $R$. For each $v\in R$ there is a sequence $\{c_{i}\}_{i\geq 1}\seq R$ of linear combinations of components of $u$ that converges  to $v$ uniformly in the norm $||\cdot||_{u}$.
\end{theorem}
\begin{proof}See
 \cite[40.2]{LuxZaan71}.
 \end{proof}

\subsection{Essential extensions and the projectable hull}\label{ss:esspol}

A monomorphism $\iota \colon (G,u)$ $\hookrightarrow$ $(H,v)$  in $\U$ will be referred to as an \emph{extension} (of $G$ by $H$). The extension is \emph{essential} if whenever a $\U$-morphism $f\colon (H,v) \to (A,a)$ is such that the composition $f\circ\iota$ is monic, then $f$ is monic.
Amongst several well-known characterisations of essential extensions we shall use the following.

\begin{lemma}\label{l:ess}Let $\iota \colon (G,u)\hookrightarrow (H,v)$ be a monomorphism in $\U$. The following are equivalent.
\begin{enumerate}
\item\label{i:ess1}  The extension $\iota$ is essential.
\item\label{i:ess2}  The map $\nu_{H}\colon P \in \Pol{H}\longmapsto P \cap \iota(G) \in \Pol{\iota(G)}$ is an isomorphism from the Boolean algebra of polars of $H$ onto that of $\iota(G)$. The inverse isomorphism is the map $\nu_{H}^{-1}\colon Q \in \Pol{\iota(G)} \longmapsto Q^{\pperp_{H}} \in \Pol{H}$.
\item\label{i:ess3} For each $y \in H$ with $y >0$ there is $x \in G$ with $0 <\iota(x) < ny$ for some integer $n >0$.
\end{enumerate}
\end{lemma}
\begin{proof}See  \cite[Prop.\ 3.1 and Thm.\ 3.7]{Conrad71} and \cite[\S 2]{Conrad73}.
 \end{proof}

\begin{definition}\label{d:ph}An essential extension $\epsilon\colon (G,u) \hookrightarrow (K,w)$  in $\U$ is said to be a \emph{projectable hull} if $K$ is projectable, and whenever $\iota\colon (G,u) \hookrightarrow (H,v)$ is another essential extension with $H$ projectable, there exists an injective $\ell$-homomorphism $\varphi\colon (K,w)\rightarrow (H,v)$ in $\U$ that makes the following diagram commute.

\[
 \begin{tikzcd}
(G,u)   \arrow[hookrightarrow]{r}{\epsilon}\arrow[hookrightarrow]{dr}{\iota}&(K,w)\arrow[rightarrow, dashed]{d}{\varphi} \\
&(H,v)
 \end{tikzcd}
\]

\end{definition}

It turns out that the $\ell$-homomorphism $\varphi$ in the preceding definition  is automatically an essential extension. Also note that a projectable hull is unique up to an isomorphism in $\U$.

\begin{remark}\label{rem:generalisations}Through the general treatment in \cite{Ball82}, hulls related to projectability properties can and have been fruitfully investigated at the level of all  lattice-ordered (not necessarily Abelian) groups, with no assumption on the existence of units. In particular, any lattice-ordered group turns out to have a strongly projectable hull in this generalised sense, \cite[Thm.\ 2.25]{Ball82}, which agrees with the usual one in the representable case.
\end{remark}

\subsection{The Yosida representation: the case of strong units}\label{ss:Yosida}
For $X$ a topological space, recall that a subset  $S\subseteq \C{(X)}$ is said to  \emph{separate  the points of $X$} if for any $x\neq y \in X$ there is $f \in S$ with $f(x)\neq f(y)$. The next result summarises the classical Yosida representation; everything is rooted and essentially proved in \cite{Yosida41}.

\begin{theorem}[The Yosida Representation]\label{t:yosida} Recall that $(G, u)$ is a $\U$-object with maximal spectral space $X_{G}$.
\begin{enumerate}
\item[(a)]For each $\mathfrak{m}\in X_G$, there exists a \emph{unique} monomorphism

\begin{align*}
\iota_{\mathfrak{m}}\colon (G/\mathfrak{m},u/\mathfrak{m})\hookrightarrow (\R,1)
\end{align*}
in $\U$. Upon setting

\begin{align*}
\widehat{g}(\mathfrak{m}):=\iota_{\mathfrak{m}}(g/\mathfrak{m}) \in \R\,,
\end{align*}

each $g \in G$ induces a  function

\begin{align*}
\widehat{g}\colon X_G\to \R
\end{align*}

that  is continuous with respect to the spectral topology on the domain and the Euclidean topology on the co-domain.
\item[(b)]The map

\begin{align*}
\widehat{\cdot} \,\,\colon (G,u) \longrightarrow (\C{(X_G)},1_{X_G})
\end{align*}
given by \textup{(a)} is a monomorphism in $\U$ whose image $\widehat{G}\subseteq \C{(X_G)}$ separates the points of $X_G$.

\item[(c)] $X_G$ is unique up to a unique homeomorphism with respect to its properties. More explicitly, if $Y$ is any compact Hausdorff space, and $e\colon (G,u)\hookrightarrow (\C{(Y)},1_{Y})$ is any monomorphism in $\U$ whose image $e(G)\subseteq \C{(Y)}$ separates the points of $Y$, then there exists a unique homeomorphism  $f \colon Y \to X_G$ such that $(e(g))(y) = \widehat{g} (f(y))$ for all $g\in G$ and $y \in Y$.
\end{enumerate}
\end{theorem}

\begin{remark}\label{r:W}For the more general Yosida representation in the category of $\ell$-groups equipped with a \emph{weak} unit, see the standard reference  \cite{HagerRobertson77}.
\end{remark}

\begin{remark}Let us explicitly observe that components of the unit $1_{X_{G}}$ in $\C{(X_G)}$ are precisely the characteristic functions $X_{G}\to\R$, i.e.\ the continuous functions with range contained in $\{0,1\}$.
\end{remark}

\section{Representing an $\ell$-group on its minimal spectrum}\label{s:repG}

 Recall that $Z_{G}$ denotes the minimal spectral space of the $\U$-object $(G,u)$. We  show in this section that $G$ may be represented as an $\ell$-subgroup of $\C{(w_{G}{Z_{G}})}$ for the zero-dimensional Wallman compactification $w_{G}Z_{G}$ of $Z_{G}$, which we  describe  below.
 Before dealing with the general case, let us pause to recall that compactness of $Z_{G}$ is  equivalent to a complementation property of $G$.

\begin{definition}[{\cite{ConradMartinez90bis}}]\label{d:complemented}An $\ell$-group $A$ is \emph{complemented} if for each $x \in A$ there exists $y \in A$ such that $|x|\wedge |y|=0$ and $|x|\vee |y|\neq 0$ is a weak unit of $A$.
\end{definition}

\noindent Throughout we write ${\cdot}{\setminus}{\cdot}$ for set-theoretic difference. We recall $\Ss_{m} = \MinS{A} \setminus \VV_{m}(g)$.

\noindent Conrad and Martinez \cite{ConradMartinez90} proved the equivalence of (i) and (ii) below.

\begin{lemma}\label{l:conmar}For an $\ell$-group $A$, the following are equivalent.
\begin{enumerate}
\item[(i)]\label{i:comp} $A$ is complemented.
\item[(ii)]  $\MinS{A}$ is compact.

\item[(iii)] There exists a weak unit in $A$, the lattice $\pPol{A}$ is bounded, and the inclusion map $\pPol{A}\hookrightarrow \Pol{A}$ is a homomorphism of Boolean algebras.
\item[(iv)] The distributive lattice $(\{\VV_{m}(g)\}_{g\in A},\cap,\cup)$ is a Boolean algebra, i.e.\ for all $g\in A$ there is $f\in A$ such that $\Ss_{m}(g)=\VV_{m}(f)$.
\end{enumerate}
\end{lemma}

\begin{proof}(i $\Leftrightarrow$ ii) is \cite[2.2]{ConradMartinez90}.
(i $\Leftrightarrow$ iii) is also stated in passing in \cite{ConradMartinez90}; its proof is an elementary application of  (\ref{e:ppolmeet}--\ref{e:ppoljoin}).\\
To prove (iii $\Leftrightarrow$ iv) we recall that $\bigcap_{\p\in\MinS{G}}\p=\{0\}$ and that a prime $\p\in\Spec{A}$ is minimal if and only if for all $g\in\p$ there is an element $h\not\in\p$ with $h\perp g$ (see \cite[3.4.13]{BKW}). We first show $\VV_{m}(g) = \VV_{m}(g^{\pperp})$ for every $g\in A$. Since $g\in g^{\pperp}$, $\VV_{m}(g^{\pperp})\seq\VV_{m}(g)$. Let $\p\in\VV_{m}(g)$. Since $g\in\p$ and $\p\in\MinS{A}$, there exists $h\not\in\p$ such that $g\perp h$, whence $h\in g^{\perp}$. Let $f\in g^{\pperp}$. Then $f\in\p$, because $|f|\land|h|=0$ and $\p$ is prime. This ensures $\VV_{m}(g^{\pperp})\seq\VV_{m}(g)$.\\

(iv $\Rightarrow$ iii) Let $g\in A$ and $f\in A$ such that $\Ss_{m}(g) = \VV_{m}(f)$. Then each $\p\in\MinS{A}$ is contained either in $\VV_{m}(g)$ or in $\VV_{m}(f)$. We show that $g^{\pperp} = f^{\perp}$.\\
  Let $a\in f^{\perp}$. By primality, $a\in\p$ for each $\p\in\VV_{m}(g)=\Ss_{m}(f)$, and hence $\VV_{m}(g)\seq\VV_{m}(a)$. For each $b\in g^{\perp}$, $b\in\p$ whenever $g\not\in\p$, whence $b\in\p$ for each $\p\in\VV_{m}(f)$. Therefore, $|a|\land |b|\in\p$ for each $\p\in\MinS{A}$, and hence $|a|\land |b|=0$. This ensures $a\in g^{\pperp}$, whence $g^{\pperp}\supseteq f^{\perp}$.\\
  For the other inclusion, let $a\in g^{\pperp}$. By way of contradiction, suppose $|a|\land|f|=c>0$. Then there exists $\p_{c}\in\MinS{A}$ such that $c\not\in\p_{c}$. As a consequence, $a\not\in\p_{c}$, $f\not\in\p_{c}$, and $g\in\p_{c}$. By minimality of $\p$, there exists $b\not\in\p_{c}$ such that $b\perp g$. Since $|a|\land|b|=0\in\p_{c}$, then $a\in\p_{c}$, a contradiction. Hence we proved $g^{\pperp}\seq f^{\perp}$.\\

 As a consequence, $g^{\pperp\perp} = f^{\pperp}$, whence the principal polar $f^{\pperp}$ is the complement of $g^{\pperp}$ in $\pPol{A}$. Moreover, by taking $g=0$, there exists $w\in A$ such that $A=0^{\pperp\perp} = w^{\pperp}$. Then $\pPol{A}$ is bounded and a Boolean algebra, and, as a direct consequence of the definition, $w$ is a weak unit for $A$.\\

 (iii $\Rightarrow$ iv) Let $g\in A$ and $f\in A$ such that $f^{\pperp}$ is the complement of $g^{\pperp}$ in $\pPol{A}$. Then $g^{\pperp\perp} = f^{\pperp}$, and $g^{\pperp} = f^{\perp}$. We show $\VV_{m}(f)=\Ss_{m}(f^{\perp})$, whence $\VV_{m}(f)=\Ss_{m}(g^{\pperp})=\Ss_{m}(g)$. Let $\p\in\VV_{m}(f)$, then $f\in\p$. Since $\p$ is minimal, there is $h\in A$ such that $h\perp f$ and $h\not\in\p$. Hence $h\in f^{\perp}$ and $\p\not\in\VV_{m}(f^{\perp})$. This ensures $\VV_{m}(f)\seq \Ss_{m}(f^{\perp})$. For the other inclusion, let $\p\in \Ss_{m}(f^{\perp})$ and $h\not\in\p$ such that $h\perp f$. Since $|h|\land|f|=0$ the primality of $\p$ ensures $f\in\p$, whence $\p\in\VV_{m}(f)$. Therefore $\Ss_{m}(f^{\perp})\seq\VV_{m}(f)$ and the proof is complete.
\end{proof}

\begin{remark}\label{r:othermincomp}Versions of Lemma \ref{l:conmar} for commutative rings, distributive lattices, and vector lattices were proved in \cite[3.4]{HenJeri65}, \cite[Prop.\ 3.2]{Speed69}, and \cite[37.4]{LuxZaan71}, respectively.
\end{remark}

We now turn to the $w_{G}$-compactification. Recall \cite[4.4(a)]{PorterWoods88}
that a \emph{Wallman base} of a Hausdorff space $X$ is a base $\mathscr{L}$ of closed sets for $X$ that is stable under finite intersections and unions (and thus contains, in particular, $\varnothing$ and $X$),  is such that if $A\in \mathscr{L}$ and $x\in X\setminus A$ then there is $B\in \mathscr{L}$ with $x\in B$ and $A\cap B=\varnothing$, and is such that for $A,B\in \mathscr{L}$ satisfying  $A\seq X\setminus B$  there exist $C,D\in \mathscr{L}$ with $A\seq X\setminus C\seq D \seq X\setminus B$. Given such a base $\mathscr{L}$, let $w_{\mathscr{L}}X$ denote the collection of inclusion-maximal lattice filters of $\mathscr{L}$. The collection of sets $\{\mathscr{F}\in w_{\mathscr{L}}X\mid A \in \mathscr{F}\}$, as $A$ ranges in $\mathscr{L}$, is a base for the closed sets of a topology on $w_{\mathscr{L}}X$. With this topology, $w_{\mathscr{L}}X$ is compact \cite[4.4(d)]{PorterWoods88}. Given $x \in X$, set $\mathscr{U}_{x}:=\{A\in \mathscr{L}\mid x \in A\}$.  Then $\mathscr{U}_{x}\in w_{\mathscr{L}}X$, and the map

\begin{align}\label{eq:wallman}
X &\longrightarrow w_{\mathscr{L}}X\\
x \in X &\longmapsto \mathscr{U}_{x}\in w_{\mathscr{L}}X\nonumber
\end{align}

is a dense embedding, called the \emph{Wallman compactification} of $X$ induced by $\mathscr{L}$.

For any zero-dimensional Hausdorff space $X$, we note that any collection of clopen sets $\mathscr{B}$ of $X$ which also forms a closed base for the space, is, almost trivially, a Wallman base for the space, and that the generated Wallman compactification $w_{\mathscr{B}}X$ is a zero-dimensional Hausdorff space \cite[4.7(b)]{PorterWoods88}.

Furthermore, we note that in the case where $X$ is zero-dimensional and $\mathscr{B}$ is a Wallman base for $X$ consisting of clopen sets, the (bounded) functions from the uniformly closed subring of $\C{(X)}$ generated by the characteristic functions for the Boolean algebra of clopen sets generated by $\mathscr{B}$ are exactly the functions which extend to $w_{\mathscr{B}}X$ \cite[4I \& 4J]{PorterWoods88}.

We next identify $G$ with its Yosida representation $\widehat{G}\subseteq \C{(X_{G})}$, as given by Theorem \ref{t:yosida}. Recall the map $\lambda\colon Z_{G}\twoheadrightarrow X_{G}$ as in (\ref{eq:lambda}--\ref{eq:action}).  If $\widehat{g}\in \widehat{G}$, the assignment

\begin{align}\label{eq:lambdapb}
\widehat{g}\in \widehat{G}\overset{\mu}{\longmapsto} \widehat{g}\circ\lambda\in \C{(Z_{G})}
\end{align}

yields a unital homomorphism of $\ell$-groups $\mu \colon \widehat{G}\to \C{(Z_{G})}$, and a straightforward  computation confirms that $\mu$ is injective because $\lambda$ is surjective. We therefore obtain a  representation of $G$ as

\begin{align}\label{eq:mu-rep}
\mu(\widehat{G})\subseteq \C{(Z_{G})}.
\end{align}

\begin{remark}\label{r:zerosets}
We notice that the Yosida representation $\widehat{g}$ of $g$ is such that, for every $\m\in X_{G}$, $\widehat{g}(\m)=0$ if, and only if, $g\in\m$. This ensures the inclusion of zero sets
\begin{equation*}
\VV_{m}(g)\seq\left\{\p\in Z_{G}\mid \mu(\widehat{g})(\p)=\widehat{g}(\lambda(\p))=0\right\}.
\end{equation*}
Hence
\begin{equation*}
\lambda^{-1}(\Ss_{M}{(g)})=\Ss{(\mu(\widehat{g}))}\seq\Ss_{m}{(g)},
\end{equation*}
where $\Ss{(\mu(\widehat{g}))}:=Z_{G}\setminus \mu(\widehat{g})^{-1}(0)$ is the support of $\mu(\widehat{g})$ at $Z_{G}$, and $\lambda^{-1}(\Ss_{M}{(g)})$ is the preimage of $\Ss_{M}{(g)}$ under $\lambda$.
\end{remark}

We consider now the Boolean algebra $\Ba{(Z_{G})}$ of subsets of $Z_{G}$ generated by the collection $\{\VV_m{(g)}\}_{g\in G}$. By \cite[10.2.1]{BKW}, for every $g\in G$, $\VV_{m}{(g)}\subseteq Z_{g}$ is a clopen set, hence all the elements of $\Ba{(Z_{G})}$ are clopens of $Z_{G}$. Moreover, since $\{\VV_m{(g)}\}_{g\in G}\seq\Ba{(Z_{G})}$, they also form a closed base for $Z_{G}$, and hence $\Ba{(Z_{G})}$ is a Wallman base of $Z_{G}$. We denote by
\begin{equation}\label{e:wG}
w_{G}Z_{G}
\end{equation}
the Wallman compactification $w_{\Ba{(Z_{G})}}Z_{G}$ given by $\Ba{(Z_{G})}$. The Boolean algebra of all clopens of $w_{G}Z_{G}$, written $\Cp{(w_{G}Z_{G})}$, can be identified with the elements of $\Ba{(Z_{G})}$:
\begin{equation}\label{e:clopens}
 \Ba{(Z_{G})} \cong \Cp{(w_{G}Z_{G})}.
\end{equation}

Let $\K{(Z_{G})}\seq\C{(Z_{G})}$ be the Boolean algebra of characteristic functions for the elements of $\Ba{(Z_{G})}$. Since each member of $\K{(Z_{G})}$ extends  uniquely to a member  of the Boolean algebra $\K{(w_{G}Z_{G})}$ of all characteristic functions in $\C{(w_{G}Z_{G})}$ by \cite[4G]{PorterWoods88},
we have the isomorphism of Boolean algebras
\begin{equation}\label{e:charfunctions}
 \K{(Z_{G})} \cong \K{(w_{G}Z_{G})}.
\end{equation}

In summary, any two Boolean algebras in (\ref{e:clopens}) and (\ref{e:charfunctions}) are isomorphic, and their dual Stone space is  (\ref{e:wG}).

\begin{remark}\label{r:beta0}It may happen that $w_{G}Z_{G}$ is strictly smaller than the \emph{Banaschewski compactification} of $Z_{G}$, the largest zero-dimensional compactification usually denoted $\beta_{0}Z_{G}$. See Section \ref{s:ex}.
\end{remark}

\begin{lemma}\label{l:freudenthal}With reference to the embedding \textup{(\ref{eq:mu-rep})}, the  uniform completion of the linear subspace of $\C{(Z_{G})}$ generated by $\K{(Z_{G})}$ contains $\mu(\widehat{G})$.
\end{lemma}
\begin{proof}Let $V$ be the Riesz space generated by $\mu(\widehat{G})\cup \K{(Z_{G})}$ in $\C{(Z_{G})}$.
\begin{claim}
 The function $1_{Z_{G}}$ is a strong unit for $V$.
\end{claim}
\begin{proof}
For each $g\in G$ the function $\widehat{g}$ is bounded in $X_{G}$. Since the image of the function $\widehat{g}\circ\lambda$ coincides with the image of $\widehat{g}$, each element $\mu(\widehat{g})\in\mu(\widehat{G})$ is bounded. The characteristic functions in $\K{(Z_{G})}$ are bounded functions by definition. Therefore each element of $V$ is bounded, because the Riesz space operations preserve the property of being bounded. Hence $1_{Z_{G}}$ is a strong unit for $V$.
\end{proof}

For each $g\in G$ let $\chi_{g}\in\K{(Z_{G})}$ be the characteristic function of $\Ss_{m}(g)$.

\begin{claim}\label{c:orthogonal}$\chi_{g}^\perp=\mu(\widehat{g})^\perp$ in $V$.
\end{claim}

\begin{proof}[Proof of Claim \ref{c:orthogonal}]
Let $v\in V$ such that $v\in\chi_{g}^{\perp}$. Hence $v(\p)\land\chi_{g}(\p)=0$ for every $\p\in Z_{G}$. As a consequence, if $\chi_{g}(\p)\neq0$, then $v(\p)=0$. Therefore, whenever $\mu(\widehat{g})(\p)\neq0$, we have $v(\p)=0$ by Remark \ref{r:zerosets}. Then $v(\p)\land\mu(\widehat{g})(\p)=0$ for all $\p\in Z_{G}$, whence $v\in\mu(\widehat{g})^{\perp}$. This proves the inclusion $\chi_{g}^\perp\seq\mu(\widehat{g})^\perp$.\\
For the other inclusion, suppose there exists $v\in V$ such that $v\in\mu(\widehat{g})^\perp\setminus\chi_{g}^\perp$. Then the supports $\Ss{(v)}$ and $\Ss{(\mu(\widehat{g}))}$ are disjoint, and there is $\p\in Z_{G}$ such that $|v(\p)|\land|\chi_{g}(\p)|\neq0$. By definition of $\chi_{g}$, $\chi_{g}^{-1}(1)=\Ss{(\chi_{g})} = \Ss_{m}{(g)}$ and therefore $\p$ is in $\Ss{(v)}\cap\Ss_{m}{(g)}$, which is open since $v$ is a continuous function on $Z_{G}$ and $\Ss_{m}{(g)}$ is a basic open subset of $Z_{G}$. Hence there exists a (nonempty) basic open set $\Ss_{m}{(h)}$ (with $0\neq h\in G$) such that $\p\in\Ss_{m}{(h)}\seq\Ss{(v)}\cap\Ss_{m}{(g)}$. Let $\m\in\Ss_{M}{(h)}$. Then $h\not\in\m$, whence $h\not\in\q$ for every $\q\in\lambda^{-1}(\m)$ and $\lambda^{-1}(\m)\seq\Ss_{m}{(h)}$. Since $\Ss_{m}{(h)}$ is disjoint from $\Ss{(\mu(\widehat{g}))}$, and $\Ss{(\mu(\widehat{g}))}=\lambda^{-1}(\Ss_{M}{(g)})$ by Remark \ref{r:zerosets}, $\q\not\in\lambda^{-1}(\Ss_{M}{(g)})$ for every $\q\in\lambda^{-1}(\m)$. Then $\m\in\VV_{M}{(g)}$ and $g\in\m$. This ensures $\Ss_{M}{(h)}\seq\VV_{M}{(g)}$, and hence $|h|\land|g|=0$. By primality of $\p$, either $h\in\p$ or $g\in\p$, in contradiction with $\p\in\Ss_{m}{(h)}\seq\Ss_{m}{(g)}$. This ensures $v\in\chi_{g}^\perp$, whence $\mu(\widehat{g})^\perp\seq\chi_{g}^\perp$.

\end{proof}

\begin{claim}\label{c:projectable}$V$ is projectable.
\end{claim}

\begin{proof}[Proof of Claim \ref{c:projectable}] We need to show that each $v \in V$ induces a product splitting  $V\cong v^{\pperp}\times v^{\perp}$.
By Claim \ref{c:orthogonal}, $\chi_{g}^\pperp=\mu(\widehat{g})^\pperp$ for all $g\in G$. But since $\chi_{g}$ is a component of $1_{Z_{G}}$ (because $\chi_{g}\vee (1_{Z_{G}}-\chi_{g})=1_{Z_{G}}$ and $\chi_{g}\wedge(1_{Z_{G}}-\chi_{g})=0$), there is an induced  splitting $V\cong \chi_{g}^\pperp \times \chi_{g}^{\perp}$. This shows that each element of $V$ that lies in the generating set  $\mu(\widehat{G})\cup \K{(Z_{G})}$  induces the splitting above. A routine induction argument now shows that the splitting property is preserved by sums, meets and joins, and products by real scalars, thus settling the claim.

\end{proof}

By the preceding claim, we may apply  Freudenthal's Spectral Theorem \ref{t:freudenthal} to $(V,1_{Z_{G}})$, and infer that each element of   $\mu(\widehat{G})$ is a $1_{Z_{G}}$-uniform limit of a sequence of elements in the linear subspace of $\C{(Z_{G})}$ generated by $\K{(Z_{G})}$. Since the norm induced by $1_{Z_{G}}$ on $\C{(Z_{G})}$ coincides with the supremum norm, this completes the proof.

\end{proof}

\begin{lemma}\label{l:ext}For each $g\in G$,  there exists a unique continuous extension of $\mu(\widehat{g}) \in \C{(Z_{G})}$ to an element $g^{\sharp}\in\C{(w_{G}Z_{G})}$. That is, $g^{\sharp}$ is the unique such element whose restriction to $Z_{G}$ is $\mu(\widehat{g})$. In symbols,

\begin{align}\label{eq:ext}
g^{\sharp}{\upharpoonright_{Z_{G}}}=\mu(\widehat{g}).
\end{align}

\end{lemma}

\begin{proof}Indeed, by Lemma \ref{l:freudenthal} there is a sequence $\{c_{i}\}_{i\geq 1}$ of linear combinations of elements of $\K{(Z_{G})}$ that converges uniformly to $\mu(\widehat{g})$.  Each member of $\K{(Z_{G})}$ extends  uniquely to a member  of $\K{(w_{G}Z_{G})}$ \cite[4G]{PorterWoods88}, and therefore each $c_{i}$ extends to a linear combination $k_{i}$ of  elements of  $\K{(w_{G}Z_{G})}$. It is now elementary to check that $\{k_{i}\}_{i\geq 1}$ is a Cauchy sequence in  $\C{(w_{G}Z_{G})}$ because $\{c_{i}\}_{i\geq 1}$ is one in $\C{(Z_{G})}$. Take $g^{\sharp}$ to be the limit of $\{k_{i}\}_{i\geq 1}$, which is of course a continuous function by the Uniform Limit Theorem. Finally, note that  $g^\sharp$ has property (\ref{eq:ext}) by construction, and is the unique  member of $\C{(w_{G}Z_{G})}$ with this property  because $Z_{G}$ is dense in its $w_{G}$-compactification, and the codomain of the functions --- namely, $\R$ --- is Hausdorff.
\end{proof}

In light of Lemma \ref{l:ext}, the function

\begin{align}\label{eq:sharp}
\cdot^{\sharp}\colon G \hookrightarrow \C{(w_{G}Z_{G})}
\end{align}

that acts by $g\mapsto g^{\sharp}$ is injective. It is elementary that this embedding preserves the lattice and group structure of $G$, and is also unit-preserving. We have therefore proved:
\begin{theorem}\label{t:min-rep}Each $\U$-object $(G,u)$ has a representation into $\C{(w_{G}Z_{G})}$ as in \textup{(\ref{eq:sharp})}.\qed
\end{theorem}

\begin{definition}\label{d:PH}We  write $\PH{(G)}$ for the $\ell$-subgroup of $\C{(w_{G}Z_{G})}$ generated by

 \begin{align}\label{eq:generators}
 G^{\sharp}\cup\K{(w_{G}Z_{G})}.
 \end{align}

 We further write

\begin{align}\label{eq:ph}
\pi \colon G \hookrightarrow \PH{(G)}
\end{align}

for the $\U$-monomorphism of $G$ into $\PH{(G)}$ obtained by restricting the codomain of (\ref{eq:sharp}) to $\PH{(G)}$.
\end{definition}

\section{Characterisation of the elements of $\PH{(G)}$}\label{s:el}
In this section we characterise the functions in $\C{(w_{G}Z_{G})}$ that lie in $\PH{(G)}$.
We begin by preparing two  lemmas.

\begin{lemma}\label{l:separation}There is a  homeomorphism $\MaxS{\PH{(G)}}\cong w_{G}Z_{G}$.
\end{lemma}

\begin{proof}Indeed, the characteristic functions $\K{(w_{G}Z_{G})}\seq \PH{(G)}$ separate the points of $w_{G}Z_{G}$, because the latter is zero-dimensional; now apply Yosida's Theorem \ref{t:yosida}.
\end{proof}

\begin{remark}\label{r:known}The next lemma is a consequence of more general results, cf.\ \cite[Proposition 2.4]{HagerKimberMcGovern03}. We provide a proof here, for the reader's convenience.
\end{remark}

\begin{lemma}\label{l:products} Let $g\in G$, and let $\chi\in G$ be a component of the unit $u$. Let us identify $G$ with  its Yosida representation $\widehat{G}\seq\C{(X_{G})}$. The pointwise product $g\chi$ defined by $(g\chi)(x)=g(x)\chi(x)$ for each $x \in X_{G}$ is a continuous function, and hence an element of $\C{(X_{G})}$. Then $g\chi\in \widehat{G}$.
\end{lemma}
\begin{proof}(Skipping all trivialities, in this proof we identify isomorphism with equality  without further warning.) Since $\chi$ is a component of $u$ we have a product splitting $\widehat{G}=\chi^{\perp}\times\chi^{\perp\perp}$ ($\perp$ computed in $\widehat{G}$), and a corresponding disjoint union decomposition $X_{G}=A\sqcup B$, $A:=\chi^{-1}(0)$, $B:=\chi^{-1}(1)$, $A$ and $B$ disjoint clopens in $X_{G}$. Then, clearly, $\C{(X_{G})}=\C{(A)}\times\C{(B)}$, $\chi^{\perp}\seq \C{(A)}$, and  $\chi^{\perp\perp}\seq \C{(B)}$.  Now since $g\in \chi^{\perp}\times\chi^{\perp\perp}$, $g$ may be uniquely expressed as a sum $g_{1}+g_{2}$, $g_{i} \in \widehat{G}$, $g_{1}\in \chi^{\perp}$, $g_{2} \in \chi^{\perp\perp}$. Then $g$ and $g_{2}$ agree over $B$, so that $g\chi=g_{2}\chi=g_{2}\in \widehat{G}$, and the lemma is proved.
\end{proof}

\begin{remark}
Let $0\leq g\in G$, and let $\chi\in G$ be a component of the unit
$u$. Identifying $G$ with its Yosida representation
$\widehat{G}\seq\C{(X_{G})}$, we notice that the function $g$ is
bounded on the support of the characteristic function
$\chi$. Therefore, there exists a (unique minimal) integer $n\geq0$
such that $g\leq n\chi$ holds on the support of $\chi$, and hence
$g\chi=g\wedge n\chi$ holds in $G$. This yields  an explicit
representation of the product $g\chi$ discussed in Lemma
\ref{l:products}, using only the operations of $G$. Any element $g\in
G$, indeed, can be written as the difference $g^{+}-g^{-}$ between its
\emph{positive part} $g^{+}:=g\vee 0$ and its \emph{negative part}
$g^{-}:=(-g)\vee 0$, with $0\leq g^{+},g^{-}\in G$. As a consequence,
there exist two (unique minimal) integers $n_{+},n_{-}\geq0$ such that
$g\chi = (g^{+}\wedge n_{+}\chi) - (g^{-}\wedge n_{-}\chi)$.

In the following, we use the product $g\chi$ for brevity, but each
such occurrence may be replaced by the equivalent expression
$(g^{+}\wedge n_{+}\chi) - (g^{-}\wedge n_{-}\chi)$.
\end{remark}

By a \emph{partition of unity} in a $\U$-object $(G,u)$ we mean in this paper a finite family of  non-zero elements $P:=\{\chi_{i}\}_{i=1}^{l}$ of $G$ such that $\sum_{i=1}^{l}\chi_{i}=u$, and $\chi_{i}\wedge\chi_{j}=0$ whenever $i\neq j$. It is elementary that each $\chi_{i}$ is a component of $u$. It follows that, in the Yosida representation $\widehat{G}$ of $G$, each $\widehat{\chi}_{i}$ is a characteristic function.

We can now prove:

\begin{theorem}\label{t:decomposition}For each $e\in \C{(w_{G}Z_{G})}$, the following are equivalent.
\begin{enumerate}
\item\label{i:decomp1} $e\in \PH{(G)}$.
\item\label{i:decomp2} There exists a partition of unity $\chi_{1},\ldots,\chi_{l}$ in $\PH{(G)}$  --- equivalently, in $\C{(w_{G}Z_{G})}$ ---  along with elements $a_{1},\ldots,a_{l}\in G$, such that

\begin{align}\label{eq:decomp}
e=\sum_{i=1}^{l}a_{i}^{\sharp}\chi_{i},
\end{align}

where $\sharp$ is the embedding \textup{(\ref{eq:sharp})}, and $a_{i}^{\sharp}\chi_{i}$ denotes the pointwise product of $a_{i}^{\sharp}$ and $\chi_{i}$ in $\C{(w_{G}Z_{G})}$.
\end{enumerate}
\end{theorem}

\begin{proof}First, let us explicitly note that $\PH{(G)}$ and  $\C{(w_{G}Z_{G})}$ have  the same collection of partitions of unity because $\K{(w_{G}Z_{G})}\seq \PH{(G)}$.

\smallskip \noindent (\ref{i:decomp1})$\Rightarrow$(\ref{i:decomp2})\
Recall that $\PH{(G)}$ is the $\ell$-subgroup of $\C{(w_{G}Z_{G})}$ generated by the set (\ref{eq:generators}). Hence by the elementary theory of lattice-groups we can write $e$ as

\[
\bigwedge_{i\in I} \bigvee_{j \in J} \left(g_{ij}^{\sharp} + c_{ij}k_{ij}\right),
\]

where $I$ and $J$ are finite sets of indices, at least one of which is non-empty, $g_{ij} \in G$, $c_{ij} \in \Z$ and
$k_{ij} \in \K{(w_{G}Z_{G})}$.  Now for each $k_{ij}$,
we obtain associated clopen subsets of $w_{G}Z_{G}$, namely their supports
and their complements. This (necessarily non-empty) collection of clopens obviously covers $w_{G}Z_{G}$. It is elementary that we can refine this cover into a
finite partition $\{D_m\}_{m \in M}$ of clopens of the space  by taking
intersections and set-theoretic differences.

On each $D_m$, each $k_{ij}$ is constant --- either zero or one --- by construction. Let us define the element $\delta_{ij}^{m}\in G$ by setting

\begin{align}\label{eq:delta}
\delta_{ij}^{m}:=\begin{cases}u &\text{if $D_{m}\seq k_{ij}^{-1}(1)$},\\
0 &\text{if $D_{m}\seq k_{ij}^{-1}(0)$}.\end{cases}
\end{align}

Now consider the element of $\PH{(G)}$

\begin{align*}
e_{m}:=\bigwedge_{i\in I} \bigvee_{j \in J} \left(g_{ij}^{\sharp} + c_{ij}(\delta_{ij}^{m})^\sharp\right).
\end{align*}

Observe that \emph{the function $e_{m}$ agrees over $D_{m}$ with the function $e$}, for each $m \in M$. This follows immediately from our definition of $\delta_{ij}^{m}$ in (\ref{eq:delta})  above.
Moreover, since $\sharp$ is an $\ell$-homomorphism, we have

\[
e_{m}=\left(\bigwedge_{i\in I} \bigvee_{j \in J} \left(g_{ij} + c_{ij}\delta_{ij}^{m}\right)\right)^{\sharp}=a_{m}^{\sharp},
\]

where  $a_{m}:=\bigwedge_{i\in I} \bigvee_{j \in J} \left(g_{ij} + c_{ij}\delta_{ij}^{m}\right)\in G$. Hence, if we let $\chi_m$ be the characteristic
function of $D_m$, we conclude

\[ e = \sum_{m\in M} a_m^{\sharp}\chi_m,\]

as was to be shown.

\smallskip \noindent (\ref{i:decomp2})$\Rightarrow$(\ref{i:decomp1})\ This follows at once from Lemmas \ref{l:separation} and \ref{l:products}.

\end{proof}

 \section{Construction of the projectable hull}\label{s:ph}

Our final aim is to show that the embedding \textup{(\ref{eq:sharp})} provides a description of the projectable hull of $G$. This is  our main result:

\begin{theorem}\label{t:ph}
For any $\U$-object $(G,u)$, the embedding $\pi \colon G \hookrightarrow \PH{(G)}$ as in \textup{(\ref{eq:ph})} is the
projectable hull of $G$.
\end{theorem}

\begin{proof} The proof that $\PH{(G)}$ is projectable is identical to that of Claim
\ref{c:projectable}.

To prove that the map $\pi$ is an essential extension, we verify (\ref{i:ess3}) in Lemma \ref{l:ess}. Pick $0<e\in\PH{(G)}$, and express it as $e=\sum_{i=1}^{l}a_{i}^{\sharp}\chi_{i}$ by Theorem \ref{t:decomposition}, for a partition of unity $\{\chi_{i}\}_{i=1}^{l}$ in $\PH{(G)}$ and elements $\{a_{i}\}_{i=1}^{l}$ in $G$. Since $e>0$, we must have $a_{i}^{\sharp}\chi_{i}\geq 0$ for each $i$, and $a_{i_{0}}^{\sharp}\chi_{i_{0}}> 0$ for some $i_{0}$. It is enough to show that there is $h\in G$ such that $0<h^{\sharp}\leq n a_{i_{0}}^{\sharp}\chi_{i_{0}}$, for then $0<h^{\sharp}\leq ne$ follows easily.  By \eqref{e:charfunctions}, we can identify the characteristic functions in $\K{(w_{G}Z_{G})}$ with the elements of $\K{(Z_{G})}$.
Set $a:=a_{i_{0}}$ and $\chi:=\chi_{i_{0}}\land\chi_{a}$, where $\chi_{a}\in\K{(Z_{G})}$ is the characteristic function of $\Ss_{m}(a)$.

Since the support $\Ss{(\chi)}=\chi^{-1}(1)$ is clopen and nonempty (because $a^{\sharp}\chi\neq0$), there exists $0<g\in G$ such that the basic open set $\Ss_{m}{(g)}$ is nonempty and contained in $\Ss{(\chi)}\seq\Ss_{m}{(a)}$. Hence $h:=g\land a>0$, and $\Ss{(h^{\sharp})}\seq\Ss_{m}{(h)}\seq\Ss{(\chi)}$.

We now claim that $h^{\sharp}\leq a^{\sharp}\chi$.

It is enough to prove that the inequality holds for a point $x\in w_{G}Z_{G}$ in the support of $\chi$, where $\chi(x)=1$. Since $\sharp$ is an $\ell$-homomorphism, $h^{\sharp}=a^{\sharp}\wedge g^{\sharp}>0$. If $a^{\sharp}(x)\leq g^{\sharp}(x)$, then $h^{\sharp}(x)=a^{\sharp}(x)$, and the inequality holds. Otherwise, we have $h^{\sharp}(x)=g^{\sharp}(x)< a^{\sharp}(x)$. This settles the claim and

completes the proof that $\pi$ is essential.

To show $\PH{(G)}$ is a hull, it suffices to show that given the (unital)
essential embedding $\iota$ into $H$ there exists an (automatically essential and unital)
embedding $\varphi$ making the diagram below commute:

\[
 \begin{tikzcd}\tag{*}\label{d:phi}
(G,u) \arrow[hookrightarrow]{r}{\pi}
   \arrow[hookrightarrow]{dr}{\iota}&
   (\PH{(G)},u^{\sharp}) \arrow[hookrightarrow,dashed]{d}{\varphi} \\
   &(H,v)
 \end{tikzcd}
\]

We  define a function $\varphi\colon \PH{(G)}\to H$ as follows. First we set

\begin{align}\label{eq:phionG}
\varphi(a^{\sharp}):=\iota(a), \text{ for each } a \in G.
\end{align}

Since $H$ is projectable, for each $g\in G$ there is a unique component $\sigma_{g}\in H$ of the unit $v$ that satisfies $\iota(g)^{\perp\perp}=\sigma_{g}^{\perp\perp}$,
where $\perp$ is computed in $H$. By \eqref{e:charfunctions}, we identify the components of the unit $u^{\sharp}$ in $\PH{(G)}$ with the characteristic functions in $\K{(Z_{G})}$. Let $\chi_{g}\in\PH{(G)}$ be the characteristic function of $\Ss_{m}(g)$ for $g\in G$. The analogous result of Claim \ref{c:orthogonal} for $\PH{(G)}$ ensures that $\chi_{g}^{\perp}=\pi(g)^{\perp}$ in $\PH{(G)}$. We set
\begin{equation}\label{eq:phionchig}
 \varphi(\chi_{g}):= \sigma_{g}.
\end{equation}
Let $\K{(H)}$ be the Boolean algebra of components of the unit $v$ in $H$.

\begin{claim}\label{c:phihom}
 The map $\varphi\colon\{\chi_{g}\}_{g\in G}\to\K{(H)}$ defined as in \eqref{eq:phionchig} is a lattice homomorphism.
\end{claim}

\begin{proof}
 Since $\iota$ is a unital homomorphism, trivially $\varphi(\chi_{0})=0$ and $\varphi(\chi_{u})=v$. Let $g_{1},g_{2}\in G$. It is easy to check that $\chi_{g_{1}}\land\chi_{g_{2}}=\chi_{g_{1}\land g_{2}}$ and $\chi_{g_{1}}\vee\chi_{g_{2}}=\chi_{g_{1}\vee g_{2}}$. Let $\varphi(\chi_{g_{1}})=\sigma_{g_{1}}$, $\varphi(\chi_{g_{2}})=\sigma_{g_{2}}$, $\varphi(\chi_{g_{1}\land g_{2}})=\sigma_{g_{1}\land g_{2}}$ and $\varphi(\chi_{g_{1}\vee g_{2}})=\sigma_{g_{1}\vee g_{2}}$ as in \eqref{eq:phionchig}. Hence, by \eqref{e:ppolmeet} and \eqref{e:ppoljoin},
 \begin{align*}
  \sigma_{g_{1}\land g_{2}}^{\pperp}&=\iota(g_{1}\land g_{2})^{\pperp} = (\iota(g_{1})\land \iota(g_{2}))^{\pperp} =\\
  &= \iota(g_{1})^{\pperp}\cap \iota(g_{2})^{\pperp} = \sigma_{g_{1}}^{\pperp}\cap \sigma_{g_{2}}^{\pperp}= (\sigma_{g_{1}}\land\sigma_{g_{2}})^{\pperp},
 \end{align*}
 and
 \begin{align*}
  \sigma_{g_{1}\vee g_{2}}^{\pperp}&=\iota(g_{1}\vee g_{2})^{\pperp} = (\iota(g_{1})\vee \iota(g_{2}))^{\pperp} =\\
  &= (\iota(g_{1})^{\pperp}\cup \iota(g_{2})^{\pperp})^{\pperp} = (\sigma_{g_{1}}^{\pperp}\cup\sigma_{g_{2}}^{\pperp})^{\pperp} = (\sigma_{g_{1}}\vee\sigma_{g_{2}})^{\pperp}.
 \end{align*}
This ensures $\sigma_{g_{1}\land g_{2}}=\sigma_{g_{1}}\land\sigma_{g_{2}}$ and $\sigma_{g_{1}\vee g_{2}}=\sigma_{g_{1}}\vee\sigma_{g_{2}}$. Therefore $\varphi(\chi_{g_{1}}\land\chi_{g_{2}})=\varphi(\chi_{g_{1}\land g_{2}})=\varphi(\chi_{g_{1}})\land\varphi(\chi_{g_{2}})$ and $\varphi(\chi_{g_{1}}\vee\chi_{g_{2}})=\varphi(\chi_{g_{1}\vee g_{2}})=\varphi(\chi_{g_{1}})\vee\varphi(\chi_{g_{2}})$, and the claim is proved.
\end{proof}
\noindent
The previous Claim \ref{c:phihom}, together with \cite[Lemma 1 in Section V.4]{BalbesDwinger74}, ensures that $\varphi$  extends  uniquely to a homomorphism $\varphi\colon \K{(Z_{G})}\to\K{(H)}$ of Boolean algebras. Specifically, since the Boolean algebra $\K{(Z_{G})}$ can be generated by the set $\{\chi_{g}\}_{g\in G}$ using the negation $u^{\sharp}-$ and the meet operation $\land$, the image of a general element of $\K{(Z_{G})}$ under $\varphi$ can be inductively described as follows. Let $\chi_{1}, \chi_{2}, \alpha, \beta_{1}, \beta_{2}\in \K{(Z_{G})}$ such that $\chi_{1} = u^{\sharp}-\alpha$ and $\chi_{2}=\beta_{1}\land\beta_{2}$. Then
\begin{equation}\label{eq:phionK}
 \varphi(\chi_{1})=v-\varphi(\alpha)\quad\text{and}\quad
 \varphi(\chi_{2})=\varphi(\beta_{1})\land\varphi(\beta_{2}).
\end{equation}

\noindent
Finally, for a general $e \in \PH{(G)}$, we first write $e$ as in (\ref{eq:decomp}) using Theorem \ref{t:decomposition}, and then, using (\ref{eq:phionG}--\ref{eq:phionK}), we set

\begin{align}\label{eq:phigeneral}
\varphi(e):=\sum_{i=1}^{l}\varphi(a_{i}^{\sharp})\varphi(\chi_{i}).
\end{align}

Since each product $\varphi(a_{i}^{\sharp})\varphi(\chi_{i})$  is an element of $H$ by Lemma \ref{l:products}, $\varphi(e)$ as in (\ref{eq:phigeneral}) is an element of $H$.

We next verify that $\varphi$  is a well-defined function. Given a decomposition  (\ref{eq:decomp}) of $e\in \PH{(G)}$ as in Theorem \ref{t:decomposition}, suppose $e=\sum_{j=1}^{t}b_{j}^{\sharp}\xi_{j}$ is another such decomposition. It suffices to show that
\begin{align}\label{eq:show}
\sum_{i=1}^{l}\varphi(a_{i}^{\sharp})\varphi(\chi_{i})=\sum_{j=1}^{t}\varphi(b_{j}^{\sharp})\varphi(\xi_{j}).
\end{align}

It is elementary to verify that the set $\{\chi_{i}\wedge \xi_{j}\mid \chi_{i}\wedge \xi_{j}\neq 0 \}$
forms a partition of unity that refines both $\{\chi_{i}\}_{i=1}^{l}$
and $\{\xi_{j}\}_{j=1}^{t}$; that is,  each $\chi_i$ and $\xi_j$ is a sum (or join, by pairwise disjointness) of elements $\chi_{i'}\wedge \xi_{j'}\neq 0$.
It follows that $e$ can  be expressed
in two ways as

\begin{align*}
e=\sum a_{i}^{\sharp}(\chi_{i}\wedge \xi_{j})=\sum b_{j}^{\sharp}(\chi_{i}\wedge \xi_{j}),
\end{align*}
and hence $a_{i}^{\sharp}(\chi_{i}\wedge \xi_{j})=b_{j}^{\sharp}(\chi_{i}\wedge \xi_{j})$ for all $i,j$.

\noindent
In the simplest case, we must prove:
\begin{claim}
Assuming $\chi\in\K{(Z_{G})}$,  suppose $a^{\sharp}\chi =
b^{\sharp}\chi$. Then $\varphi(a^{\sharp}\chi) = \iota(a)\varphi(\chi)=\iota(b)\varphi(\chi) = \varphi(b^{\sharp}\chi)$.
\end{claim}
\begin{proof}

Let $\nu_{\PH{(G)}}\colon\Pol{\PH{(G)}}\to\Pol{\pi(G)}$ and $\nu_{H}^{-1}\colon\Pol{\iota(G)}\to\Pol{H}$ be the isomorphisms of Boolean algebras given in Lemma \ref{l:ess}. Then also $\nu_{\pi}:=\pi^{-1}\circ\nu_{\PH{(G)}}\colon\Pol{\PH{(G)}}\to\Pol{G}$ and $\nu_{\iota}:=\nu_{H}^{-1}\circ\iota\colon\Pol{G}\to\Pol{H}$ are isomorphisms of Boolean algebras. Since $\chi^{\perp_{\PH{(G)}}}$ is a
polar, there exists a unique polar $P_{\chi}\in\Pol{G}$ such that $P_{\chi}^{\perp_{G}}=\nu_{\pi}(\chi^{\perp_{\PH{(G)}}})$.
From $a^{\sharp}\chi =
b^{\sharp}\chi$, we have $\pi(a-b) = (a^{\sharp} -
b^{\sharp}) \in \chi^{\perp_{\PH{(G)}}}$. Hence $a-b \in
P_{\chi}^{\perp_G}$ and $\iota(a)-\iota(b)=\iota(a-b) \in \iota(P_{\chi})^{\perp_{\iota(G)}} =
\iota(P_{\chi})^{\perp_H} \cap \iota(G)$.
To complete the proof we show $\iota(P_{\chi})^{\perp_H}=\varphi(\chi)^{\perp_H}$, whence $(\iota(a)-\iota(b))\in\varphi(\chi)^{\perp_H}$ and $\iota(a)\varphi(\chi)=\iota(b)\varphi(\chi)$. \\
We proceed by induction on the structure of $\chi$ using \eqref{eq:phionK}.
In the basic case we suppose $\chi:=\chi_{g}$, for $g\in G$.
Then $\chi_{g}^{\perp_{\PH{(G)}}}=\pi(g)^{\perp_{\PH{(G)}}}$ by Claim \ref{c:orthogonal}, whence $P_{\chi}^{\perp_{G}} = \nu_{\pi}(\pi(g)^{\perp_{\PH{(G)}}}) = \pi^{-1}(\nu_{\PH{(G)}}(\pi(g)^{\perp_{\PH{(G)}}})) = \pi^{-1}(\pi(g)^{\perp_{\PH{(G)}}}\cap\pi(G)) = g^{\perp_{G}} = (g^{\pperp_{G}})^{\perp_{G}}$, and $P_{\chi} = g^{\pperp_{G}}$.
By definition of $\varphi$ on $\{\chi_{g}\}_{g\in G}$, we have $\varphi(\chi)^{\perp_{H}} = \iota(g)^{\perp_{H}} = \iota(g^{\pperp_{G}})^{\perp_{H}}$, where the last equality is a direct consequence of the essentiality of $\iota$ (see \cite[\S 2]{Conrad73}). \\
We now consider the case $\chi := u^{\sharp} - \alpha$, and suppose $\iota(P_{\alpha})^{\perp_H}=\varphi(\alpha)^{\perp_H}$.
Since $\chi$ and $\alpha$ are components of unity, $\chi^{\perp_{\PH{(G)}}} = \alpha^{\pperp_{\PH{(G)}}}$, whence $P_{\chi}^{\perp_{G}} = P_{\alpha}^{\pperp_{G}}$ and $\iota(P_{\chi})^{\perp_{H}} = \iota(P_{\alpha})^{\pperp_{H}}$, because both $\nu_{\pi}$ and $\nu_{\iota}$ are  isomorphisms of Boolean algebras. Since also $\varphi\colon\K{(Z_{G})}\to\K{(H)}$ is an isomorphism of Boolean algebras, $\varphi(\chi)=v-\varphi(\alpha)$.
Therefore, $\iota(P_{\chi})^{\perp_H}=\iota(P_{\alpha})^{\pperp_H}=\varphi(\alpha)^{\pperp_H}=\varphi(\chi)^{\perp_H}$. \\
The case $\chi:=\beta_{1}\land\beta_{2}$ is proved in a similar way. Suppose $\iota(P_{\beta_{1}})^{\perp_H}=\varphi(\beta_{1})^{\perp_H}$ and $\iota(P_{\beta_{2}})^{\perp_H}=\varphi(\beta_{2})^{\perp_H}$.
By \eqref{e:ppolmeet}, $\chi^{\pperp_{\PH{(H)}}}=(\beta_{1}\land\beta_{2})^{\pperp_{\PH{(H)}}}=\beta_{1}^{\pperp_{\PH{(H)}}}\cap\beta_{2}^{\pperp_{\PH{(H)}}}$, whence $\chi^{\perp_{\PH{(H)}}} = (\beta_{1}^{\pperp_{\PH{(H)}}}\cap\beta_{2}^{\pperp_{\PH{(H)}}})^{\perp_{\PH{(H)}}} = \beta_{1}^{\pperp_{\PH{(H)}}}\vee\beta_{2}^{\pperp_{\PH{(H)}}}$. As a consequence, $P_{\chi}^{\perp_{\PH{(H)}}} = P_{\beta_{1}}^{\perp_{\PH{(H)}}} \vee P_{\beta_{2}}^{\perp_{\PH{(H)}}}$ and $\iota(P_{\chi})^{\perp_{H}} = \iota(P_{\beta_{1}})^{\perp_{H}} \vee \iota(P_{\beta_{2}})^{\perp_{H}}$.
Since $\varphi(\chi) = \varphi(\beta_{1})\land\varphi(\beta_{2})$, we have $\varphi(\chi)^{\pperp_{\PH{(H)}}} = \varphi(\beta_{1})^{\pperp_{\PH{(H)}}}\cap\varphi(\beta_{2})^{\pperp_{\PH{(H)}}}$, whence $\varphi(\chi)^{\perp_{\PH{(H)}}} = \varphi(\beta_{1})^{\perp_{\PH{(H)}}}\vee\varphi(\beta_{2})^{\perp_{\PH{(H)}}} = \iota(P_{\beta_{1}})^{\perp_{H}} \vee \iota(P_{\beta_{2}})^{\perp{H}} = \iota(P_{\chi})^{\perp_{H}}$. The claim is settled.
\end{proof}
Since $\varphi\colon \K{(Z_{G})}\to\K{(H)}$ is a homomorphism of Boolean algebras, $\varphi$ preserves
partitions of unity and $\varphi(\chi \wedge \xi) = \varphi(\chi) \wedge
\varphi(\xi)$. Therefore
\begin{align*}
\sum \varphi(a_{i}^{\sharp})\varphi(\chi_{i}\wedge \xi_{j})&=\sum \varphi(b_{j}^{\sharp})\varphi(\chi_{i}\wedge \xi_{j}), \\
\sum_j \varphi(a_{i}^{\sharp})\varphi(\chi_{i}\wedge \xi_{j}) &= \varphi(a_{i}^{\sharp})\varphi(\chi_{i}) \mbox{ for all  $i$'s,} \\
\sum_i \varphi(b_{j}^{\sharp})\varphi(\chi_{i}\wedge \xi_{j}) &= \varphi(b_{j}^{\sharp})\varphi(\xi_{j}) \mbox{ for all $j$'s.}
\end{align*}

These together prove (\ref{eq:show}).

The map $\varphi$ makes the diagram (\ref{d:phi}) commute by
construction. To show that it is  an $\ell$-homomorphism one argues as follows. Given $e+f \in \PH{(G)}$, to prove $\varphi(e + f) =
\varphi(e) + \varphi(f)$, we first take decompositions of $e+f$, $e$ and
$f$ as in (\ref{eq:decomp}) of Theorem \ref{t:decomposition}. We then pick a joint refinement of the three partitions of unity involved.  We finally proceed as in the preceding argument   that shows  $\varphi$ is well-defined.
We omit the  elementary  details. The argument for the remaining operations is analogous.

To show $\varphi$ is  injective, consider $e \neq f \in
\PH{(G)}$. Using again decompositions as in (\ref{eq:decomp}) of  Theorem \ref{t:decomposition}, and a
common refinement of the associated partitions, we see that $e$ and
$f$ must differ on some element of the common refinement. Injectivity
of $\varphi$ then follows at once from the injectivity of $\iota$ and $\pi$.
\end{proof}

\section{Examples}\label{s:ex}

We close the paper with two examples. Example \ref{ex:parab} shows that the space $Z_{G}$ of minimal primes can fail to be compact even though its $G$-indexed compactification $w_{G}Z_{G}$ is actually $\beta_{0}Z_{G}$, the largest zero-dimensional compactification of $Z_{G}$. This happens when the base of clopens of $Z_{G}$ indexed by elements of $G$ is not a Boolean algebra --- because it fails to be closed under complements --- and yet it is large enough that the Boolean algebra it generates consists of all clopens of $Z_{G}$.  We note in passing that the $\ell$-group in this first example can be shown to be finitely generated.  Example \ref{ex:ref}, by contrast, is an instance where the $G$-indexed compactification $w_{G}Z_{G}$ is strictly smaller than $\beta_{0}Z_{G}$. Here $G$ is of the form $\C({X)}$ for $X$ a compact Hausdorff space.

\begin{example}\label{ex:parab}
Let $\mathscr{M}$ be the set of continuous and piecewise linear functions $f\colon[0,1]^{2}\to\R$ with integer coefficients. This means that $f$ is continuous, and that there are finitely many triplets of integers $(a_{i},b_{i}, c_{i}) \in \Z^{3}$ such that for all $(x,y)\in[0,1]^{2}$ we have  $f(x,y) = a_{i}x+b_{i}y+c_{i}$ for some $i$. When equipped with the pointwise operations of minimum, maximum, and addition, $\mathscr{M}$ is an $\ell$-group. The function identically equal to $1$ is a strong unit.

 Let
 \begin{equation*}
P:=\{(x,y)\in[0,1]^{2}\mid y=x^{2}\}\cup\{(x,y)\in[0,1]^{2}\mid y=0\},
 \end{equation*}
 and let $(G,1_{P})$ be the unital $\ell$-group obtained by restricting each element of $\mathscr{M}$ to ${P}$, where $1_{P}\colon P \to \R$ is the function constantly equal to $1$. We now show that
  $Z_{G}:=\MinS{G}$ fails to be compact. For this,  consider the projection function $\pi_{y}\colon(x,y)\mapsto y$, along with  the set of functions $h_{n}$ ($0<n\in\N$) which are equal to $nx-1$ on the segment $\{(x,y)\in[0,1]^2\mid y=0\text{ and }x\in[\frac{1}{n},1]\}$ and to $0$ elsewhere. These are elements of $G$.
  Then $\{\Ss_{m}(\pi_{y})\}\cup\{\Ss_{m}(h_{n})\}_{n>0}$ is an open cover of $Z_{G}$ without a finite subcover. According to Lemma \ref{l:conmar}, $G$ is not complemented. Indeed, by piecewise linearity and continuity, there can be no $f\in \mathscr{M}$ that restricts to $P$ so as to satisfy $\pi_{y}^{\pperp}=f^{\perp}$. By the same lemma, $\{\VV_{m}(g)\}_{g\in G}$
  is not a Boolean algebra. Indeed,  it does not contain the complement $\Ss_{m}(\pi_{y})$ of $\VV_{m}(\pi_{y})$. In particular, $\{\VV_{m}(g)\}_{g\in G}$ is not the collection of all clopens of $Z_{G}$, because $\Ss_{m}(\pi_{y})$ is clopen --- it is both a
  basic open set and the intersection of the
  closed sets in $\{\VV_{m}(h_{n})\}_{n>0}$. However, the Boolean algebra $\Ba{(Z_G)}$ generated by $\{\VV_{m}(g)\}_{g\in G}$ is the collection of \emph{all} clopens of $Z_{G}$, as can be seen by direct inspection. Therefore $\MaxS{\PH{(G)}}\cong w_{G}Z_{G}$ is homeomorphic to $\beta_{0}Z_{G}$, the maximal zero-dimensional compactification of $Z_{G}$.
\end{example}

\begin{example}\label{ex:ref}
Let $D$ be the discrete space with $|D| =
\aleph_1$, let $\alpha D = D \cup \{\alpha\}$ be the one-point
compactification of $D$, and let $G = \C{(\alpha D)}$ so that $\MaxS{G} \cong \alpha
D$.  Then $Z_{G}:=\MinS{G}$ can be identified with the collection of fixed ultrafilters on $D$
together with the free ultrafilters on $D$ which contain a countable
subset.
The basic closed sets of $Z_{G}$, of the form
\[\{\mathscr{U} \in Z_{G} \mid g^{-1}(0)\in \mathscr{U}\},\]
are actually clopen.  We show that the Boolean algebra
generated by these basic clopen sets is {\em not} the
Boolean algebra of all clopens, but a proper subalgebra. For, let
$D_1$ and $D_2$ be elements of a partition of $D$ with $|D_i| = \aleph_1$, $i=1,2$. Then we have an induced partition of $Z_{G}$ into two clopens:

\[
Z_{G} = \{\mathscr{U} \mid \exists C \in \mathscr{U} \mbox{ with }
C \subseteq D_1\} \biguplus \{\mathscr{U} \mid \exists C \in \mathscr{U}
\mbox{ with } C \subseteq D_2\}.
\]
Neither summand in the partition above is a
basic closed set of $Z_{G}$. Therefore $\MaxS{\PH{(G)}}\cong
w_{G}Z_{G} \not\cong \beta_0 Z_{G}$.
\end{example}

\begin{remark}\label{r:c(x)} In  case $G=\C{(X)}$ for some  compact Hausdorff space $X$, the main result of this paper  is related to several others  in the literature, and especially to the construction of zero-dimensional covers of $X$. Postponing fuller accounts to further research, we mention at least the following important
connection. In \cite{Vermeer84}, Vermeer constructs the
basically disconnected  cover of $X$ (in the more general case of  a completely regular Hausdorff space) through an inverse limit \cite[Theorem 4.4]{Vermeer84}. The inverse limit is defined by means of an auxiliary space  $\Lambda_1\,(X)$ --- itself a cover of $X$ --- for which see in particular \cite[Theorem 3.5]{Vermeer84}. Now, the
results by Hager et al.\ in \cite[Section 3]{HagerKimberMcGovern03} entail amongst other things that, in our notation,  there is a  homeomorphism $\MaxS{\PH{(\C{(X)})}}\cong \Lambda_1(X)$. By   Lemma \ref{l:separation} we conclude
 $\Lambda_1\,(X) \cong w_{\C{(X)}}Z_{\C{(X)}}$. We thus obtain a description of Vermeer's cover $\Lambda_1\,(X)$ as the Wallman compactification of the \emph{minimal} spectral space of $\C{(X)}$ described in (\ref{e:wG}--\ref{e:charfunctions}) above. Contrast Vermeer's \cite[Theorem 3.5]{Vermeer84}, which describes  $\Lambda_1\,(X)$ using regularised  zero sets in  the \emph{maximal} spectral space $X$ of $\C{(X)}$. Further,  the space $Z_{\C{(X)}}$ is compact if, and only if, $X$ is \emph{cozero-complemented}, i.e.\ the lattice of cozero sets of $X$ is complemented; this is well-known and also follows  from Lemma \ref{l:conmar}. In this
 case we  have $w_{\C{(X)}}Z_{\C{(X)}}\cong Z_{\C{(X)}}$.
 Consequently, $\Lambda_1\,(X)\cong Z_{\C{(X)}}$ --- when $X$ is cozero-complemented, Vermeer's  $\Lambda_1\,(X)$ is the minimal spectral space of $\C{(X)}$, and further work  reveals Vermeer's cover $\Lambda_1\,(X)\to X$ in \cite{Vermeer84} as the ``push-up'' map $\lambda\colon Z_{\C{(X)}}\to X$ in
 (\ref{eq:lambda}--\ref{eq:action}).
\end{remark}

\part*{Acknowledgements.}
\noindent We gratefully acknowledge  partial support by the Italian FIRB ``Futuro in Ricerca'' grant RBFR10DGUA. The grant partially supported a visit of R. N. Ball to the Universit\`a degli Studi di Milano, Italy, during which the fundamental ideas of the present paper were developed.
We would also like to express our thanks to an anonymous referee for detecting a serious blunder in an earlier version of this paper, and for providing us with  Example \ref{ex:ref}. The same referee pointed out to us the relevance of  \cite{Vermeer84} and \cite{HagerKimberMcGovern03} for our results (cf.\ Remark \ref{r:c(x)}).
 Finally, we are grateful to S. J. van Gool for pointing out to us that the use of  the standard extension result  \cite[Lemma 1 in Section V.4]{BalbesDwinger74} could shorten the proof of Theorem \ref{t:ph}.

\bibliographystyle{plain}
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\end{document}

config: arxiv


TITLE: Do exist constant curvature manifolds (hyperbolic or elliptic) with torsion?
QUESTION [0 upvotes]: I would say no. Because a constant curvature manifold is symmetric. Any 2 dimensional subspace should be like a hyperboloid or a sphere. These objects do not have torsion.

REPLY [9 votes]: You should consider the following example:
Let $M^3=\mathrm{SU}(2)\simeq S^3$ endowed with its biïnvariant metric $g$ (unique up to a constant multiple, let's fix this by requiring that the $g$ has constant sectional curvature equal to $1$).  Now consider the unique connection $\nabla$ for which the left-invariant vector fields on $SU(2)$ are parallel.  This connection is compatible with the biïnvariant metric and it has torsion, since $T(X,Y) = -[X,Y]\not=0$ when $X$ and $Y$ are linearly independent left-invariant vector fields.  This metric has the same geodesics as the Levi-Civita connection, and every $2$-plane has sectional-curvature $1$.
The symmetry group of $(M^3,g,\nabla)$ is $\mathrm{SO}(4)$, the maximum possible. So this is an example of a constant curvature manifold with torsion.
As É. Cartan and J. A. Schouten showed in 1926 (On Riemannian geometries admitting an absolute parallelism), this example generalizes to biïnvariant metrics on compact simple Lie groups endowed with the unique connection for which the left-invariant vector fields are parallel, though they no long actually have constant sectional curvature.
However, they also show in that paper that there is a connection $\nabla$ on the $7$-sphere $S^7$ of unit octonions that is flat, compatible with the natural metric of constant curvature on $S^7$, and invariant under the group $\mathrm{Spin}(7)\subset\mathrm{SO}(8)$.  Because $\mathrm{Spin}(7)$ acts transitively on $3$-planes in $\mathbb{O}\simeq\mathbb{R}^8$, it follows that this geometry $(S^7, g,\nabla)$ also has constant curvature.


TITLE: Understanding Parameters
QUESTION [2 upvotes]: My textbook (New Tertiary Mathematics, Volume 1 Part 1, Pure Mathematics: The Core, by C Plumpton & P S W Macilwaine) introduces Parameters in the following manner:
"The coordinates of a point on a curve can usually be expressed in terms of a third variable which is called a parameter. For example $(t^{3}, t^{2})$, where $t$ varies, are the coordinates of a point on the curve $y^{3} = x^{2}$."
I really don't understand this at all and I thought here might be the place to ask for help understanding it.
Perhaps I'm thinking about this wrong, but what I take from a sentence like that is: if I choose some value of $t$ - let's say $t = 2$ - I should get the coordinates of a point on the curve $y^{3} = x^{2}$. But if we try that with $t = 2$, we get $(8, 4)$. But if I try letting $x = 4$, I get $y = \sqrt[3]{16} \neq 8$.
So, if I may, I'd like to ask what I'm supposed to understand by that sentence.

REPLY [2 votes]: The parametrization given is $(x,y)=(t^{3}, t^{2})$.
You have inverted the roles of $x$ and $y$, probably mislead by $y$ appearing first in the equation $y^{3} = x^{2}$.


TITLE: Rescaling solution in matrix equation
QUESTION [1 upvotes]: Consider the following equation:
\begin{equation}
\frac{ \mathrm{d} \mathbf{p}(t)}{ \mathrm{d} t}=(\mathbf{D}-\mathbf{A})\mathbf{p}(t)
\end{equation}
Where $\mathbf{A}$ is a symmetric $N\times N$ matrix and $\mathbf{D}$ is a diagonal matrix.
The solution is simply:
\begin{equation}
\mathbf{p}(t)=e^{(\mathbf{D}-\mathbf{A})t}\mathbf{p}(0)
\end{equation}
Now consider the following equation:
\begin{equation}
\frac{ \mathrm{d} \mathbf{q}(t)}{ \mathrm{d} t}=(\mathbf{I}-\mathbf{A}\mathbf{D}^{-1})\mathbf{q}(t)
\end{equation}
My question: Can I express $\mathbf{q}(t)$ in terms of $\mathbf{p}(t)$?

REPLY [0 votes]: $
\def\L{\left}\def\R{\right}
\def\LR#1{\L(#1\R)}
\def\BR#1{\Big(#1\Big)}
\def\vec#1{\operatorname{vec}\LR{#1}}
\def\Unvec#1{\operatorname{Unvec}\BR{#1}}
\def\qiq{\quad\implies\quad}
$
Assume that a matrix $X$ exists such that
$\,q = Xp.\;$
This ansatz yields
$$\eqalign{
\dot q &= X\dot p \\
 &= X\LR{D-A}p \\
 &= X\LR{D-A}\LR{X^{-1}q} \\
 &= \LR{XFX^{-1}}q \\
}$$
You would like the matrix in parentheses to equal the matrix
$\,G=(I-AD^{-1})$
Therefore
$$\eqalign{
G &= XFX^{-1} \qiq
GX &= XF \\
}$$
which is a Sylvester Equation which can sometimes be solved for the unknown $X$ matrix.
One approach is to use vectorization
$$\eqalign{
0 &= \BR{I\otimes G - F^T\otimes I}\,\vec{X} \\
0 &= M\,\vec{X} \\
}$$
This means that $\vec{X}$ lies in the nullspace of $M$.
Continuing with an arbitrary vector $y$ and the pseudoinverse $M^+$
$$\eqalign{
\vec{X} &= \LR{I-M^+M}y \\
X &= \Unvec{y-M^+My} \\
}$$
If $\,M^+M=I\,$ then this approach fails, and a solution of the assumed form does not exist.