Limits of vector lattices]Limits of vector lattices

Walt van Amstel

Jan Harm van der Walt

Department of Mathematics and Applied Mathematics, University of Pretoria, Corner of Lynnwood Road and Roper Street,
Hatfield 0083, Pretoria, South Africa and DSI-NRF Centre of Excellence in
Mathematical and Statistical Sciences (CoE-MaSS), South Africa

Department of Mathematics and Applied Mathematics, University of Pretoria, Corner of Lynnwood Road and Roper Street,
Hatfield 0083, Pretoria, South Africa

The first author was supported by a grant from the DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS),
South Africa. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS.  The second author was supported by the NRF of South Africa, grant number 115047.  The results in this paper were obtained, in part, while both authors visited Leiden University from September 2021 to January 2022.  This visit was funded by the European Union Erasmus+ ICM programme.  The authors thank Prof. Marcel de Jeu and the Mathematical Institute at Leiden University for their hospitality.  The authors thank the reviewer for a meticulous reading of the article along with a number of helpful suggestions.

[2010]Primary 46M40; Secondary 46A40, 46E05

If $K$ is a compact Hausdorff space so that the Banach lattice $\cont(K)$ is isometrically lattice isomorphic to a dual of some Banach lattice, then $\cont(K)$ can be decomposed as the $\ell^\infty$-direct sum of the carriers of a maximal singular family of order continuous functionals on $\cont(K)$.  In order to generalise this result to the vector lattice $\contX$ of continuous, real valued functions on a realcompact space $X$, we consider direct and inverse limits in suitable categories of vector lattices.  We develop a duality theory for such limits and apply this theory to show that $\contX$ is lattice isomorphic to the order dual of some vector lattice $\vlat{F}$ if and only if $\contX$ can be decomposed as the inverse limit of the carriers of all order continuous functionals on $\contX$.  In fact, we obtain a more general result:  A Dedekind complete vector lattice $\vlat{E}$ is perfect if and only if it is lattice isomorphic to the inverse limit of a suitable family of order continuous functionals on $\vlat{E}$.  A number of other applications are presented, including a decomposition theorem for order dual spaces in terms of spaces of Radon measures.

§ INTRODUCTION

Let $K$ be a compact Hausdorff space. A basic question concerning the Banach lattice $\cont(K)$ is the following:  Does there exist a Banach space (lattice) $\vlat{E}$ so that $\cont(K)$ is isometrically (lattice) isomorphic to the dual $\vlat{E}^\ast$ of $\vlat{E}$?  That is, does $\cont(K)$ have a Banach space (lattice) predual?  In general, the answer to this question is `no'. The unit ball of $\cont[0,1]$ has only two extreme points, but the unit ball of the dual of an infinite dimensional Banach space has infinitely many extreme points.  Hence $\cont[0,1]$ is not the dual of any Banach space; hence also not of any Banach lattice. On the other hand, $\cont(\beta \N)$ is the dual of $\ell^1$.  The problem is therefore to characterise those spaces $K$ for which $\cont(K)$ is a dual Banach space (lattice). Combining two classic results of Dixmier [17] and Grothendieck [22], respectively, gives an answer to this question in the setting of Banach spaces, see also [14] for a recent presentation.  The Banach lattice case is treated in [34].

In order to formulate this result we recall the following.  A Radon measure $\mu$ on $K$ is called normal if $|\mu|(B)=0$ for every closed nowhere dense subset $B$ of $K$.  The space of all normal measures on $K$ is denoted $\vlat{N}(K)$.  The space $K$ is called Stonean if it is extremally disconnected; that is, the closure of every open set is open.  $K$ is hyper-Stonean[We feel obligated to recall Kelley's remark [28]: `In spite of my affection and admiration for Marshall Stone, I find the notion of a Hyper-Stone downright appalling.'] if it is Stonean and the union of the supports of the normal measures on $K$ is dense in $K$.

Let $K$ be a compact Hausdorff space.  Consider the following statements.

(i) $\cont(K)$ has a Banach lattice predual.

(ii) $\cont(K)$ has a Banach space predual.

(iii) $K$ is hyper-Stonean.

(iv) Let $\cal{F}$ be a maximal singular family of normal probability measures on $K$, and for each $\mu\in \cal{F}$ let $S_\mu$ denote its support.  Then

\[
\cont(K)\ni u\longmapsto \left( \left.u\right|_{S_\mu} \right)_{\mu\in \cal{F}}\in \bigoplus_{\infty} \cont(S_\mu)
\]

is an isometric lattice isomorphism.

Statements (i), (ii) and (iii) are equivalent, and each implies (iv).  If $K$ is Stonean, then all four statements are equivalent.

Furthermore, in case $\cont(K)$ has a Banach space predual $\vlat{E}$, this predual is also a Banach lattice predual and is unique up to isometric lattice isomorphism.  In particular, $\vlat{E}$ is isometrically lattice isomorphic to $\vlat{N}(K)$.

This result can be reformulated by identifying $\vlat{N}(K)$ with the order continuous dual of $\cont(K)$, via the isometric lattice isomorphism between the dual of $\cont(K)$ and the space of Radon measures on $K$, and $\cont(S_\mu)$ with the carrier of the corresponding functional on $\cont(K)$.

Let $K$ be a compact Hausdorff space.  Consider the following statements.

(i) $\cont(K)$ has a Banach lattice predual.

(ii) $\cont(K)$ has a Banach space predual.

(iii) $\cont(K)$ is Dedekind complete and has a separating order continuous dual.

(iv) Let $\cal{F}$ be a maximal singular family of order continuous functionals on $\cont(K)$, and for each $\varphi\in\cal{F}$ let $\vlat{C}_\varphi$ denote its carrier and $P_\varphi$ the band projection onto $\vlat{C}_\varphi$.  Then

\[
\cont(K)\ni u \longmapsto (P_\varphi u)_{\varphi\in \cal{F}}\in \bigoplus_{\infty} \vlat{C}_\varphi
\]

is an isometric lattice isomorphism.

Statements (i), (ii) and (iii) are equivalent, and each implies (iv).  If $K$ is Stonean, then all four statements are equivalent.

Furthermore, in case $\cont(K)$ has a Banach space predual $\vlat{E}$, this predual is also a Banach lattice predual and is unique up to isometric lattice isomorphism.  In particular, $\vlat{E}$ is isometrically lattice isomorphic to the order continuous dual $\ordercontn{\cont(K)}$ of $\cont(K)$.

The above problem may be generalised to the class of realcompact spaces.  Recall that a realcompact space is a Tychonoff space $X$ which is homeomorphic to a closed subset of some product of $\R$.  Equivalently, $X$ is realcompact if it is a Tychonoff space and for every point $x \in \beta X\setminus X$ (where $\beta X$ denotes the Stone-Čech compactification of $X$) there exists a real-valued, continuous function $u$ on $X$ which does not extend to a continuous, real-valued function on $X\cup\{x\}$.  For every Tychonoff space $X$ there exists a unique (up to homeomorphism) realcompact space $\upsilon X$ so that $\contX$ and $\cont(\upsilon X)$ are isomorphic vector lattices, see for instance [23], <cit.> and <cit.>. The realcompact space $\upsilon X$ is called the realcompactification of $X$.

Let $X$ be a realcompact space.  Then $\cont(X)$ is a vector lattice but, in general, not a Banach lattice.  Hence we ask the following question: Does there exist a vector lattice $\vlat{E}$ so that $\orderdual{\vlat{E}}$ is lattice isomorphic to $\cont(X)$?  That is, does $\contX$ have an order predual?  Xiong [37] obtained the following answer to this question.

Let $X$ be realcompact space.  Denote by $S$ the union of the supports of all compactly supported normal measures[See Section <ref>.] on $X$.  The following statements are equivalent.

(i) There exists a vector lattice $\vlat{E}$ so that $\orderdual{\vlat{E}}$ is lattice isomorphic to $\cont(X)$.

(ii) $\contX$ is lattice isomorphic to $\orderdual{(\ordercontn{\cont(X)})}$.

(iii) $X$ is extremally disconnected and $\upsilon S=X$.

This result differs from the corresponding result for compact spaces in the following respects.  Unlike in the Banach lattice setting, $\cont(X)$ may have more than one order predual, see [37].  Secondly, the condition that $\contX$ is Dedekind complete and has a separating order continuous dual does not imply that $\contX$ has an order predual.  Indeed, in <cit.> an example is provided of a realcompact space $X$ so that $\contX$ is Dedekind complete and has a separating order continuous dual, but is not the order dual of any vector lattice.  Furthermore, we have no counterpart of the decomposition

\[
\cont(K)\ni u \longmapsto (P_\varphi u)_{\varphi\in \cal{F}}\in \bigoplus_{\infty} \vlat{C}_\varphi.
\]

The naive extension of this decomposition to the class of extremally disconnected realcompact spaces does not provide a characterization of those spaces $\contX$ which admit an order predual.  It will be shown in Section <ref>, Proposition <ref>, that if $X$ is an extremally disconnected realcompact space and $\cal{F}$ is a maximal singular family in $\ordercontn{\contX}$ so that

\[
\cont(X)\ni u \longmapsto (P_\varphi u)_{\varphi\in \cal{F}}\in \prod_{\varphi\in\cal{F}} \vlat{C}_\varphi
\]

is a lattice isomorphism, then $\ordercontn{\contX}$ is an order predual for $\contX$.  The converse, however, is false, see Example <ref>.

In view of the above, we formulate the following problem.  Let $X$ be an extremally disconnected realcompact space.  Can the property `$\contX$ admits an order predual' be characterised in terms of a suitable decomposition of $\contX$ in terms of the carriers of order continuous functionals on $\contX$?  We solve this problem using direct and inverse limits in suitable categories of vector lattices.[In the literature, direct and inverse limits are also referred to as inductive and projective limits, respectively.]

Such limits are common in analysis, see for instance [6], <cit.>, <cit.> and [12].  Direct limits of vector lattices were introduced by Filter [19] and inverse limits of vector lattices have appeared sporadically in the literature, see for instance [16, 29], but no systematic study of this construction has been undertaken in the context of vector lattices.  We therefore take the opportunity to clarify the question of existence of inverse limits in certain categories of vector lattices. We also establish the permanence of a number of vector lattice properties under the inverse limit construction. Our treatment of direct and inverse limits of vector lattices is found in Sections <ref> and <ref> respectively.  Inspired by results in the theory of convergence spaces [6] we obtain duality results for direct and inverse limits of vector lattices, see Section <ref>.  These results are roughly of the following form:  If a vector lattice $\vlat{E}$ can be expressed as the direct (inverse) limit of some system of vector lattices, then the order (continuous) dual of $\vlat{E}$ can be expressed in a natural way as the inverse (direct) limit of a system of order (continuous) duals. In addition to a solution of the mentioned decomposition problem, a number of applications of the general theory of direct and inverse limits of vector lattices are presented in Section <ref>.  These include the computations of order (continuous) duals of function spaces and a structural characterisation of order dual spaces in terms of spaces of Borel measures.

In the next section, we state some preliminary definitions and results which are used in the rest of the paper.

§ PRELIMINARIES

§.§ Vector lattices

In order to make the paper reasonably self-contained we recall a few concepts and facts from the theory of vector lattices.  For undeclared terms and notation we refer to the reader to any of the standard texts in the field, for instance [3, 4, 31, 38].  Let $\vlat{E}$ and $\vlat{F}$ be real vector lattices. For $u,v\in \vlat{E}$ we write $u<v$ if $u\leq v$ and $u\neq v$. In particular, $0 < u$ means $u$ is positive but not zero. We note that if $\vlat{E}$ is a space of real-valued functions on a set $X$, then $0 < v$ does not mean that $0 < v(x)$ for every $x\in X$.

For sets $A,B\subseteq \vlat{E}$ let $A\vee B \defeq \{u\vee v ~:~ u\in A,~ v\in B\}$. The sets $A\wedge B$, $A^+$, $A^-$ and $|A|$ are defined similarly.  Lastly, $A^d \defeq \{u\in \vlat{E} ~:~ |u|\wedge |v|=0 \text{ for all } v\in A\}$.  We write $A\downarrow u$ if $A$ is downward directed and $\inf A=u$. Similarly, we write $B\uparrow u$ if $B$ is upward directed and $\sup B = u$.

Let $T:\vlat{E}\to \vlat{F}$ be a linear operator.  Recall that $T$ is positive if $T\left[ \vlat{E}^+ \right] \subseteq \vlat{F}^+$, and regular if $T$ is the difference of two positive operators.  $T$ is order bounded if $T$ maps order bounded sets in $\vlat{E}$ to order bounded sets in $\vlat{F}$.  If $\vlat{F}$ is Dedekind complete, $T$ is order bounded if and only if $T$ is regular <cit.>.  Further, $T$ is order continuous if $\inf |T[A]|=0$ whenever $A\downarrow 0$ in $\vlat{E}$.  Every order continuous operator is necessarily order bounded <cit.>.  $T$ is a lattice homomorphism if it preserves suprema and infima of finite sets, and a normal lattice homomorphism if it preserves suprema and infima of arbitrary sets; equivalently, if it is an order continuous lattice homomorphism, see <cit.>.  A lattice isomorphism is a bijective lattice homomorphism $T:\vlat{E}\to\vlat{F}$.  An operator $T$ is a lattice isomorphism if and only if it is bijective and both $T$ and $T^{-1}$ are positive <cit.>.  We say that $T$ is interval preserving if for all $0\leq u\in\vlat{E}$, $T[[0,u]]=[0,T(u)]$.  An interval preserving map need not be a lattice homomorphism, nor is a (normal) lattice homomorphism in general interval preserving, see for instance <cit.>.  However, the following holds.  We have not found this result in the literature, and therefore we include the simple proof.

Let $\vlat{E}$ and $\vlat{F}$ be vector lattices and $T:\vlat{E}\to\vlat{F}$ a positive operator.  The following statements are true.

(i) If $T$ is injective and interval preserving then $T$ is a lattice isomorphism onto an ideal in $\vlat{F}$, hence a normal lattice homomorphism into $\vlat{F}$.

(ii) If  $T$ is a lattice homomorphism and $T[\vlat{E}]$ is an ideal in $\vlat{F}$ then $T$ is interval preserving.

Assume that $T$ is injective and interval preserving.  $T[\vlat{E}]$ is an ideal in $\vlat{F}$ by <cit.>.  Therefore, because $T$ is injective, it suffices to show that $T$ is a lattice homomorphism.  To this end, consider $u,v\in \vlat{E}^+$.  Then $0\leq T(u) \wedge T(v) \leq T(u)$ and $0\leq T(u)\wedge T(v) \leq T(v)$.  Since $T$ is interval preserving and injective there exists $w\in [0,u]\cap[0,v] = [0, u\wedge v]$ so that $T(w)=T(u)\wedge T(v)$.  We have

\[
T(w)\leq T(u\wedge v)\leq T(u) \text{ and } T(w)\leq T(u\wedge v)\leq T(v).
\]

Hence $T(u)\wedge T(v) = T(w) \leq T(u\wedge v) \leq T(u)\wedge T(v)$ so that $T(u\wedge v) = T(w) = T(u)\wedge T(v)$.

To see that $T$ is a normal lattice homomorphism, let $A\downarrow 0$ in $\vlat{E}$.  Then $T[A]\downarrow 0$ in $T[\vlat{E}]$ because $T$ is a lattice isomorphism onto $T[\vlat{E}]$.  But $T[\vlat{E}]$ is and ideal in $\vlat{F}$, so $T[A]\downarrow 0$ in $\vlat{F}$.

Assume that $T$ is a lattice homomorphism and $T[\vlat{E}]$ is an ideal in $\vlat{F}$.  Let $0\leq u \in \vlat{E}$ and $0\leq v\leq T(u)$.  Because $T[\vlat{E}]$ is an ideal in $\vlat{F}$ there exists $w\in \vlat{E}$ so that $T(w) = v$.  Let $w'= (w\vee 0)\wedge u$.  Then $0\leq w'\leq u$ and $T(w') = (v\vee 0)\wedge T(u) = v$.

Let $\vlat{E}$ be a vector lattice, $\vlat{A}$ and $\vlat{B}$ projection bands in $\vlat{E}$, $P_\vlat{A}$ and $P_{\vlat{B}}$ the band projections of $\vlat{E}$ onto $\vlat{A}$ and $\vlat{B}$, respectively, and $I_{\vlat{E}}$ the identity operator on $\vlat{E}$.  Assume that $\vlat{A}\subseteq \vlat{B}$. The following statements are true.

(i) $P_\vlat{A}$ is an order continuous lattice homomorphism.

(ii) $P_\vlat{A} \leq I_{\vlat{E}}$.

(iii) $P_\vlat{A} P_\vlat{B} = P_\vlat{B} P_\vlat{A} = P_\vlat{A}$.

(iv) $P_\vlat{A}$ is interval preserving.

For (i), see <cit.>. For (ii) and (iii), see <cit.> and <cit.> respectively. Lastly, (iv) follows from Proposition <ref> (ii), since $P_{\vlat{A}}[\vlat{E}]=\vlat{A}$ is a band, hence an ideal, in $\vlat{E}$.

The order dual of $\vlat{E}$ is $\orderdual{\vlat{E}}\defeq \{\varphi: \vlat{E}\to\R ~:~ \varphi \text{ is order bounded}\}$, and the order continuous dual of $\vlat{E}$ is $\ordercontn{\vlat{E}} \defeq \{\varphi\in \orderdual{\vlat{E}} ~:~ \varphi \text{ is order continuous}\}$.  If $A\subseteq \vlat{E}$ and $B\subseteq \orderdual{\vlat{E}}$ we set

\[
\ann{A} \defeq \{\varphi\in \orderdual{\vlat{E}} ~:~ \varphi(u)=0,~u\in A\},~~ \preann{B} \defeq \{u\in\vlat{E} ~:~ \varphi(u) = 0,~ \varphi\in B\}.
\]

For $\varphi\in \orderdual{\vlat{E}}$ the null ideal (or absolute kernel) of $\varphi$ is

\[
\nullid{\varphi} \defeq \{u\in \vlat{E} ~:~ |\varphi|(|u|)=0\}.
\]

The carrier of $\varphi$ is $\carrier{\varphi} \defeq \nullid{\varphi}^d$.  The null ideal $\nullid{\varphi}$ of $\varphi$ is an ideal in $\vlat{E}$ and its carrier $\carrier{\varphi}$ is a band; if $\varphi$ is order continuous then $\nullid{\varphi}$ is also a band in $\vlat{E}$, see for instance <cit.>.

Define $\sigma:\vlat{E}\ni u\mapsto \Psi_u\in \ordercontnbidual{\vlat{E}}$ by setting $\Psi_u(\varphi)\defeq \varphi(u)$ for all $u\in\vlat{E}$ and $\varphi\in \ordercontn{\vlat{E}}$.  Then $\sigma$ is a lattice homomorphism, and, if $\preann{\ordercontn{\vlat{E}}}=\{0\}$, $\sigma$ is injective, see <cit.>.   We call $\vlat{E}$ perfect if $\sigma[\vlat{E}]=\ordercontnbidual{\vlat{E}}$.

In the following theorem, we briefly recall some basic facts concerning the order adjoint of a positive operator $T:E\to F$ which we make use of in the sequel.

Let $\vlat{E}$ and $\vlat{F}$ be vector lattices and $T:\vlat{E}\to\vlat{F}$ a positive operator. Denote by $T^\sim :\vlat{F}^\sim \to \vlat{E}^\sim$ its order adjoint, $\varphi\mapsto \varphi\circ T$.  The following statements are true.

(i) $T^\sim$ is positive and order continuous.

(ii) If $T$ is order continuous then $T^\sim[\ordercontn{\vlat{F}}]\subseteq \ordercontn{\vlat{E}}$.

(iii) If $T$ is interval preserving then $T^\sim$ is a lattice homomorphism.

(iv) If $T$ is a lattice homomorphism then $T^\sim$ is interval preserving.  The converse is true if $\preann{\orderdual{\vlat{F}}}=\{0\}$.

For (i), see <cit.>. The statement in (ii) follows directly from the fact that composition of order continuous operators is order continuous. For (iii), see <cit.>. The first statement in (iv) is proven in <cit.>. The second statement is proven in <cit.>. We note that although [4] declares a blanket assumption at the start of the book that all vector lattices under consideration in [4] are Archimedean, the proofs of <cit.> do not make use of this assumption.

The statements in (iii) and (iv) are special cases of  We note that although [4] declares a blanket assumption at the start of the book that all vector lattices under consideration in [4] are Archimedean, the proof of <cit.> does not make use of this assumption.

the positivity of $T^\sim$ is easily verified and order continuity follows directly from the proof of <cit.>.  The statements in (iii) and (iv) are special cases of <cit.>. We note that although [4] declares a blanket assumption at the start of the book that all vector lattices under consideration in [4] are Archimedean, the relevant part of the proof of <cit.> along with the proof of <cit.> does not make use of this assumption.

For (i), the positivity of $T^\sim$ is easily verified and order continuity follows directly from the proof of <cit.>. The statement in (ii) follows directly from the fact that composition of order continuous operators is order continuous. The statements in (iii) and (iv) are special cases of <cit.>. We note that although [4] declares a blanket assumption at the start of the book that all vector lattices under consideration in [4] are Archimedean, the relevant part of the proof of <cit.> along with the proof of <cit.> does not make use of this assumption.

Let $\vlat{E}$ and $\vlat{F}$ be vector lattices and $T:\vlat{E}\to\vlat{F}$ a linear lattice homomorphism onto $\vlat{F}$.  The following statements are true.

(i) $T^\sim[\vlat{F}^\sim] = \ann{\ker(T)}$.

(ii) If $\vlat{E}$ is Archimedean and $T$ is order continuous then $T^\sim[\ordercontn{\vlat{F}}] = \ann{\ker(T)}\cap\ordercontn{\vlat{E}}$.

Let $\varphi\in\vlat{F}^\sim$.  If $u\in\ker(T)$ then $T^\sim(\varphi) (u) = \varphi (T(u)) = \varphi(0) = 0$.  Hence $\varphi\in\ann{\ker(T)}$.

Let $\psi\in \ann{\ker(T)}$.  Define $\varphi:\vlat{F}\to\R$ by setting $\varphi(v) = \psi(u)$ if $v=T(u)$.  Then $\varphi\in \vlat{F}^\sim$ and $T^\sim(\varphi) = \psi$.

It follows from (i) and Theorem <ref> (ii) that $T^\sim[\ordercontn{\vlat{F}}] \subseteq \ann{\ker(T)}\cap\ordercontn{\vlat{E}}$.  We show that if $T^\sim(\varphi) \in \ordercontn{\vlat{E}}$ for some $\varphi\in \vlat{F}^\sim$ then $\varphi\in \ordercontn{\vlat{F}}$.  From this and (i) it follows that $T^\sim[\ordercontn{\vlat{F}}] = \ann{\ker(T)}\cap\ordercontn{\vlat{E}}$.  We observe that it suffices to consider positive $\varphi\in\vlat{F}^\sim$.  Indeed, $T$ is a surjective lattice homomorphism and therefore also interval preserving.  Hence by Theorem <ref> (iii), $T^\sim$ is a lattice homomorphism.

Suppose that $0\leq \varphi\in\vlat{F}^\sim$ and that $T^\sim(\varphi) \in \ordercontn{\vlat{E}}$.  Let $A\downarrow 0$ in $\vlat{F}$.  Define $B\defeq T^{-1}[A]\cap \vlat{E}_+$.  Then $B$ is downward directed and $T[B]=A$.  In particular, $\varphi[A] = T^\sim(\varphi)[B]$.  Let $C\defeq \{w\in \vlat{E} ~:~ 0\leq w\leq v \text{ for all } v\in B\}$.  If $w\in C$ then $0\leq T(w) \leq u$ for all $u\in A$ so that $T(w) = 0$.  Hence $C\subseteq \ker(T)$.  Since $\vlat{E}$ is Archimedean, we have $B-C\downarrow 0$ in $\vlat{E}$, see <cit.>.  Since $T^\sim(\varphi)$ is order continuous, $T^\sim(\varphi)[B-C]\downarrow 0$; that is, for every $\epsilon>0$ there exists $v\in B$ and $w\in C$ so that $\varphi(T(v)) = \varphi (T(v-w)) = T^\sim(\varphi)(v-w)<\epsilon$.  Hence, for every $\epsilon>0$ there exists $u\in A$ so that $\varphi(u)<\epsilon$.  This shows that $\varphi[A]\downarrow 0$ so that $\varphi\in \ordercontn{\vlat{F}}$ as required.
Let $I$ be a non-empty set and let $\vlat{E}_\alpha$ be a vector lattice for every $\alpha\in I$. Then $\dsprod_{\alpha\in I} \vlat{E}_\alpha$ is a vector lattice with respect to the coordinate-wise operations.  If the index set is clear form the context, we omit it and write $\dsprod \vlat{E}_\alpha$.  For $\beta\in I$ let $\pi_\beta:\dsprod\vlat{E}_\alpha\to \vlat{E}_\beta$ be the coordinate projection onto $\vlat{E}_\beta$ and $\iota_\beta:\vlat{E}_\beta\to\dsprod \vlat{E}_\alpha$ the right inverse of $\pi_\beta$ given by

\[
\pi_\alpha(\iota_\beta(u))= \left\{\begin{array}{lll}
u & \text{if} & \alpha=\beta \smallskip \\
0 & \text{if} & \alpha\neq \beta.\\
\end{array}\right.
\]

We denote by $\displaystyle\bigoplus \vlat{E}_\alpha$ the ideal in $\dsprod \vlat{E}_\alpha$ consisting of $u\in \dsprod \vlat{E}_\alpha$ for which $\pi_\alpha(u)\neq 0$ for only finitely many $\alpha\in I$.  The following properties of $\dsprod \vlat{E}_\alpha$ and $\displaystyle \bigoplus \vlat{E}_\alpha$ are used frequently in the sequel.

Let $I$ be a non-empty set and $\vlat{E}_\alpha$ a vector lattice for every $\alpha\in I$.  The following statements are true.

(i) The coordinate projections $\pi_\beta$ and their right inverses $\iota_\beta$ are normal, interval preserving lattice homomorphisms.
(ii) $\dsprod \vlat{E}_\alpha$ is Archimedean if and only if each $\vlat{E}_\alpha$ is Archimedean.
(iii) $\dsprod \vlat{E}_\alpha$ is Dedekind complete if and only if each $\vlat{E}_\alpha$ is Dedekind complete.
(iv) If $I$ has non-measurable cardinal, then the order dual of $\dsprod \vlat{E}_\alpha$ is $\displaystyle \bigoplus \vlat{E}_\alpha^\sim$.
(v) The order continuous dual of $\dsprod\vlat{E}_\alpha$ is $\displaystyle \bigoplus \ordercontn{(\vlat{E}_\alpha)}$.
(vi) The order dual of $\displaystyle \bigoplus \vlat{E}_\alpha$ is $\dsprod \vlat{E}_\alpha^\sim$.
(vii) The order continuous dual of $\displaystyle \bigoplus \vlat{E}_\alpha$ is $\dsprod \ordercontn{\left( \vlat{E}_\alpha \right)}$.

We leave the straightforward proofs of (i), (ii), (iii), (vi) and (vii) to the reader.

Assume that $I$ has non-measurable cardinal.  By (i) of this theorem and Theorem <ref> (iii) and (iv), $\iota_\beta^\sim : \orderdual{\left(\dsprod \vlat{E}_\alpha\right)}\to \orderdual{\vlat{E}}_\beta$ is an interval preserving normal lattice homomorphism for every $\beta\in I$.  Because each $\varphi\in \orderdual{\left(\dsprod \vlat{E}_\alpha\right)}$ is linear and order bounded, the set $I_\varphi \defeq \{\beta \in I ~:~ \iota_\beta^\sim(\varphi)\neq 0 \}$ is finite.  Define $S:\orderdual{\left(\dsprod \vlat{E}_\alpha\right)} \to \displaystyle \bigoplus \orderdual{\vlat{E}}_\alpha$ by setting

\[
S(\varphi) \defeq (\iota_\alpha^\sim(\varphi))_{\alpha\in I},~~ \varphi\in \orderdual{\left(\dsprod \vlat{E}_\alpha\right)}.
\]
Then $S$ is a lattice homomorphism. It remains to verify that $S$ is bijective.

We show that $S$ is injective.  Let $0\neq \varphi \in \orderdual{\left(\dsprod \vlat{E}_\alpha\right)}$.  Fix $0\leq u\in\dsprod\vlat{E}_\alpha$ so that $\varphi(u)\neq 0$.  For $f\in \R^I$ let $fu\in\dsprod \vlat{E}_\alpha$ be defined by $\pi_\alpha (fu) = f(\alpha)\pi_\alpha (u)$, $\alpha\in I$.  Define $\hat\varphi : \R^I\to \R$ by setting

\[
\hat \varphi (f) \defeq \varphi(fu),~~ f\in \R^I.
\]

Then $\hat\varphi$ is a non-zero order bounded linear functional on $\R^I$.  Because $I$ has nonmeasurable cardinal, $I$ with the discrete topology is realcompact, see <cit.>.  Therefore there exists a non-zero finitely supported and countably additive measure $\mu$ on the powerset $2^I$ of $I$ so that

\[
\hat\varphi (f) = \int_I f \thinspace d\mu = \sum_{\alpha \in I} f(\alpha)\mu(\alpha),~~ f\in \R^I,
\]

see <cit.>.  Let $\alpha$ be in the support of $\mu$, and let $g$ be the indicator function of $\{\alpha\}$.  Then $0\neq \mu(\alpha)=\hat \varphi(g) = \varphi(gu) = \iota_\alpha^\sim(\varphi)(\pi_\alpha(u))$.  Therefore $S(\varphi)\neq 0$ so that $S$ is injective.

To see that $S$ is surjective, observe that for every $\beta\in I$, $\pi_\beta^\sim :\orderdual{\vlat{E}}_\beta\to \orderdual{\left(\dsprod \vlat{E}_\alpha\right)}$ is an interval preserving normal lattice homomorphism by (i) of this theorem and Theorem <ref> (iii) and (iv).  Define $T: \displaystyle \bigoplus \orderdual{\vlat{E}}_\alpha\to \orderdual{\left(\dsprod \vlat{E}_\alpha\right)}$ by setting

\[
T(\psi) \defeq \sum \pi_\alpha^\sim(\psi_\alpha),~~ \psi=(\psi_\alpha)\in \bigoplus \orderdual{\vlat{E}}_\alpha.
\]

Then $T$ is a positive operator.  We claim that $S\circ T$ is the identity on $\displaystyle \bigoplus \orderdual{\vlat{E}}_\alpha$.  Indeed, for any $\psi\in \displaystyle \bigoplus \orderdual{\vlat{E}}_\alpha$ we have

\[
S(T(\psi)) = \sum_{\alpha\in I}(\iota_\beta^\sim(\pi_\alpha^\sim(\psi_\alpha)))_{\beta\in I} = \sum_{\alpha\in I}(\psi_\alpha\circ \pi_\alpha\circ\iota_\beta )_{\beta\in I}.
\]
By definition of the $\iota_\beta$ it follows that $S(T(\psi)) = \psi$ which verifies our claim. Therefore $S$ is a lattice isomorphism.

Define $S:\orderdual{\left(\dsprod \vlat{E}_\alpha\right)} \to \displaystyle \bigoplus \orderdual{\vlat{E}}_\alpha$ as in the proof of (iv).  By Theorem <ref> (ii), $S$ maps $\ordercontn{\left(\dsprod \vlat{E}_\alpha\right)}$ into $\displaystyle \bigoplus \ordercontn{(\vlat{E}_\alpha)}$.  A similar argument to that given in proof of (iv) shows that $S$ is a surjective lattice homomorphism.  Hence it remains to show that $S$ is injective.

Let $0\leq \varphi \in \ordercontn{\left(\dsprod \vlat{E}_\alpha\right)}$ and suppose that $S(\varphi) = 0$.  Then $\iota_\beta^\sim(\varphi) = 0$ for every $\beta\in I$.  But for any $0\leq u \in \dsprod\vlat{E}_\alpha$,

\[
u = \sup\left\lbrace \sum_{\alpha \in F}\iota_\alpha(u) ~:~F\subseteq I \text{ is finite} \right\rbrace.
\]

Therefore by the order continuity of $\varphi$,

\[
\varphi(u) = \sup \left\lbrace \sum_{\alpha \in F}\iota_\alpha^\sim (\varphi)(u) ~:~ F\subseteq I \text{ is finite}\right\rbrace = 0
\]

for all $0\leq u \in \dsprod\vlat{E}_\alpha$; hence $\varphi=0$.  Because $S$ is a lattice homomorphism it follows that, for all $\varphi\in \ordercontn{\left(\dsprod \vlat{E}_\alpha\right)}$, if $S(\varphi) = 0$ then $\varphi=0$; that is, $S$ is injective.

We note that the statement in Theorem 2.5 (iv) is not true if $I$ has measurable cardinal:  In this case the map $S$ in the proof of Theorem 2.5 (iv) is not injective.  To see this, suppose that $I$ has measurable cardinal.  Then $I$ with the discrete topology is not realcompact.  We identify $\R^I$ with $\cont (\upsilon I)$. Let $x\in \upsilon I\setminus I$.  Then $\delta_x :\R^I \ni u\mapsto u(x)\in \R$ is a non-zero, positive linear functional on $\R^I$, but $S(\delta_x)=0$.

We now define the categories which are the setting of this paper.  It is readily verified that these are indeed categories.

${}$       Objects        Morphisms

$\vlc$     Vector lattices     Lattice homomorphisms

$\nvlc$     Vector lattices     Normal lattice homomorphisms

$\vlic$     Vector lattices     Interval preserving lattice homomorphisms

$\nvlic$     Vector lattices     Normal, interval preserving lattice homomorphisms

We refer to these four categories as categories of vector lattices.  If ${\bf C}$ is a category of vector lattices, then a ${\bf C}$-morphism is a morphism within the category ${\bf C}$. Below we depict the subcategory relationships between the categories of vector lattices under consideration.

[row sep = 4pt , column sep = 7pt]

[symbol=⊇]ru [symbol=⊇]rd

§.§ Measures on topological spaces

Because the terminology related to measures on topological spaces varies across the literature, we declare our conventions.  Let $X$ be a Hausdorff topological space.  For a function $u:X\to\R$ we denote by $\zeroset{u}$ the zero set of $u$ and by $\cozeroset{u}$ its co-zero set, that is, the complement of $\zeroset{u}$.  If $A\subseteq X$ then $\onefunction_A$ denotes the indicator function of $A$.

Denote by $\borel{X}$ the Borel $\sigma$-algebra generated by the open sets in $X$. A (signed) Borel measure on $X$ is a real-valued and $\sigma$-additive function on $\borel{X}$.  We denote the space of all signed Borel measures on $X$ by $\vlat{M}_\sigma(X)$.  This space is a Dedekind complete vector lattice with respect to the pointwise operations and order <cit.>.  In particular, for $\mu,\nu\in \vlat{M}_\sigma(X)$,

\[
(\mu\vee \nu)(B) = \sup\left\lbrace \mu(A)+\nu(B\setminus A) ~:~ A\subseteq B,~~ A \in \borel{X} \right\rbrace,~~ B \in \borel{X}.
\]
For any upward directed set $D\subseteq \vlat{M}_\sigma(X)^+$ with $\sup D=\nu$ in $\vlat{M}_\sigma(X)$,

\[
\nu(B) = \sup\{\mu(B) ~:~ \mu\in D\},~~ B \in \borel{X}.
\]
Following Bogachev [9], we call a Borel measure $\mu$ on $X$ a Radon measure if for every $B \in \borel{X}$,

\[
|\mu|(B)=\sup\{|\mu|(K) ~:~ K\subseteq B \text{ is compact}\}.
\]

Equivalently, $\mu$ is Radon if for every $B \in \borel{X}$ and every $\epsilon>0$ there exists a compact set $K\subseteq B$ so that $|\mu|(B\setminus K)<\epsilon$.  Observe that if $\mu$ is Radon, then also

\[
|\mu|(B)=\inf\{ |\mu|(U) ~:~ U\supseteq B \text{ is open}\}.
\]

Denote the space of Radon measures on $X$ by $\vlat{M}(X)$.

Recall that the support of a Borel measure $\mu$ on $X$ is defined as

\[
S_\mu \defeq \{x\in X ~:~ |\mu|(U)>0 \text{ for all } U\ni x \text{ open}\}.
\]

A non-zero Borel measure $\mu$ may have empty support, and even if $S_\mu\neq \emptyset$, it may have measure zero <cit.>.  However, if $\mu$ is a non-zero Radon measure, then $S_\mu \neq \emptyset$ and $|\mu|(S_\mu)=|\mu|(X)$; in fact, for every $B \in \borel{X}$, $|\mu|(B)=|\mu|(B\cap S_\mu)$.  We list the following useful properties of the support of a measure; the proofs are straightforward and therefore omitted.

Let $\mu$ and $\nu$ be Radon measures on $X$.  The following statements are true.

(i) If $|\mu|\leq |\nu|$ then $S_\mu\subseteq S_\nu$.

(ii) $S_{\mu+\nu}\subseteq S_{|\mu|+|\nu|}$

(iii) $S_{|\mu|+|\nu|} = S_\mu \cup S_\nu$.

A Radon measure $\mu$ is called compactly supported if $S_\mu$ is compact.  We denote the space of all compactly supported Radon measures on $X$ as $\vlat{M}_c(X)$. Further, a Radon measure $\mu$ on $X$ is called a normal measure if $|\mu|(L)=0$ for all closed nowhere dense sets $L$ in $X$.  The space of all normal Radon measures on $X$ is denoted $\vlat{N}(X)$, and the space of compactly supported normal Radon measures by $\vlat{N}_c(X)$.

The following statements are true.

(i) $\vlat{M}(X)$ is an band in $\vlat{M}_\sigma(X)$

(ii) $\vlat{M}_c(X)$ is an ideal in $\vlat{M}(X)$.

(iii) $\vlat{N}(X)$ is a band in $\vlat{M}(X)$.

(iv) $\vlat{N}_c(X)$ is a band in $\vlat{M}_c(X)$.

For the proof of (i), let $\mu,\nu\in\vlat{M}(X)$.  Consider a Borel set $B$ and a real number $\epsilon>0$.  There exists a compact set $K\subseteq B$ so that $|\mu|(B\setminus K)<\epsilon/2$ and $|\nu|(B\setminus K)<\epsilon/2$.  We have $|\mu+\nu|(B\setminus K)\leq |\mu|(B\setminus K) + |\nu|(B\setminus K)<\epsilon$.  Therefore $\mu+\nu \in \vlat{M}(X)$.  A similar argument shows that $a\mu\in \vlat{M}(X)$ for all $a\in\R$.  It also follows in this way that for all $\nu\in\vlat{M}_\sigma(X)$ and $\mu\in \vlat{M}(X)$, if $|\nu|\leq |\mu|$ then $\nu\in \vlat{M}(X)$.  By definition of a Radon measure, $|\mu|\in\vlat{M}(X)$ whenever $\mu\in\vlat{M}(X)$.  Therefore $\vlat{M}(X)$ is an ideal in $\vlat{M}_\sigma(X)$.

To see that $\vlat{M}(X)$ is a band in $\vlat{M}_\sigma(X)$, consider an upward directed subset $D$ of $\vlat{M}(X)^+$ so that $\sup D=\nu$ in $\vlat{M}_\sigma(X)$.  Fix a Borel set $B$ and a real number $\epsilon>0$.  There exists $\mu\in D$ so that $\nu(B)-\epsilon/2<\mu(B)$.  But $\mu$ is a Radon measure, so there exists a compact subset $K$ of $B$ so that $\mu(K)>\mu(B)-\epsilon/2$.  Therefore $\nu(K)\geq \mu(K)>\mu(B)-\epsilon/2> \nu(B)-\epsilon$.  Therefore $\nu \in \vlat{M}(X)$ so that $\vlat{M}(X)$ is a band in $\vlat{M}_\sigma(X)$.

The statement in (ii) follows immediately from the definition of the support of a measure and Proposition <ref>. It is clear that $\vlat{N}(X)$ is an ideal in $\vlat{M}(X)$, and that it is a band follows from the expression for suprema in $\vlat{M}_\sigma(X)$.  Hence (iii) is true.  That (iv) is true follows immediately from (iii).

Unsurprisingly, there is a close connection between Radon measures on $X$ and order bounded linear functionals on $\contX$.  Theorem <ref> to follow is implicit in <cit.>, see also [24] where a treatment is given in terms of Baire measures.  In order to facilitate the discussion of order continuous functionals, we include a short proof.

Let $X$ be a realcompact space.  There is a lattice isomorphism $\orderdual{\contX}\ni \varphi \longmapsto \mu_\varphi \in \vlat{M}_c(X)$ so that for every $\varphi\in \orderdual{\contX}$,

\[
\varphi(u) = \int_X u \thinspace d\mu_\varphi,~~ u\in\contX.
\]

We identify the space $\contbX$ with $\cont\left(\beta X\right)$. Because $\contbX$ is an ideal in $\contX$, the restriction map from $\orderdual{\contX}$ to $\orderdual{\contbX}$ is a lattice homomorphism <cit.>.  It follows from <cit.> that this map is injective.  Thus by the Riesz Representation Theorem <cit.>, for every $\varphi\in\orderdual{\contX}$ there exists a unique Radon measure $\nu_\varphi$ on $\beta X$ so that

\[
\varphi(u) = \int_{\beta X}u \thinspace d\nu_\varphi,~~ u\in\contbX.
\]

Furthermore, the map $\varphi\mapsto \nu_\varphi$ is a lattice isomorphism onto its range.

We claim that the range of this map is $\vlat{M}_0(\beta X)\defeq \{\nu \in \vlat{M}(\beta X) ~:~ S_\nu \subseteq X\}$.  According to <cit.>, $S_{\nu_\varphi}\subseteq X$ for every $\varphi\in \orderdual{\contX}$.  Hence $\nu_\varphi\in \vlat{M}_0(\beta X)$.  Conversely, let $\nu\in \vlat{M}_0(\beta X)$.  Since $S_\nu \subseteq X$ is compact in $\beta X$, hence also in $X$,

\[
\psi(u) \defeq \int_{S_{\nu}} u \thinspace d\nu,~~ u\in\contX
\]

defines an order bounded functional on $\contX$.  For every $u\in \contbX$ we have

\[
\psi(u) = \int_{S_{\nu}} u \thinspace d\nu = \int_{\beta X} u d\nu.
\]

Therefore $\nu_\psi = \nu$ which establishes our claim.

We have shown that $\orderdual{\contX}\ni \varphi\mapsto \nu_\varphi \in \vlat{M}_0(\beta X)$ is a lattice isomorphism.  We now show that $\vlat{M}_0(\beta X)$ is isomorphic to $\vlat{M}_c(X)$.

Let $\nu\in \vlat{M}_0(\beta X)$.  The Borel sets in $X$ are precisely the traces on $X$ of Borel sets in $\beta X$ <cit.>.  Furthermore, if $B', B'' \in \borel{\beta X}$ so that $B'\cap X = B''\cap X$ then $\nu(B')=\nu(B'\cap S_\nu)=\nu(B''\cap S_\nu)=\nu(B'')$.  For $B \in \borel{X}$ define

\[
\nu^\ast (B) \defeq \nu(B') \text{ with }B'\in \borel{\beta X} \text{ so that }B'\cap X=B.
\]

It follows from the previous observation that $\nu^\ast$ is well-defined.  It follows easily that $\nu^\ast\in \vlat{M}_c(X)$, and that the map $\vlat{M}_0(\beta X)\ni \nu\mapsto \nu^\ast\in \vlat{M}_c(X)$ is injective, linear, and bipositive.  Let $\mu\in \vlat{M}_c(X)$.  For every $B\in \borel{\beta X}$ let $\nu(B) \defeq \mu(B\cap X)$.  Then $\nu\in \vlat{M}_0(\beta X)$ and $\nu^\ast = \mu$.  Therefore $\vlat{M}_0(\beta X)\ni \nu\mapsto \nu^\ast\in \vlat{M}_c(X)$ is a lattice isomorphism.

For $\varphi \in \orderdual{\contX}$ let $\mu_\varphi \defeq (\nu_\varphi)^\ast$.  Then $\orderdual{\contX}\ni \varphi\mapsto \mu_\varphi \in \vlat{M}_c(X)$ is a lattice isomorphism. It remains to show that, for every $\varphi \in \orderdual{\contX}$,

\[
\varphi(u) = \int_{X} u \thinspace d\mu_\varphi,~~ u\in\contX.
\]

Fix $0\leq \varphi \in \orderdual{\contX}$ and $u\in\contX^+$.  A minor modification of the proof of <cit.> shows that there exists a natural number $N$ so that $\varphi( u )=\varphi( u \wedge n\onefunction_X)$ for every $n\geq N$.  But

\[
\int_X u \thinspace d\mu_\varphi = \sup_{n\in\N} \int_X u \wedge n\onefunction_X \thinspace d\mu_\varphi,
\]

and, for every $n\in\N$,

\[
\int_X u \wedge n\onefunction_X \thinspace d\mu_\varphi = \int_{\beta X} u \wedge n\onefunction_X \thinspace d\nu_\varphi = \varphi(u\wedge n\onefunction_X)
\]

\[
\varphi(u) = \int_X u \thinspace d\mu_\varphi,
\]

as desired.

Let $X$ be a realcompact space.  Let $\varphi$ be an order bounded functional on $\contX$.  Then $\varphi$ is order continuous if and only if $\mu_\varphi$ is a normal measure.  The map

\[
\ordercontn{\contX}\ni \varphi \longmapsto \mu_{\varphi} \in \vlat{N}_c(X)
\]
is a lattice isomorphism onto $\vlat{N}_c(X)$.

We make use of the notation introduced in the proof of Theorem <ref>.  It suffices to show that for any $0\leq \varphi\in \orderdual{\contX}$, $\varphi$ is order continuous if and only if $\mu_\varphi$ is normal.  Let $0\leq \varphi \in \ordercontn{\contX}$.  Because $\contbX$ is an ideal in $\contX$ the restriction of $\varphi$ to $\contb(X)$ is order continuous.  Hence the measure $\nu_\varphi\in\vlat{M}_0(\beta X)$ so that

\[
\varphi(u)=\int_{\beta X}u \thinspace d\nu_\varphi,~~ u\in\contb(X)
\]

is a normal measure on $\beta X$, see for instance <cit.>.  It therefore follows that the measure $\mu_\varphi = (\nu_\varphi)^\ast \in\vlat{M}_c(X)$ is a normal measure on $X$.

Conversely, let $0\leq \varphi \in \orderdual{\contX}$ be such that $\mu_\varphi$ is a normal measure on $X$.  Then the Borel measure $\nu$ on $\beta X$ given by

\[
\nu(B) = \mu_\varphi(B\cap X),~~ B\in \borel{\beta X}
\]
is a normal measure on $\beta X$.  Hence $S_{\nu}$ is regular-closed in $\beta X$, see <cit.>.  But $S_\nu = S_{\mu_\varphi}\subseteq X$ so that $S_{\mu_\varphi}$ is regular-closed in $X$.  Therefore, if $D\downarrow 0$ in $\contX$ then $\left.D\right|_{S_{\mu_\varphi}} = \{  \left. u \right|_{S_{\mu_\varphi}} ~:~ u\in D \} \downarrow 0$ in $\cont(S_{\mu_\varphi})$, see <cit.>.  Also, $\mu_\varphi$ restricted to the Borel sets in $S_{\mu_\varphi}$ is a normal measure on $S_{\mu_\varphi}$.  Hence

\[
\inf_{u\in D} \varphi(u) = \inf_{u\in D}\int_{S_{\mu_\varphi}}u \thinspace d\mu_\varphi = 0.
\]

Therefore $\varphi$ is order continuous.

§ DIRECT LIMITS

We recall the definitions of a direct system in a category of vector lattices, and of the direct limit of such a system.  These definitions are specializations of the corresponding definitions in general categories, see for instance <cit.> and <cit.> where direct limits are referred to as colimits. We summarise some existence results and list vector lattice properties that have permanence under the direct limit construction.  Addition results are found in [19].  Lastly, we give a number of examples of direct limits which we will make use of later.

Let ${\bf C}$ be a category of vector lattices, $I$ a directed set, $\vlat{E}_\alpha$ a vector lattice for each $\alpha\in I$, and $e_{\alpha, \beta}: \vlat{E}_\alpha\to \vlat{E}_\beta$ a ${\bf C}$-morphism for all $\alpha \precc \beta$ in $I$.  The ordered pair $\cal{D} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ is called a direct system in ${\bf C}$ if, for all $\alpha \precc \beta \precc \gamma$ in $I$, the diagram

\[
\begin{tikzcd}[cramped]
\vlat{E}_\alpha \arrow[rd, "e_{\alpha, \beta}"'] \arrow[rr, "e_{\alpha, \gamma}"] & & \vlat{E}_\gamma\\
& \vlat{E}_\beta\arrow[ru, "e_{\beta, \gamma}"']
\end{tikzcd}
\]

commutes in ${\bf C}$.

Let ${\bf C}$ be a category of vector lattices and $\cal{D} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ a direct system in ${\bf C}$. Let $\vlat{E}$ be a vector lattice and for every $\alpha\in I$, let $e_\alpha: \vlat{E}_\alpha \to \vlat{E}$ be a ${\bf C}$-morphism.  The ordered pair $\cal{S} \defeq (\vlat{E}, (e_\alpha)_{\alpha\in I})$ is a  compatible system of $\cal{D}$ in ${\bf C}$ if, for all $\alpha\precc\beta$ in $I$, the diagram

\[
\begin{tikzcd}[cramped]
\vlat{E}_\alpha \arrow[rd, "e_{\alpha, \beta}"'] \arrow[rr, "e_{\alpha}"] & & \vlat{E}\\
& \vlat{E}_\beta\arrow[ru, "e_{\beta}"']
\end{tikzcd}
\]

commutes in ${\bf C}$.

Let ${\bf C}$ be a category of vector lattices and $\cal{D} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ a direct system in ${\bf C}$.  The direct limit of $\cal{D}$ in ${\bf C}$ is a compatible system $\cal{S}\defeq(\vlat{E}, (e_\alpha)_{\alpha\in I})$ of $\cal{D}$ in ${\bf C}$ so that for any compatible system $\tilde{S} \defeq (\tilde{\vlat{E}}, (\tilde{e}_\alpha)_{\alpha\in I})$ of $\cal{D}$ in ${\bf C}$ there exists a unique ${\bf C}$-morphism $r: \vlat{E}\to \tilde{\vlat{E}}$ so that, for every $\alpha\in I$, the diagram

\[
\begin{tikzcd}[cramped]
\vlat{E} \arrow[rr, "r"] & & \tilde{\vlat{E}}\\
& \vlat{E}_\alpha \arrow[lu, "e_{\alpha}"] \arrow[ru, "\tilde{e}_{\alpha}"']
\end{tikzcd}
\]

commutes in ${\bf C}$.  We denote the direct limit of a direct system $\cal{D}$ by $\ind{\cal{D}}$ or $\ind{\vlat{E}_\alpha}$.

Since the direct limit of a direct system is in fact an initial object in a certain derived category, it follows that the direct limit, when it exists, is unique up to a unique isomorphism, see for instance <cit.>.

§.§ Existence and permanence properties of direct limits

Filter [19] shows that any direct system $\cal{D} \defeq ((\vlat{E}_\alpha)_{\alpha\in I},(e_{\alpha , \beta})_{\alpha \precc\beta})$ in $\vlc$ has a direct limit in $\vlc$.[It is not formulated in exactly these terms.]  In particular, the set-theoretic direct limit <cit.> of $\cal{D}$ equipped with suitable vector space and order structures is also the direct limit of $\cal{D}$ in $\vlc$.  We briefly recall the details.

For $u$ in the disjoint union $\biguplus \vlat{E}_\alpha$ of the collection $\left( \vlat{E}_\alpha \right)_{\alpha\in I}$, denote by $\alpha(u)$ that element of $I$ so that $u\in \vlat{E}_{\alpha(u)}$.  Define an equivalence relation on $\biguplus \vlat{E}_\alpha$ by setting $u\sim v$ if and only if there exists $\beta \scc \alpha (u),\alpha (v)$ in $I$ so that $e_{\alpha(u) , \beta}(u) = e_{\alpha(v) , \beta}(v)$.  Let $\vlat{E} \defeq  \biguplus \vlat{E}_\alpha / \sim$ and denote the equivalence class generated by $u\in \biguplus\vlat{E}_\alpha$ by $\dot u$.

Let $\dot u,\dot v\in \vlat{E}$.  We set $\dot u\leq \dot v$ if and only if there exists $\beta \scc \alpha (u),\alpha(v)$ in $I$ so that $e_{\alpha(u) , \beta}(u) \leq e_{\alpha(v) , \beta}(v)$.  Further, for $a,b\in\R$ define

\[
a\dot u + b\dot v \defeq \dot{\overbrace{a e_{\alpha(u) , \beta}(u) + b e_{\alpha(v) , \beta}(v)}},
\]

where $\beta \scc \alpha(u),\alpha(v)$ in $I$ is arbitrary.  With addition, scalar multiplication and the partial order so defined, $\vlat{E}$ is a vector lattice.  The lattice operations are given by

\[
\dot u\wedge\dot v\ = \ \dot{\overbrace{e_{\alpha(u) , \beta}(u) \wedge e_{\alpha(v) , \beta}(v)}}
\]

\[
\dot u\vee\dot v\ = \ \dot{\overbrace{e_{\alpha(u) , \beta}(u) \vee e_{\alpha(v) , \beta}(v)}},
\]
with $\beta \scc \alpha(u),\alpha(v)$ in $I$ arbitrary.

For each $\alpha\in I$ define $e_\alpha : \vlat{E}_\alpha \to \vlat{E}$ by setting $e_\alpha (u) \defeq \dot u$ for $u\in \vlat{E}_\alpha$. Each $e_\alpha$ is a lattice homomorphism and the diagram

\[
\begin{tikzcd}[cramped]
\vlat{E}_\alpha \arrow[rd, "e_{\alpha, \beta}"'] \arrow[rr, "e_{\alpha}"] & & \vlat{E}\\
& \vlat{E}_\beta\arrow[ru, "e_{\beta}"']
\end{tikzcd}
\]

commutes in $\vlc$ for all $\alpha \precc\beta$ in $I$ so that $\cal{S}\defeq (\vlat{E},(e_\alpha)_{\alpha\in I})$ is a compatible system of $\cal{D}$ in $\vlc$.  Further, if $\tilde{\cal{S}} = (\tilde{\vlat{E}},(\tilde{e}_\alpha)_{\alpha\in I}$ is another compatible system of $\cal{D}$ in $\vlc$ then

\[
r: \vlat{E}\ni \dot u \longmapsto \tilde{e}_{\alpha(u)}(u)\in \tilde{\vlat{E}}
\]

is the unique lattice homomorphism so that the diagram

\[
\begin{tikzcd}[cramped]
\vlat{E} \arrow[rr, "r"] & & \tilde{\vlat{E}}\\
& \vlat{E}_\alpha \arrow[lu, "e_{\alpha}"] \arrow[ru, "\tilde{e}_{\alpha}"']
\end{tikzcd}
\]

commutes for every $\alpha\in I$.  Hence $\cal{S}$ is indeed the direct limit of $\cal{D}$ in $\vlc$.

We give two further existence results for direct limits of direct systems in other categories of vector lattices.

Let $\cal{D} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ be a direct system in $\vlic$, and let $\cal{S} \defeq (\vlat{E},(e_\alpha)_{\alpha \in I})$ be the direct limit of $\cal{D}$ in $\vlc$. Then $\cal{S}$ is the direct limit of $\cal{D}$ in $\vlic$.

We show that each $e_\alpha$ is interval preserving.  To this end, fix $\alpha\in I$ and $0<u\in \vlat{E}_\alpha$.  Suppose that $\dot 0 \leq \dot v \leq e_\alpha (u)=\dot u$.  Then there exists a $\beta \scc \alpha , \alpha(v)$ in $I$ so that $0\leq e_{\alpha(v) , \beta}(v) \leq e_{\alpha , \beta} (u)$.  But $e_{\alpha , \beta}$ is interval preserving, so there exists $0\leq w\leq u$ in  $\vlat{E}_\alpha$ so that $e_{\alpha , \beta}(w)=e_{\alpha(v) , \beta}(v)$.  Therefore $e_\alpha(w) = \dot w = \dot v$.  Hence $e_\alpha$ is interval preserving. Therefore $\cal{S}$ is a compatible system of $\cal{D}$ in $\vlic$.

Let $\tilde{\cal{S}}=(\tilde{\vlat{E}},(\tilde{e}_\alpha)_{\alpha\in I})$ be a compatible system of $\cal{D}$ in $\vlic$, thus also in $\vlc$.  We show that the unique linear lattice homomorphism $r:\vlat{E}\to \tilde{\vlat{E}}$ is interval preserving.  Consider $\dot u \in \vlat{E}^+$.  Let $0\leq v \leq r(\dot u)$ in $\tilde{\vlat{E}}$, that is, $0\leq v\leq \tilde{e}_{\alpha(u)}(u)$.  But $\tilde{e}_{\alpha(u)}$ is interval preserving so there exists $0\leq w\leq u$ in $\vlat{E}_{\alpha(u)}$ so that $v=\tilde{e}_{\alpha(u)}(w)$.  Thus $\dot 0 \leq \dot w \leq \dot u$ and $r(\dot w) = v$ in $\vlat{E}$.  Therefore $r$ is interval preserving.

Let $\cal{D} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ be a direct system in $\nvlic$, and let $\cal{S} \defeq (\vlat{E},(e_\alpha)_{\alpha \in I})$ be the direct limit of $\cal{D}$ in $\vlc$.  Assume that $e_{\alpha , \beta}$ is injective for all $\alpha \precc \beta$ in $I$.  Then $\cal{S}$ is the direct limit of $\cal{D}$ in $\nvlic$.

We start by proving that $e_\alpha :\vlat{E}_\alpha\to \vlat{E}$ is injective for every $\alpha\in I$.  Fix $\alpha \in I$ and $u\in \vlat{E}_\alpha$ so that $e_\alpha (u ) = \dot 0$ in $\vlat{E}$.  Then there exists $\beta \scc \alpha$ in $I$ so that $e_{\alpha , \beta} (u)=0$.  But $e_{\alpha , \beta}$ is injective, so $u=0$.  Hence $e_\alpha$ is injective.

By Theorem <ref>, $e_\alpha :\vlat{E}_\alpha\to \vlat{E}$ is an injective interval preserving lattice homomorphism for every $\alpha\in I$. It follows from Proposition <ref> (i) that $e_\alpha$ is a $\nvlic$-morphism for every $\alpha\in I$. Therefore $\cal{S}$ is a compatible system of $\cal{D}$ in $\nvlic$.

Let $\tilde{S} = (\tilde{\vlat{E}},(\tilde{e}_\alpha)_{\alpha\in I})$ be a compatible system of $\cal{D}$ in $\nvlic$.  By Theorem <ref> the canonical map $r:\vlat{E}\to \tilde{\vlat{E}}$ is an interval preserving lattice homomorphism.  We claim that $r$ is a normal lattice homomorphism.  To this end, let $A\downarrow \dot 0$ in $\vlat{E}$.  Without loss of generality we may suppose that $A$ is bounded from above in $\vlat{E}$, say by $\dot u_0$.  There exists $\alpha \in I$ and $u_0 \in \vlat{E}_\alpha$ so that $\dot u_0 = e_\alpha (u_0)$.  Because $e_\alpha$ is injective and interval preserving, there exists for every $\dot u\in A$ a unique $u\in [0,u_0]\subseteq \vlat{E}_\alpha$ so that $e_\alpha (u) = \dot u$.  In particular, $e_\alpha^{-1}[A]\subseteq [0,u_0]$.  We claim that $\inf e_{\alpha}^{-1}[A] = 0$ in $\vlat{E}_\alpha$.  Let $0\leq v\in \vlat{E}_\alpha $ be a lower bound for $e_{\alpha}^{-1}[A]$.  Then $e_{\alpha}(v)\geq0$ is a lower bound for $A$ in $\vlat{E}$, hence $e_{\alpha}(v)=0$.  But $e_\alpha$ is injective, so $v=0$.  This verifies our claim.  By definition, $r[A]=\tilde{e}_\alpha[e_{\alpha}^{-1}[A]]$.  Because $\tilde{e}_{\alpha}$ is a normal lattice homomorphism it follows that $\inf r[A] = 0$ in $\tilde{\vlat{E}}$.

We recall the following result on permanence of vector lattice properties under the direct limit construction from [19].

Let $\cal{D} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ be a direct system in a category ${\bf C}$ of vector lattices.  Assume that $e_{\alpha , \beta}$ is injective for all $\alpha \precc \beta$ in $I$.  Let $\cal{S} \defeq (\vlat{E},(e_\alpha)_{\alpha \in I})$ be the direct limit of $\cal{D}$ in $\vlc$.  Then the following statements are true.

(i) $\vlat{E}$ is Archimedean if and only if $\vlat{E}_\alpha$ is Archimedean for all $\alpha\in I$.

(ii) If ${\bf C}$ is $\vlic$ then $\vlat{E}$ is order separable if and only if $\vlat{E}_\alpha$ is order separable for every $\alpha\in I$.

(iii) If ${\bf C}$ is $\vlic$ then $\vlat{E}$ has the (principal) projection property if and only if $\vlat{E}_\alpha$ has the (principal) projection property for every $\alpha\in I$.

(iv) If ${\bf C}$ is $\vlic$ then $\vlat{E}$ is ($\sigma$-)Dedekind complete if and only if $\vlat{E}_\alpha$ is ($\sigma$-)Dedekind complete for every $\alpha\in I$.

(v) If ${\bf C}$ is $\vlic$ then $\vlat{E}$ is relatively uniformly complete if and only if $\vlat{E}_\alpha$ is relatively uniformly complete for every $\alpha\in I$.

Before we proceed to discuss examples of direct limits we make some clarifying remarks about the structure of the direct limit of vector lattices.

Let $\cal{D} \defeq ((\vlat{E}_\alpha)_{\alpha\in I},(e_{\alpha , \beta})_{\alpha \precc\beta})$ be a direct system in $\vlc$ and let $\cal{S} \defeq (\vlat{E},(e_\alpha)_{\alpha \in I})$ be the direct limit of $\cal{D}$ in $\vlc$.

(i) Unless clarity demands it, we henceforth cease to explicitly express elements of $\vlat{E}$ as equivalence classes; that is, we write $u\in\vlat{E}$ instead of $\dot u\in\vlat{E}$.

(ii) For every $u\in \vlat{E}$ there exists at least one $\alpha\in I$ and $u_\alpha \in\vlat{E}_\alpha$ so that $u = e_\alpha (u_\alpha)$.  If $u = e_\beta (u_\beta)$ for some other $\beta\in I$ and $u_\beta\in \vlat{E}_\beta$ then there exists $\gamma \scc \alpha,\beta$ in $I$ so that $e_{\alpha , \gamma}(u_\alpha) = e_{\beta , \gamma}(u_\beta)$, and hence

\[
e_\gamma ( e_{\alpha , \gamma}(u_\alpha)) = u = e_\gamma ( e_{\beta , \gamma}(u_\beta)).
\]

(iii) It is proven in Theorem <ref> that if $e_{\alpha , \beta}$ is injective for all $\alpha \precc \beta$ in $I$ then $e_\alpha$ is injective for all $\alpha\in I$. In this case we identify $\vlat{E}_\alpha$ with the sublattice $e_\alpha[\vlat{E}_\alpha]$ of $\vlat{E}$.

(iv) An element $u\in \vlat{E}$ is positive if and only if there exist $\alpha \precc \beta$ in $I$ and $u_\alpha\in\vlat{E}_\alpha$ so that $e_\alpha(u_\alpha) = u$ and $e_{\alpha , \beta}(u_\alpha)\geq 0$ in $\vlat{E}_\beta$.  Combining this observation with (ii) we see that $u\geq 0$ if and only if there exist $\alpha\in I$ and $0\leq u_\alpha \in\vlat{E}_\alpha$ so that $u=e_\alpha(u_\alpha)$.

§.§ Examples of direct limits

In [19] a number of examples are presented of naturally occurring vector lattices which can be expressed as direct limits of vector lattices.  We provide further examples which will be used in Section <ref>.

Let $\vlat{E}$ be a vector lattice.  Let $(\vlat{E}_\alpha)_{\alpha\in I}$ be an upward directed collection of ideals in $\vlat{E}$ such that $\vlat{E}_\alpha \subseteq \vlat{E}_\beta$ if and only if $\alpha \precc \beta$.  Assume that $\displaystyle\bigcup \vlat{E}_\alpha=\vlat{E}$.  For all $\alpha\precc \beta$ in $I$, let $e_{\alpha , \beta}:\vlat{E}_\alpha\to \vlat{E}_\beta$ and $e_\alpha:\vlat{E}_\alpha \to \vlat{E}$ be the inclusion mappings.  Then $\cal{D} \defeq ((\vlat{E}_{\alpha})_{\alpha\in I},(e_{\alpha , \beta})_{\alpha\precc \beta})$ is a direct system in $\nvlic$ and $\cal{S}\defeq (\vlat{E},(e_{\alpha})_{\alpha\in I})$ is the direct limit of $\cal{D}$ in $\nvlic$.

It is clear that $\cal{D}$ is a direct system in $\nvlic$ and that $\cal{S}$ is a compatible system of $\cal{D}$ in $\nvlic$.  Let $\tilde{\cal{S}} = (\tilde{\vlat{E}},(\tilde{e}_{\alpha})_{\alpha\in I})$ be any compatible system of $\cal{D}$ in $\nvlic$.  We show that there exists a unique $\nvlic$-morphism $r:\vlat{E}\to\tilde{\vlat{E}}$ so that for all $\alpha\in I$, the diagram

\[
\begin{tikzcd}[cramped]
\vlat{E} \arrow[rr, "r"] & & \tilde{\vlat{E}}\\
& \vlat{E}_\alpha \arrow[lu, "e_{\alpha}"] \arrow[ru, "\tilde{e}_{\alpha}"']
\end{tikzcd}
\]

If $u\in\vlat{E}$ and $\alpha,\beta\in I$ are such that $u\in \vlat{E}_\alpha,\vlat{E}_\beta$, then $\tilde{e}_\alpha (u) = \tilde{e}_\beta(u)$.  Indeed, for any $\gamma \scc \alpha,\beta$ in $I$

\[
\tilde{e}_{\gamma}(u) = \tilde{e}_{\gamma}(e_{\alpha, \gamma}(u)) = \tilde{e}_{\alpha}(u)
\]

\[
\tilde{e}_{\gamma}(u) = \tilde{e}_{\gamma}(e_{\beta, \gamma}(u)) = \tilde{e}_{\beta}(u)
\]

Therefore the map $r:\vlat{E}\to \tilde{\vlat{E}}$ given by

\[
r(u) = \tilde{e}_{\alpha}(u) ~\text{if}~ u \in \vlat{E}_\alpha
\]
is well-defined.  It is clear that this map makes the diagram above commute.  Further, if $u,v\in \vlat{E}$ then there exists $\alpha\in I$ so that $u,v \in \vlat{E}_\alpha$.  Then for all $a,b\in\R$ we have $au+bv,u\vee v \in\vlat{E}_\alpha$ so that

\[
r(au+bv) = \tilde{e}_{\alpha}(au+bv) = a \tilde{e}_{\alpha}(u) + b \tilde{e}_{\alpha}(v) = a\thinspace r(u)+b\thinspace r(v)
\]

\[
r(u\vee v) = \tilde{e}_{\alpha}(u\vee v) = \tilde{e}_{\alpha}(u)\vee \tilde{e}_{\alpha}(v) = r(u)\vee r(v).
\]

Hence $r$ is a lattice homomorphism.  A similar argument shows that $r$ is interval preserving.  To see that $r$ is a normal lattice homomorphism, let $A\downarrow 0$ in $\vlat{E}$.  Without loss of generality, assume that there exists $0<u_0 \in \vlat{E}$ so that $u\leq u_0$ for all $u\in A$.  Then $A\subseteq \vlat{E}_\alpha$ for some $\alpha\in I$ so that $r[A]=\tilde{e}_\alpha [A]$. Hence, because $\tilde{e}_\alpha$ is a normal lattice homomorphism,  $ \inf r[A]  = 0$.  Therefore $r$ is a $\nvlic$-morphism.

It remains to show that $r$ is the unique $\nvlic$-morphism making the diagram above commute.  Suppose that $\tilde{r}$ is another such morphism.  Let $u\in \vlat{E}$.  There exists $\alpha\in I$ so that $u\in \vlat{E}_\alpha$.  We have $\tilde{r}(u) = \tilde{r}(e_\alpha(u)) = \tilde{e}_\alpha(u) = r(u)$, which completes the proof.

The remaining examples in this section may readily been seen to be special cases of Example <ref>.  Therefore we omit the proofs.

Let $\vlat{E}$ be a vector lattice.  For every $0<u\in \vlat{E}$ let $\vlat{E}_u$ be the ideal generated by $u$ in $\vlat{E}$.  For all $0<u\leq v$ let $e_{u , v}:\vlat{E}_u\to \vlat{E}_v$ and $e_u:\vlat{E}_u\to \vlat{E}$ be the inclusion mappings.  Let $I$ be an upward directed subset of $\vlat{E}^+\mysetminus\{0\}$ so that $\vlat{E}=\displaystyle \bigcup \vlat{E}_u$.  Then $\cal{D} \defeq ((\vlat{E}_{u})_{u\in I},(e_{u , v})_{u\leq v})$ is a direct system in $\nvlic$ and $\cal{S}\defeq (\vlat{E},(e_{u})_{u\in I})$ is the direct limit of $\cal{D}$ in $\nvlic$.

Let $(X,\Sigma,\mu)$ be a complete $\sigma$-finite measure space.  Let $\Xi \defeq (X_n)$ be an increasing sequence (w.r.t. inclusion) of measurable sets with positive measure so that $X =\displaystyle \bigcup X_n$. For $n\leq m$ in $\N$ let $e_{n,m}:\vlat{L}^p(X_n)\to \vlat{L}^p(X_m)$ be defined (a.e.) by setting

\[
e_{n,m}(u)(t)\defeq \left\{ \begin{array}{lll}
u(t) & \text{ if } & t\in X_n \smallskip \\
0 & \text{ if } & t\in X_m\!\setminus X_n \\
\end{array}\right.
\]

for each $u\in\vlat{L}^p(X_n)$.  Further, define

\[
\vlat{L}^p_{\Xi-c}(X) \defeq \left\lbrace u\in\vlat{L}^p(X) ~:~ u=0 \text{ a.e. on } X\setminus X_n \text{ for some } n\in\N \right\rbrace.
\]

For $n\in\N$ let $e_n:\vlat{L}^p(X_n)\to \vlat{L}^p_{\Xi-c}(X)$ be given by

\[
e_{n}(u)(t)\defeq \left\{ \begin{array}{lll}
u(t) & \text{ if } & t\in X_n \smallskip \\
0 & \text{ if } & t\in X\setminus X_n \\
\end{array}\right.
\]

The following statements are true.

(i) $\cal{D}^p_{\Xi-c}\defeq \left((\vlat{L}^p(X_n))_{n\in\N}, (e_{n , m})_{n\leq m}\right)$ is a direct system in $\nvlic$.
(ii) $\cal{S}^p_{\Xi-c}\defeq \left(\vlat{L}^p_{\Xi-c}(X),(e_n)_{n\in\N}\right)$ is the direct limit of $\cal{D}^p_{\Xi-c}$ in $\nvlic$.

Let $X$ be a locally compact Hausdorff space.  Let $\Gamma \defeq (X_\alpha)_{\alpha\in I}$ be an upward directed (with respect to inclusion) collection of non-empty open precompact subsets of $X$ so that $\displaystyle\bigcup X_\alpha = X$.  For each $\alpha\in I$, let $\vlat{M}(\bar X_\alpha)$ be the space of Radon measures on $\bar X_\alpha$ and $\vlat{M}_c(X)$ the space of compactly supported Radon measures on $X$.  For all $\alpha \precc \beta$ in $I$, let $e_{\alpha , \beta}:\vlat{M}(\bar X_\alpha)\to \vlat{M}(\bar X_\beta)$ be defined by setting

\[
e_{\alpha , \beta}(\mu)(B) \defeq \mu(B\cap \bar X_\alpha) \text{ for all } \mu\in \vlat{M}(\bar X_\alpha) \text{ and } B\in\borel{\bar X_{\beta}}.
\]

Likewise, for $\alpha\in I$, define $e_\alpha:\vlat{M}(\bar X_\alpha)\to\vlat{M}_c(X)$ by setting

\[
e_{\alpha}(\mu)(B) \defeq \mu(B\cap \bar X_\alpha) \text{ for all } \mu\in \vlat{M}(X_\alpha) \text{ and } B\in\borel{X}.
\]

The following statements are true.

(i) $\cal{D}_{\Gamma}\defeq \left((\vlat{M}(\bar X_\alpha)_{\alpha\in I},(e_{\alpha , \beta})_{\alpha\precc \beta}\right)$ is a direct system in $\nvlic$ and $e_{\alpha , \beta}$ is injective for all $\alpha\precc \beta$ in $I$.

(ii) $\cal{S}_{\Gamma}\defeq \left(\vlat{M}_c(X),(e_\alpha)_{\alpha \in I}\right)$ is the direct limit of $\cal{D}_{\Gamma}$ in $\nvlic$.

Let $X$ be a locally compact Hausdorff space.  Let $\Gamma \defeq (X_\alpha)_{\alpha\in I}$ be an upward directed (with respect to inclusion) collection of open precompact subsets of $X$ so that $\displaystyle\bigcup X_\alpha  =X$.  For each $\alpha\in I$, let $\vlat{N}(\bar X_\alpha)$ be the space of normal Radon measures on $\bar X_\alpha$ and $\vlat{N}_c(X)$ the space of compactly supported normal Radon measures on $X$.  For all $\alpha\precc \beta$ in $I$, let $e_{\alpha , \beta}: \vlat{N}(\bar X_\alpha)\to \vlat{N}(\bar X_\beta)$ be defined by setting

\[
e_{\alpha , \beta}(\mu)(B) \defeq \mu(B\cap \bar X_\alpha) \text{ for all } \mu\in \vlat{N}(\bar X_\alpha) \text{ and } B\in\borel{\bar X_\beta}.
\]
Likewise, for $\alpha\in I$, define $e_\alpha:\vlat{N}(\bar X_\alpha)\to\vlat{N}_c(X)$ by setting

\[
e_{\alpha}(\mu)(B) \defeq \mu(B\cap \bar X_\alpha) \text{ for all } \mu\in \vlat{N}(X_\alpha) \text{ and } B\in\borel{X}.
\]

The following statements are true.

(i) $\cal{E}_{\Gamma} \defeq \left((\vlat{N}(\bar X_\alpha)_{\alpha\in I},(e_{\alpha , \beta})_{\alpha\precc \beta}\right)$ is a direct system in $\nvlic$ and $e_{\alpha , \beta}$ is injective for all $\alpha\precc \beta$ in $I$.

(ii) $\cal{T}_{\Gamma}\defeq \left(\vlat{N}_c(X),(e_\alpha)_{\alpha \in I}\right)$ is the direct limit of $\cal{J}_{\Gamma}$ in $\nvlic$.

§ INVERSE LIMITS

In this section we discuss inverse systems and inverse limits in categories of vector lattices, which are the categorical dual concepts of direct systems and direct limits.  Below we present the definitions of inverse systems and inverse limits in these categories.  As is the case in the previous section, these definitions are specializations of the corresponding definitions in general categories, see for instance <cit.> or <cit.>.

Let ${\bf C}$ be a category of vector lattices, $I$ a directed set, $\vlat{E}_\alpha$ a vector lattice for each $\alpha\in I$, and $p_{\beta , \alpha}:\vlat{E}_\beta\to \vlat{E}_\alpha$ a ${\bf C}$-morphism for all $\beta \scc \alpha$ in $I$.  The ordered pair $\cal{I} \defeq \left((\vlat{E}_\alpha)_{\alpha \in I},(p_{\beta , \alpha})_{\beta \scc \alpha}\right)$ is an inverse system in ${\bf C}$ if, for all $\alpha \precc\beta\precc\gamma$ in $I$, the diagram

\[
\begin{tikzcd}[cramped]
\vlat{E}_\gamma \arrow[rd, "p_{\gamma, \beta}"'] \arrow[rr, "p_{\gamma, \alpha}"] & & \vlat{E}_\alpha\\
& \vlat{E}_\beta\arrow[ru, "p_{\beta, \alpha}"']
\end{tikzcd}
\]

commutes in ${\bf C}$.

Let ${\bf C}$ be a category of vector lattices and $\cal{I} \defeq \left((\vlat{E}_\alpha)_{\alpha \in I},(p_{\beta , \alpha})_{\beta \scc \alpha}\right)$ an inverse system in ${\bf C}$.  Let $\vlat{E}$ be a vector lattice and for every $\alpha\in I$, let $p_\alpha: \vlat{E} \to \vlat{E}_\alpha$ be a ${\bf C}$-morphism.  The ordered pair $\cal{S} \defeq (\vlat{E}, (p_\alpha)_{\alpha\in I})$ is a compatible system of $\cal{I}$ in ${\bf C}$ if, for all $\alpha\precc\beta$ in $I$, the diagram

\[
\begin{tikzcd}[cramped]
\vlat{E} \arrow[rd, "p_\beta"'] \arrow[rr, "p_{\alpha}"] & & \vlat{E}_\alpha\\
& \vlat{E}_\beta\arrow[ru, "p_{\beta, \alpha}"']
\end{tikzcd}
\]

commutes in ${\bf C}$.

Let ${\bf C}$ be a category of vector lattices and $\cal{I} \defeq \left((\vlat{E}_\alpha)_{\alpha \in I},(p_{\beta , \alpha})_{\beta \scc \alpha}\right)$ an inverse system in ${\bf C}$.  The inverse limit of $\cal{I}$ in ${\bf C}$ is a compatible system $\cal{S}\defeq (\vlat{E}, (p_\alpha)_{\alpha\in I})$ so that for any compatible system $\tilde{\cal{S}}\defeq (\tilde{\vlat{E}}, (\tilde{p}_\alpha)_{\alpha\in I})$ in ${\bf C}$ there exists a unique ${\bf C}$-morphism $s: \tilde{\vlat{E}}\to \vlat{E}$ so that, for all $\alpha\in I$, the diagram

\[
\begin{tikzcd}[cramped]
\tilde{\vlat{E}} \arrow[rd, "\tilde{p}_{\alpha}"']\arrow[rr, "s"] & & \vlat{E}\arrow[ld, "p_{\alpha}"]\\
& \vlat{E}_\alpha
\end{tikzcd}
\]

commutes in ${\bf C}$.  The inverse limit of $\cal{I}$ is denoted by $\proj{\cal{I}}$ or simply $\proj \vlat{E}_\alpha$.

Since inverse limits are terminal objects in a certain derived category, they are unique up to a unique isomorphism when they exist, see for instance <cit.>

§.§ Existence of inverse limits

Our first task is to establish the existence of inverse limits in various categories of vector lattices.  The basic result, akin to Filter's result for direct systems, is the following.

Let $\cal{I} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta \scc\alpha}\right)$ be an inverse system in $\vlc$. Define the set

\[
\vlat{E} \defeq \left\lbrace u \in \dsprod \vlat{E}_\alpha ~:~ \pi_\alpha(u) = p_{\beta,\alpha}(\pi_\beta(u)) \text{ for all } \alpha\precc \beta \text{ in } I \right\rbrace. %\tag{$\star$}
\]

For every $\alpha\in I$ define $p_\alpha \defeq \left.\pi_\alpha\right|_{\vlat{E}}$. The following statements are true.

(i) $\vlat{E}$ is a vector sublattice of $\dsprod \vlat{E}_\alpha$.

(ii) The pair $\cal{S}\defeq ( \vlat{E} , (p_\alpha)_{\alpha\in I})$ is the inverse limit of $\cal{I}$ in $\vlc$.

We verify that $\vlat{E}$ is a sublattice of $\dsprod \vlat{E}_\alpha$; that it is a linear subspace follows by a similar argument, as the reader may readily verify.  Consider $u$ and $ v$ in $\vlat{E}$.  Then $\pi_\alpha(u \vee v) = \pi_\alpha(u) \vee \pi_\alpha(v)$ for all $\alpha\in I$.  Fix any $\alpha,\beta \in I$ so that $\beta \scc \alpha$.  Then

\[
p_{\beta , \alpha}(\pi_\beta(u \vee v)) = p_{\beta , \alpha}(\pi_\beta(u))\vee p_{\beta , \alpha}(\pi_\beta(u)) = \pi_\alpha(u) \vee \pi_\alpha(v)=\pi_\alpha(u\vee v).
\]

Therefore $u \vee v \in \vlat{E}$.  Similarly, $u\wedge v \in \vlat{E}$ so that $\vlat{E}$ is a sublattice of $\dsprod \vlat{E}_\alpha$.

From the definitions of $\vlat{E}$ and the $p_\alpha$ it is clear that $\cal{S}$ is a compatible system of $\cal{I}$ in $\vlc$.  Let $\tilde{\cal{S}} = (\tilde{\vlat{E}},(\tilde{p}_\alpha)_{\alpha\in I})$ be any compatible system of $\cal{I}$ in $\vlc$.  Define $s:\tilde{\vlat{E}}\to \vlat{E}$ by setting $s(u) \defeq (\tilde{p}_{\alpha}(u))_{\alpha\in I}$.  Let $\beta \scc \alpha$ in $I$.  Because $\tilde{\cal{S}}$ is a compatible system

\[
p_{\beta , \alpha} (\tilde{p}_\beta (u)) = \tilde{p}_\alpha (u), ~ u\in \tilde{\vlat{E}}.
\]

Therefore $s(u)\in \vlat{E}$ for all $u\in \tilde{\vlat{E}}$.  Because each $\tilde{p}_\alpha$ is a lattice homomorphism, so is $s$.  By the definitions of $s$ and the $p_\alpha$, respectively, it follows that $p_\alpha \circ s = \tilde{p}_\alpha$ for every $\alpha \in I$.  We show that $s$ is the unique lattice homomorphism with this property.  To this end, let $\tilde{s}:\tilde{\vlat{E}}\to \vlat{E}$ be a lattice homomorphism so that $p_\alpha \circ \tilde{s}=\tilde{p}_\alpha$ for every $\alpha\in I$.  Fix $u\in \tilde{\vlat{E}}$.  Then for every $\alpha\in I$,

\[
\pi_\alpha (\tilde{s}(u)) = p_\alpha (\tilde{s}(u)) = \tilde{p}_\alpha (u) = \pi_\alpha (s(u)).
\]

Hence $s=\tilde{s}$ and therefore $\proj{\cal{I}} = (\vlat{E},(p_\alpha)_{\alpha \in I})$ in $\vlc$.

Let $\cal{I} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta\scc\alpha}\right)$ be an inverse system in $\nvlc$ and $\cal{S} \defeq ( \vlat{E}, (p_\alpha)_{\alpha\in I} )$ its inverse limit in $\vlc$.  If $\vlat{E}_\alpha$ is Dedekind complete for every $\alpha\in I$ then $\cal{S}$ is the inverse limit of $\cal{I}$ in $\nvlc$.

Let $\cal{I} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta\scc\alpha}\right)$ be an inverse system in $\nvlc$ and $\cal{S} \defeq \left(\vlat{E},(p_\alpha)_{\alpha\in I}\right)$ its inverse limit in $\vlc$.  Let $A\subseteq \vlat{E}$ and assume that $\inf A = u$ or $\sup A = u$ in $\dsprod\vlat{E}_\alpha$.  Then $u\in \vlat{E}$.

First, we prove that the $p_\alpha$ are normal homomorphisms: Fix $\alpha\in I$ and let $A\downarrow 0$ in $E$.  Since $E_\alpha$ is Dedekind complete for every $\alpha\in I$, by Theorem <ref> (iii), the product $\dsprod E_\alpha$ is also Dedekind complete.  Therefore $A\downarrow u$ in $\dsprod E_\alpha$ for some $u\in \dsprod E_\alpha$. By (i), we have  $u\in E$ which implies $A\downarrow u$ in $E$.  Since $A\downarrow 0$ in $E$, we conclude that $A\downarrow 0$ in $\dsprod E_\alpha$. By the normality of the coordinate projections $\pi_\alpha$, it follows that

\[
\inf p_\alpha \left[ A \right] = \inf \pi_\alpha \left[ A \right] = \pi_\alpha \left[ \inf A \right] = 0.
\]
Hence the pair $\cal{S}$ is a compatible system over $\cal{I}$ in $\nvlc$.  It remains to verify that $\cal{S}$ satisfies the universal property of the inverse limit in $\nvlc$: Let $\tilde{\cal{S}} \defeq (\tilde{E},(\tilde{p}_\alpha)_{\alpha\in I})$ be a compatible system in $\nvlc$. Following the proof of Theorem <ref>, we need only show that $s:\tilde{E}\to E$ where $s(v) \defeq (\tilde{p}_{\alpha}(v))_{\alpha\in I}$ is a $\nvlc$-morphism: Let $A\downarrow 0$ in $\tilde{E}$, then since each $\tilde{p}_\alpha$ is a normal homomorphism, we have $p_{\alpha}[s[A]] = \tilde{p}_\alpha [A]\downarrow 0$ in $E_\alpha$ for each $\alpha \in I$. By Proposition <ref>, it follows that $s[A]\downarrow 0$ in $\dsprod E_\alpha$, and by (i) above $s[A]\downarrow 0$ in $E$. Therefore $s$ is a $\nvlc$-morphism.

For this, it is sufficient to consider infima of downward directed subsets of $\vlat{E}$.

Let $A\subseteq \vlat{E}$ and assume that $A\downarrow u$ in $\dsprod\vlat{E}_\alpha$.  Then for every $\alpha\in I$, $p_\alpha [A]=\pi_\alpha[A]\downarrow \pi_\alpha(u)$ in $\vlat{E}_\alpha$.  For $\beta \scc \alpha$ in $I$,

\[
\pi_\alpha(u) = \inf p_\alpha [A] = \inf p_{\beta , \alpha}[p_\beta[A]] = p_{\beta , \alpha}(\inf p_\beta[A]) = p_{\beta , \alpha}(\pi_\beta(u));
\]

the second to last identity follows form the fact that $p_{\beta , \alpha}$ is a normal lattice homomorphism.  Therefore $u\in \vlat{E}$.

Assume that $\vlat{E}_\alpha$ is Dedekind complete for every $\alpha\in I$.  We show that the $p_\alpha$ are normal lattice homomorphisms.  Our proof proceeds in two steps.  The first step is to show that $\vlat{E}$ is closed under the formation of arbitrary infima and suprema in $\dsprod\vlat{E}_\alpha$.[We remark that this fact does not depend on the $\vlat{E}_\alpha$ being Dedekind complete.] For this, i

Let $\cal{I} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta\scc\alpha}\right)$ be an inverse system in $\nvlc$ and $\cal{S} \defeq ( \vlat{E}, (p_\alpha)_{\alpha\in I} )$ its inverse limit in $\vlc$. The following statements are true.

(i) Let $A\subseteq \vlat{E}$ and assume that $\inf A = u$ or $\sup A = u$ in $\dsprod \vlat{E}_\alpha$, then $u\in E$.

(ii) If $\vlat{E}_\alpha$ is Dedekind complete for every $\alpha\in I$ then $\cal{S}$ is the inverse limit of $\cal{I}$ in $\nvlc$.

It is sufficient to consider infima of downward directed subsets of $\vlat{E}$. Let $A\subseteq \vlat{E}$ and assume that $A\downarrow u$ in $\dsprod\vlat{E}_\alpha$. By Theorem <ref> (i), for every $\alpha\in I$, $p_\alpha [A]=\pi_\alpha[A]\downarrow \pi_\alpha(u)$ in $\vlat{E}_\alpha$.  For $\beta \scc \alpha$ in $I$,

\[
\pi_\alpha(u) = \inf p_\alpha [A] = \inf p_{\beta , \alpha}[p_\beta[A]] = p_{\beta , \alpha}(\inf p_\beta[A]) = p_{\beta , \alpha}(\pi_\beta(u));
\]

the second to last identity follows from the fact that $p_{\beta , \alpha}$ is a normal lattice homomorphism.  Therefore $u\in \vlat{E}$.

First, we prove that the $p_\alpha$ are normal lattice homomorphisms.  Let $A\downarrow 0$ in $\vlat{E}$.  Since $\vlat{E}_\alpha$ is Dedekind complete for every $\alpha\in I$, so is $\dsprod \vlat{E}_\alpha$.  Therefore $A\downarrow u$ in $\dsprod \vlat{E}_\alpha$ for some $u\in \dsprod \vlat{E}_\alpha$.  Then $u\in \vlat{E}$ so that $A\downarrow u$ in $\vlat{E}$.  But $A\downarrow  0$ in $\vlat{E}$, hence $u =  0$.  Therefore $\inf p_\alpha [A] = \pi_\alpha(u) = 0$ for every $\alpha \in I$.

From the above it follows that $\cal{S}$ is a compatible system in $\nvlc$.  It remains to show that $\cal{S}$ satisfies Definition <ref> in $\nvlc$.  Let $\tilde{\cal{S}} = (\tilde{\vlat{E}},(\tilde{p}_\alpha)_{\alpha\in I})$ be a compatible system in $\nvlc$.  Based on Theorem <ref> we need only show that $s:\tilde{\vlat{E}}\to \vlat{E}$ defined by setting $s(u) \defeq (\tilde{p}_{\alpha}(u))_{\alpha\in I}$ for every $u \in \tilde{\vlat{E}}$ is a normal lattice homomorphism.

Let $A\downarrow 0$ in $\tilde{\vlat{E}}$.  Then, since each $\tilde{p}_\alpha$ is a normal lattice homomorphism, $p_{\alpha}[s[A]] = \pi_\alpha[s[A]] = \tilde{p}_\alpha [A]\downarrow 0$ in $\vlat{E}_\alpha$ for every $\alpha \in I$.  Hence $s[A]\downarrow  0$ in $\dsprod \vlat{E}_\alpha$, therefore also in $\vlat{E}$.  Therefore $s$ is a normal lattice homomorphism, hence a $\nvlc$-morphism, so that $\proj{\cal{I}}=(\vlat{E},(p_\alpha)_{\alpha \in I})$ in $\nvlc$.

Let $\cal{I} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta\scc \alpha}\right)$ be an inverse system in a category of vector lattices, and $\cal{S} \defeq \left(\vlat{E},(p_\alpha)_{\alpha\in I}\right)$ its inverse limit in $\vlc$.  We occasionally suppress the projections $p_\alpha$ and simply write $\vlat{E}=\proj{\cal{I}}$ or `$\vlat{E}$ is the inverse limit of $\cal{I}$'.

§.§ Permanence properties

In this section we establish some permanence properties for inverse limits, along the same vein as those for direct limits given in Theorem <ref>.  These follow easily from the construction of inverse limits given in Theorem <ref> and the properties of products of vector lattices given in Theorem <ref>

Let $\cal{I} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta \scc \alpha}\right)$ be an inverse system in $\vlc$ and $\cal{S} \defeq \left(\vlat{E},(p_\alpha)_{\alpha\in I}\right)$ its inverse limit in $\vlc$.  The following statements are true.

(i) If $\vlat{E}_\alpha$ is Archimedean for every $\alpha\in I$ then so is $\vlat{E}$.

(ii) If $\vlat{E}_\alpha$ is Archimedean and relatively uniformly complete for every $\alpha\in I$ then $\vlat{E}$ is relatively uniformly complete.

We note that (i) follows immediately from Theorems <ref> (ii) and the construction of an inverse limit in $\vlc$.

For (ii), assume that $\vlat{E}_\alpha$ is Archimedean and relatively uniformly complete for every $\alpha \in I$.  We show that every relatively uniformly Cauchy sequence in $\vlat{E}$ is relatively uniformly convergent.  Because $\vlat{E}$ is Archimedean by (i), it follows from <cit.> that it suffices to consider increasing sequences.  Let $(u_n)$ be an increasing, relatively uniformly Cauchy sequence in $\vlat{E}$.  Then for every $\alpha\in I$, $(p_\alpha(u_n))$ is an increasing sequence in $\vlat{E}_\alpha$.  According to <cit.>, $(p_\alpha(u_n))$ is relatively uniformly Cauchy in $\vlat{E}_\alpha$.  Because each $\vlat{E}_\alpha$ is relatively uniformly complete, there exists $u_\alpha\in \vlat{E}_\alpha$ so that $(p_\alpha(u_n))$ converges relatively uniformly to $u_\alpha$.  In fact, because $(p_\alpha (u_n))$ is increasing, $u_\alpha = \sup\{p_\alpha(u_n) ~:~n\in\N\}$.  Therefore $u=(u_\alpha)=\sup \{u_n ~:~n\in\N\}$ in $\dsprod \vlat{E}_\alpha$.  By Theorem <ref> (i), $u\in \vlat{E}$ so that $u=\displaystyle\sup\{ u_n ~:~ n\in \N\}$ in $\vlat{E}$.  Therefore $(u_n)$ converges relatively uniformly to $u$  by <cit.>.  We conclude that $\vlat{E}$ is relatively uniformly complete.

Let $\cal{I} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta\scc\alpha}\right)$ be an inverse system in $\nvlc$ and $\cal{S} \defeq \left(\vlat{E},(p_\alpha)_{\alpha\in I}\right)$ its inverse limit in $\vlc$.  The following statements are true.

(i) If $\vlat{E}_\alpha$ is $\sigma$-Dedekind complete for every $\alpha\in I$ then so is $\vlat{E}$.

(ii) If $\vlat{E}_\alpha$ is Dedekind complete for every $\alpha\in I$ then so is $\vlat{E}$.

(iii) If $\vlat{E}_\alpha$ is laterally complete for every $\alpha\in I$ then so is $\vlat{E}$.

(iv) If $\vlat{E}_\alpha$ is universally complete for every $\alpha\in I$ then so is $\vlat{E}$.

We prove (ii). The statements in (i) and (iii) follow by almost identical arguments, and (iv) follows immediately from (ii) and (iii).

Let $D\subseteq \vlat{E}$ be an upwards directed set bounded above by $u \in \vlat{E}$.  For every $\alpha\in I$ the set $D_\alpha \defeq p_\alpha\left[ D \right]$ is bounded above in $\vlat{E}_\alpha$ by $\pi_\alpha(u) \in \vlat{E}_\alpha$.  Since $\vlat{E}_\alpha$ is Dedekind complete for every $\alpha\in I$, $v_\alpha \defeq \sup D_\alpha$ exists in $\vlat{E}_\alpha$ for all $\alpha\in I$.  We have that $\sup D = \left( v_\alpha \right)$ in $\dsprod \vlat{E}_\alpha$.  By Theorem <ref> (i), $v\defeq \left( v_\alpha \right) \in \vlat{E}$.  Because $\vlat{E}$ is a sublattice of $\dsprod \vlat{E}_\alpha$ it follows that $v=\sup D$ in $\vlat{E}$.

§.§ Examples of inverse limits

In this section we present a number of examples of inverse systems in categories of vector lattices and their limits.  These will be used in Section <ref>.  Our first example is related to Example <ref>.

Let $(X,\Sigma,\mu)$ be a complete $\sigma$-finite measure space.  Let $\Xi \defeq (X_n)$ be an increasing sequence (w.r.t. inclusion) of measurable sets with positive measure so that $X = \displaystyle \bigcup X_n$.  For $1 \leq p \leq \infty$ let $\vlat{L}^p_{\Xi-\loc}(X)$ denote the set of (equivalence classes of) measurable functions $u:X\to\R$ so that $u\onefunction_{X_n}\in \vlat{L}^p(X)$ for every $n\in\N$.  For $m \geq n$ in $\N$ let $r_{m , n}:\vlat{L}^p(X_m)\to \vlat{L}^p(X_n)$ and $r_n:\vlat{L}^p_{\Xi-\loc}(X)\to \vlat{L}^p(X_n)$ be the restriction maps.  The following statements are true.

(i) $\cal{I}^p_{\Xi-\loc} \defeq ((\vlat{L}^p(X_n))_{n\in\N},(r_{m , n})_{m \geq n})$ is an inverse system in $\nvlc$.

(ii) $\cal{S}^p_{\Xi-\loc}\defeq (\vlat{L}^p_{\Xi-\loc}(X),(r_n)_{n\in \N})$ is a compatible system of $\cal{I}^p_{\Xi-\loc}$ in $\nvlc$.

(iii) $\cal{S}^p_{\Xi-\loc}$ is the inverse limit of $\cal{I}^p_{\Xi-\loc}$ in $\nvlc$.

That (i) and (ii) are true is clear.  We prove (iii).

Because $\vlat{L}^p(X_n)$ is Dedekind complete for every $n\in\N$, $\proj{\cal{I}^p_{\Xi-\loc}}\defeq (\vlat{F},(p_n)_{n\in\N})$ exists in $\nvlc$ by Theorem <ref> (ii).  Since $\cal{S}^p_{\Xi-\loc}$ is a compatible system of $\cal{I}^p_{\Xi-\loc}$ in $\nvlc$ there exists a unique normal lattice homomorphism $s:\vlat{L}^p_{\Xi-\loc}(X)\to \vlat{F}$ so that the diagram

\[
\begin{tikzcd}[cramped]
\vlat{L}^p_{\Xi-\loc}(X) \arrow[rd, "r_{n}"']\arrow[rr, "s"] & & \vlat{F}\arrow[ld, "p_{n}"]\\
& \vlat{L}^p(X_n)
\end{tikzcd}
\]

commutes for every $n\in\N$.  We show that $s$ is bijective.  To see that $s$ is injective, suppose that $s(u)=0$ for some $u\in \vlat{L}^p_{\Xi-\loc}(X)$.  Then $r_n(u)=0$ for every $n\in\N$; that is, the restriction of $u$ to each set $X_n$ is $0$.  Since $\displaystyle\bigcup X_n = X$ it follows that $u=0$.  To see that $s$ is surjective, consider $u\in\vlat{F}$.  If $m \geq n$ then $p_n(u)=r_{m,n}(p_m(u))$; that is, $p_n(u)=p_m(u)$ a.e. on $X_n$.  Therefore $v:X\to\R$ given by

\[
v(x) \defeq p_n(u)(x) \text{ if } x\in X_n
\]

is a.e. well-defined on $X=\displaystyle \bigcup X_n$.  For $n\in\N$, $v$ restricted to $X_n$ is $p_n(u)\in \vlat{L}^p(X_n)$.  Therefore $v\in\vlat{L}^p_{\Xi-\loc}(X)$.  Furthermore, $p_n(s(v)) = r_n(v) = p_n(u)$ for all $n\in\N$ so that $s(v)=u$. We conclude that $s$ is a lattice isomorphism.

Our second example is a companion result for Examples <ref> and <ref>.

Let $X$ be a topological space and $\cal{O}\defeq \{O_\alpha ~:~ \alpha\in I\}$ collection of non-empty open subsets of $X$ which is upward directed with respect to inclusion; that is, $\alpha \precc \beta$ if and only if $O_\alpha \subseteq O_\beta$.  Assume that $\bigcup O_\alpha$ is dense and $\cont$-embedded in $X$.  For $\beta\scc \alpha$, denote by $r_{\beta , \alpha}:\cont(\bar O_\beta)\to \cont(\bar O_\alpha)$ and $r_\alpha:\cont(X)\to\cont(\bar O_\alpha)$ the restriction maps.  The following statements are true.

(i) $\cal{I}_\cal{O} \defeq ((\cont(\bar O_\alpha))_{\alpha\in I},(r_{\beta , \alpha})_{\beta \scc \alpha})$ is an inverse system in $\vlc$.

(ii) $\cal{S}_\cal{O} \defeq (\cont(X),(r_\alpha)_{\alpha\in I})$ is a compatible system of $\cal{I}_{\cal{O}}$ in $\vlc$.

(iii) $\cal{S}_\cal{O}$ is the inverse limit of $\cal{I}_{\cal{O}}$ in $\vlc$.

(iv) If $X$ is a Tychonoff space and $O_\alpha$ is precompact for every $\alpha\in I$ then $\cal{I}_\cal{O}$ is an inverse system in $\nvlic$, and $\cal{S}_\cal{O}$ is a compatible system of $\cal{I}_{\cal{O}}$ in $\nvlic$.

That (i), (ii) and (iii) are true follows from arguments similar to those used in the proof of Example <ref>.  We therefore omit the proofs of these statements.  We only note that for (iii), we use the fact that every $u\in\cont(\displaystyle\bigcup O_\alpha)$ has a unique continuous and real-valued extension to $X$; that is, restriction from $X$ to $\displaystyle\bigcup O_\alpha$ defines a lattice isomorphism from $\cont(\displaystyle\bigcup O_\alpha)$ onto $\contX$.

To verify (iv) it is sufficient to show that the $r_\alpha$ and $r_{\alpha,\beta}$ are order continuous and interval preserving.  That these maps are order continuous follows from <cit.>.  That they are interval preserving follows from the fact that every compact subset of a Tychonoff space is $\cont^\ast$-embedded.  We show that the $r_\alpha$ are interval preserving, the proof for $r_{\alpha,\beta}$ being identical.  Consider an $\alpha\in I$, $u\in \cont(X)^+$ and $v\in\cont(\bar O_\alpha)$ so that $0\leq v\leq r_\alpha(u)$.  Because $\bar O_\alpha$ is $\cont^\ast$-embedded in $X$ there exists a continuous function $v'\in \cont (X)$ so that $r_\alpha (v')=v$.  Let $w\defeq (0\vee v')\wedge u$.  Then $0\leq w\leq u$ and, because $r_\alpha$ is a lattice homomorphism, $r_\alpha(w)=v$.  Therefore $[0,r_\alpha (u)] = r_\alpha [[0,u]]$.

Our next example is of a more general nature.  It is an essential ingredient in our solution of the decomposition problem for $\cont(X)$ mentioned in Section <ref>.

Let $\vlat{E}$ be an Archimedean vector lattice.  Denote by $\bands{\vlat{E}}$ the Boolean algebra of projection bands in $\vlat{E}$.[$\bands{\vlat{E}}$ are ordered by inclusion.]  Let $\mathrm{M}$ be a non-trivial ideal in $\bands{\vlat{E}}$; that is, $\mathrm{M}\subset \bands{\vlat{E}}$ is downward closed, upward directed and does not consist of the trivial band $\{0\}$ only.  For notational convenience we express $\mathrm{M}$ as indexed by a directed set $I$, $\mathrm{M} = \{\vlat{B}_\alpha ~:~ \alpha \in I\}$, so that $\alpha \precc \beta$ if and only if $\vlat{B}_{\alpha}\subseteq \vlat{B}_{\beta}$.

For $\vlat{B}_\alpha \subseteq \vlat{B}_\beta$ in $\mathrm{M}$, denote by $P_\alpha$ the band projection of $\vlat{E}$ onto $\vlat{B}_\alpha$ and by $P_{\beta , \alpha}$ the band projection of $\vlat{B}_\beta$ onto $\vlat{B}_\alpha$; that is, $P_{\beta , \alpha} = \left.P_{\alpha}\right|_{\vlat{B}_\beta}$.  The following statements are true.

(i) $\cal{I}_{\mathrm{M}} \defeq (\mathrm{M} , (P_{\beta,\alpha})_{\beta\scc \alpha})$ is an inverse system in $\nvlic$ and $\tilde{\cal{S}} \defeq (\vlat{E},(P_\alpha)_{\alpha\in I})$ is a compatible system of $\cal{I}_\vlat{M}$ in $\nvlic$.

(ii) $\proj{\cal{I}_{\mathrm{M}}}\defeq \left(\vlat{F},(p_\alpha)_{\alpha \in I}\right)$ exists in $\vlc$.  If $\vlat{E}$ is Dedekind complete then $\left(\vlat{F},(p_\alpha)_{\alpha \in I}\right)$ is the inverse limit of $\cal{I}_{\mathrm{M}}$ in $\nvlc$.

(iii)  $P_{\mathrm{M}}:\vlat{E}\ni u \mapsto (P_\alpha(u))_{\alpha\in I}\in \vlat{F}$ is the unique lattice homomorphism so that the diagram

\[
\begin{tikzcd}[cramped]
\vlat{E} \arrow[rd, "P_{\alpha}"']\arrow[rr, "P_{\mathrm{M}}"] & & \vlat{F}\arrow[ld, "p_{\alpha}"]\\
& \vlat{B}_\alpha
\end{tikzcd}
\]

commutes for every $\alpha\in I$.  Furthermore, $P_{\mathrm{M}}[\vlat{E}]$ an order dense sublattice of $\vlat{F}$.  If $\vlat{E}$ is Dedekind complete then $P_\vlat{M}[\vlat{E}]$ is an ideal in $\vlat{F}$.

(iv) $P_{\mathrm{M}}$ is injective if and only if $\{P_\alpha : \alpha\in I\}$ separates the points of $\vlat{E}$.  In this case, $P_{\mathrm{M}}$ is a lattice isomorphism onto an order dense sublattice of $\vlat{F}$.

Since band projections are both interval preserving and order continuous, (i) follows immediately from Proposition <ref>.  The statement in (ii) follows immediately from (i) and Theorems <ref> and <ref> (ii).  That (iv) is true is a direct consequence of the definition of $P_{\mathrm{M}}$.

We proceed to prove (iii).  Since $P_\alpha$ is a lattice homomorphism for every $\alpha\in I$, $P_\vlat{M}$ is a lattice homomorphism into $\dsprod \vlat{B}_\alpha$.  If $u\in\vlat{E}$ and $\alpha \precc \beta$ then $P_{\beta,\alpha}(P_\beta ( u)) = P_\alpha (u)$ by Proposition <ref> (iii).  Hence $P_\vlat{M}[\vlat{E}]$ is a sublattice of $\vlat{F}$.  It follows from the construction of $\vlat{F}$ as a sublattice of $\dsprod \vlat{B}_\alpha$ given in Theorem <ref> that $p_\alpha \circ P_{\vlat{M}}=P_\alpha$ for all $\alpha\in I$.

Let $0 < u=(u_\alpha)\in \vlat{F}$.  There exists $\alpha_0\in I$ so that $u_{\alpha_0}>0$ in $\vlat{B}_{\alpha_0}\subseteq \vlat{E}$.  Then $0<P_\vlat{M}(u_{\alpha_0}) \leq u$ in $\vlat{F}$.  Hence $P_\vlat{M}[\vlat{E}]$ is order dense in $\vlat{F}$.

Assume that $\vlat{E}$ is Dedekind complete.  We show that $P_\vlat{M}[\vlat{E}]$ is an ideal in $\vlat{F}$.  Consider $v\in \vlat{E}^+$ and $u=(u_\alpha)\in \vlat{F}^+$ so that $0\leq u \leq P_\vlat{M}(v)$.  Then $u_\alpha \leq P_\alpha(v) \leq v$ for all $\alpha\in I$.  Let $w=\sup \{u_\alpha ~:~ \alpha \in I\}$ in $\vlat{E}$.  We claim that $P_\vlat{M}(w)=u$.  Because $u_\alpha \leq w$ for all $\alpha \in I$, $u_\alpha = P_\alpha(u_\alpha) \leq P_\alpha(w)$.  Therefore $u\leq P_\vlat{M}(w)$.  For the reverse inequality we note that for all $\beta\in I$,

\[
P_\beta(w) = \sup\{ P_\beta u_\alpha ~:~ \alpha \in I\}.
\]

We claim that $P_{\beta}(u_\alpha) \leq u_\beta$ for all $\alpha,\beta\in I$.  It follows from this claim that $P_\beta(w) \leq u_\beta$ so that $P_\vlat{M}(w) \leq u$.  Thus we need only verify that, indeed, $P_{\beta}(u_\alpha) \leq u_\beta$ for all $\alpha,\beta \in I$.  To this end, fix $\alpha,\beta\in I$.  Let $\gamma\in I$ be a mutual upper bound for $\alpha$ and $\beta$.  Because $u=(u_\alpha) \in\vlat{F}$, $\tilde{\cal{S}}$ is compatible with $\cal{I}_\vlat{M}$ and $u_\gamma,u_\alpha\in \vlat{E}$ we have

\[
P_\beta(u_\alpha) = P_\beta (P_{\gamma, \alpha}(u_\gamma)) \leq P_\beta(u_\gamma) = P_{\gamma , \beta} (P_\gamma(u_\gamma)) = P_{\gamma , \beta}(u_\gamma) = u_\beta.
\]

This completes the proof.

Let $\vlat{E}$, $\bands{\vlat{E}}$, $\mathrm{M}$, $\cal{I}_{\mathrm{M}}$, $P_{\mathrm M}$ and $\tilde{\cal{S}}$ be as in Example <ref>.  Assume that $\{P_\alpha ~:~ \alpha\in I\}$ separates the points of $\vlat{E}$.  It may happen that $P_{\mathrm M}$ maps $\vlat{E}$ onto $\proj{\cal{I}_\vlat{M}}$, but this is not always the case.  If this is the case, then $\proj{\cal{I}_\vlat{M}}=\tilde{\cal{S}}$.  A sufficient, but not necessary, condition for $P_\mathrm{M}$ to map $\vlat{E}$ onto $\vlat{F}$ is that $\vlat{E}\in\mathrm{M}$.

(i) Consider the vector lattice $\R^\omega$ of all functions from $\N$ to $\R$. For $F\subseteq \N$ let

\[
\vlat{B}_F \defeq \{u\in \R^\omega ~:~ \supp (u) \subseteq F\}.
\]

Then $\mathrm{M} \defeq \{\vlat{B}_F ~:~ \emptyset\neq F\subseteq \N \text{ finite} \}$ is a proper, non-trivial ideal in $\rm{ {\bf B}}_{\R^\omega}$ and $\{P_F : \emptyset\neq F\subseteq \N \text{ finite} \}$ separates the points of $\R^\omega$.  It is easy to see that $P_{\mathrm{M}}$ maps $\R^\omega$ onto $\proj{\cal{I}_{\mathrm{M}}}$.

(ii) Consider the vector lattice $\ell^1$.  As in (i), for $F\subseteq \N$ define

\[
\vlat{B}_F \defeq \{u\in \ell^1 ~:~ \supp (u) \subseteq F\}
\]

Then $\mathrm{M} \defeq  \{\vlat{B}_F ~:~ \emptyset\neq F\subseteq \N \text{ finite} \}$ is a proper, non-trivial ideal in $\rm{ {\bf B}}_{\ell^1}$ and $\proj{\cal{I}_\mathrm{M}}$ is $\R^\omega$.  In this case, $P_{\mathrm{M}}[\ell^1]$ is a proper subspace of $\proj{\cal{I}_\mathrm{M}}$.

Based on Remark <ref> we ask the following question:  Given a Dedekind complete vector lattice $\vlat{E}$, does there exist a proper ideal $\mathrm{M}$ in $\bands{\vlat{E}}$ so that $P_\mathrm{M}:\vlat{E}\to \proj{\cal{I}_\mathrm{M}}$ is an isomorphism onto $\proj{\cal{I}_\mathrm{M}}$?  We do not pursue this question any further here, except to note the following example.

Let $X$ be an extremally disconnected Tychonoff space.  Let $\cal{O} \defeq \{O_\alpha : \alpha \in I\}$ be a proper, non-trivial ideal in the Boolean algebra ${\bf R}_X$ of clopen subsets of $X$.  Assume that $\bigcup O_\alpha$ is dense and $\cont$-embedded in $X$.  Then $\vlat{M}\defeq \{\cont (O_\alpha) ~:~ \alpha\in I\}$ is a proper, non-trivial ideal in $\bands{\contX}$ and $P_\vlat{M}:\contX \to \proj{\cal{I}_\vlat{M}}$ is a lattice isomorphism onto $\proj{\cal{I}_\vlat{M}}$.

The Boolean algebras ${\bf R}_X$ and $\bands{\contX}$ are isomorphic.  In particular, the isomorphism is given by

\[
{\bf R}_X\ni O \longmapsto \vlat{B}_O=\{u\in \contX ~:~ \supp(u)\subseteq O\},
\]

see  <cit.>.  Moreover, for $O\in{\bf R}_X$ the band projections onto $\vlat{B}_O$ is given by restriction to $O$.  Finally, we note that for $O\in{\bf R}_X$ the band $\vlat{B}_O$ may be identified with $\cont(O)$.  Therefore $\vlat{M}$ is a proper, non-trivial ideal in $\bands{\contX}$.  It follows from Example <ref> that $\proj{\cal{I}_\vlat{M}}=\contX$, i.e. $\cal{I}_\vlat{M}:\contX \to \proj{\cal{I}_\vlat{M}}$ is a lattice isomorphism onto $\proj{\cal{I}_\vlat{M}}$.

§ DUAL SPACES

The results presented in this section form the technical heart of the paper.  Roughly speaking, we will show, under fairly general assumptions, that the order (continuous) dual of a direct limit is an inverse limit.  On the other hand, more restrictive conditions are needed to show that the order (continuous) dual of an inverse limit is a direct limit.  These results form the basis of the applications given in Section <ref>.

§.§ Duals of direct limits

Let $\cal{D}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ be a direct system in $\vlic$.  The dual system of $\cal{D}$ is the pair $\cal{D}^\sim \defeq \left( (\vlat{E}^\sim_\alpha)_{\alpha\in I}, (e^\sim_{\alpha, \beta})_{\alpha\precc\beta}\right)$.

If $\cal{D}$ is a direct system in $\nvlic$, define the order continuous dual system of $\cal{D}$ as the pair $\ordercontn{\cal{D}} \defeq \left( (\ordercontn{(\vlat{E}_\alpha)})_{\alpha\in I}, (e^\sim_{\alpha, \beta})_{\alpha\precc\beta}\right)$ with $e^\sim_{\alpha, \beta}: \ordercontn{(E_\beta)} \to \ordercontn{(\vlat{E}_\alpha)}$.

Let $\cal{D}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ be a direct system in $\vlc$.  The following statements are true.

(i) If $\cal{D}$ is a direct system in $\vlic$ then the dual system $\cal{D}^\sim$ is an inverse system in $\nvlic$.

(ii) If $\cal{D}$ is a direct system in $\nvlic$ then the order continuous dual system $\ordercontn{\cal{D}}$ is an inverse system in $\nvlic$.

We present the proof of (i).  That (ii) is true follows by a similar argument, so we omit the proof.

Assume that $\cal{D}$ is a direct system in $\vlic$. Then the maps $e_{\alpha, \beta}: \vlat{E}_\alpha\to \vlat{E}_\beta$ are interval preserving lattice homomorphisms for all $\alpha \precc \beta$. By Theorem <ref> the adjoint maps $e^\sim_{\alpha, \beta}:\vlat{E}_\beta^\sim \to \vlat{E}_\alpha^\sim$ are all normal interval preserving lattice homomorphisms.  Fix $\alpha, \beta, \gamma\in I$ such that $\alpha \precc \beta \precc \gamma$.  Since $\cal{D}$ is a direct system in $\vlic$, $e_{\alpha , \gamma} = e_{\beta , \gamma } \circ e_{\alpha , \beta}$ so that $e_{\alpha , \gamma}^\sim = e_{\alpha , \beta}^\sim \circ e_{\beta , \gamma }^\sim$. Thus the dual system $\cal{D}^\sim = \left( (\vlat{E}^\sim_\alpha)_{\alpha\in I}, (e^\sim_{\alpha, \beta})_{\alpha\precc\beta}\right)$ is an inverse system in $\nvlic$.

Let $\cal{D} \defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta}) \right)$ be a direct system in $\vlic$ and  $\cal{S}\defeq \left( \vlat{E}, (e_\alpha)_{\alpha\in I} \right)$ a compatible system of $\cal{D}$ in $\vlic$. The following statements are true.

(i) $\cal{S}^\sim \defeq \left( \vlat{E}^\sim, (e^\sim_\alpha)_{\alpha\in I} \right)$ is a compatible system for the inverse system $\cal{D}^\sim$ in $\nvlic$.

(ii) If $\cal{D}$ is a direct system in $\nvlic$ then $\ordercontn{\cal{S}} \defeq \left( \ordercontn{\vlat{E}}, (e^\sim_\alpha)_{\alpha\in I} \right)$ is a compatible system for the inverse system $\ordercontn{\cal{D}}$ in $\nvlic$.

Again, we only prove (i) as the proof of (ii) is similar.  By Theorem <ref>, $e_{\alpha}^\sim: \vlat{E}^\sim \to \vlat{E}^\sim_\alpha$ is a normal interval preserving lattice homomorphism for every $\alpha\in I$.  Furthermore, if $\alpha \precc \beta $ then $e_\alpha = e_\beta \circ e_{\alpha , \beta}$ so that $e_\alpha^\sim = e_{\alpha , \beta}^\sim \circ e_\beta^\sim$.  Therefore $\cal{S}^\sim$ is a compatible system of $\cal{D}^\sim$ in $\nvlic$.

The main results of this section are the following.

Let $\cal{D}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ be a direct system in $\vlic$, and let $\cal{S} \defeq \left( \vlat{E},(e_\alpha)_{\alpha\in I}\right)$ be the direct limit of $\cal{D}$ in $\vlic$.  The following statements are true.

(i) $\proj{\cal{D}^\sim}\defeq \left(\vlat{F},(p_\alpha)_{\alpha\in I}\right)$ exists in $\nvlc$.

(ii) $\left(\ind{\cal{D}}\right)^\sim \cong \proj{\cal{D}^\sim}$ in $\nvlc$; that is, there exists a lattice isomorphism $T: \vlat{E}^\sim \to \vlat{F}$ such that the following diagram commutes for all $\alpha\in I$.

\begin{eqnarray}\label{EQ:  Thm-Dual of ind sys in VLIC is proj of duals in NVL diagram}
\begin{tikzcd}[cramped]
\vlat{E}^\sim \arrow[rd, "e^\sim_\alpha"'] \arrow[rr, "T"] & & \vlat{F} \arrow[dl, "p_\alpha"]\\
& \orderdual{\vlat{E}}_\alpha
\end{tikzcd}
\end{eqnarray}

That (i) is true follows from Proposition <ref> and Theorem <ref> (ii) because $\orderdual{\vlat{E}}_\alpha$ is Dedekind complete for every $\alpha\in I$.

We prove (ii).  By Proposition <ref>, $\cal{S}^\sim \defeq \left( \orderdual{\vlat{E}},(e^\sim_\alpha)_{\alpha\in I}\right)$ is a compatible system for $\cal{D}^\sim$ in $\nvlic$, hence also in $\nvlc$.  Therefore there exists a unique normal lattice homomorphism $T:\orderdual{\vlat{E}}\to \vlat{F}$ so that the diagram (<ref>) commutes.  We show that $T$ is bijective.

To see that $T$ is injective, let $\psi\in \vlat{E}^\sim$ and suppose that $T(\psi)=0$.  Consider any $u\in \vlat{E}$.  There exist $\alpha\in I$ and $u_\alpha\in \vlat{E}_\alpha$ so that $u=e_\alpha(u_\alpha)$, see Remark <ref>.  Then $\psi(u) = \psi(e_\alpha(u_\alpha)) = e^\sim_\alpha(\psi)(u_\alpha) = p_\alpha(T(\psi))(u) = 0$.  This holds for all $u\in\vlat{E}$ so that $\psi=0$.  Therefore $T$ is injective.

It remains to show that $T$ maps $\vlat{E}^\sim$ onto $\vlat{F}$.  To this end, consider $(\varphi_\alpha) \in \vlat{F}^+$.  We construct a functional $0 \leq \varphi\in \vlat{E}^\sim$ so that $T(\varphi) = (\varphi_\alpha)$.

Let $u\in\vlat{E}$.  Consider any $\alpha,\beta\in I$, $u_\alpha\in\vlat{E}_\alpha$ and $u_\beta\in\vlat{E}_\beta$ so that $e_\alpha(u_\alpha)=u=e_\beta(u_\beta)$, see Remark <ref>.   We claim that $\varphi_\alpha(u_\alpha) = \varphi_\beta(u_\beta)$.  Indeed, there exists $\gamma \scc \alpha,\beta$ in $I$ so that $e_{\alpha , \gamma}(u_\alpha) = e_{\beta , \gamma}(u_\beta)$.  Furthermore, $e_{\gamma}(e_{\alpha , \gamma}(u_\alpha))=u=e_{\gamma}(e_{\beta , \gamma}(u_\beta))$.  Because $(\varphi_\alpha) \in\vlat{F}$ we have $\varphi_\alpha = e_{\alpha , \gamma}^\sim (\varphi_\gamma)$ and $\varphi_\beta = e_{\beta , \gamma}^\sim (\varphi_\gamma)$; that is,

\[
\varphi_\alpha (u_\alpha) = \varphi_\gamma (e_{\alpha , \gamma}(u_\alpha)) = \varphi_\gamma (e_{\beta , \gamma}(u_\beta)) = \varphi_\beta (u_\beta).
\]

Thus our claim is verified.

For $u\in\vlat{E}$ define $\varphi(u) \defeq \varphi_\alpha(u_\alpha)$ if $u=e_\alpha (u_\alpha)$.  By our above claim, $\varphi$ is a well-defined map from $\vlat{E}$ into $\R$. To see that $\varphi$ is linear, consider $u,v\in \vlat{E}$ and $a,b\in\R$.  Let $u=e_\alpha (u_\alpha)$ and $v=e_\beta (v_\beta)$ where $\alpha,\beta\in I$, $u_\alpha\in\vlat{E}_\alpha$ and $v_\beta\in \vlat{E}_\beta$.  There exists $\gamma \scc \alpha,\beta$ in $I$ so that

\[
au+bv = e_{\gamma}(a e_{\alpha , \gamma}(u_\alpha) + b e_{\beta , \gamma}(v_\beta)).
\]

\[
\varphi(au+bv) = \varphi_{\gamma}(a e_{\alpha , \gamma}(u_\alpha) + b e_{\beta , \gamma}(v_\beta)) = a\varphi_\gamma(e_{\alpha , \gamma}(u_\alpha)) + b \varphi_\gamma(e_{\beta , \gamma}(v_\beta)).
\]

But $e_\gamma (e_{\alpha , \gamma}(u_\alpha)) = e_\alpha(u_\alpha)=u$ and $e_\gamma (e_{\beta , \gamma}(v_\beta)) = e_\beta(v_\beta)=v$.  Hence $\varphi_\gamma(e_{\alpha , \gamma}(u_\alpha))= \varphi(u)$ and $\varphi_\gamma(e_{\beta , \gamma}(v_\beta)) = \varphi(v)$.  Therefore $\varphi(au+bv) = a\varphi(u) + b\varphi(v)$.

We show that $\varphi$ is positive.  If $0\leq u\in \vlat{E}$ then there exist $\alpha\in I$ and $0\leq u_\alpha \in\vlat{E}_\alpha$ so that $u=e_\alpha(u_\alpha)$, see Remark <ref>.  Then $\varphi(u) = \varphi_\alpha( u_\alpha )\geq 0$, the final inequality following from the fact that $(\varphi_\alpha)\in\vlat{F}^+$.

It follows from the definition of $\varphi$ and the commutativity of the diagram (<ref>) that $p_{\alpha}(T(\varphi))= e_\alpha^\sim(\varphi)=\varphi_\alpha$ for every $\alpha\in I$.  Hence $T(\varphi) = (\varphi_\alpha)$ so that $T$ is surjective.

Let $\cal{D}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ be a direct system in $\nvlic$, and let $\cal{S} \defeq \left( \vlat{E},(e_\alpha)_{\alpha\in I}\right)$ be the direct limit of $\cal{D}$ in $\vlic$.  The following statements are true.

(i) $\proj{\ordercontn{\cal{D}}}\defeq \left(\vlat{G},(p_\alpha)_{\alpha\in I}\right)$ exists in $\nvlc$.

(iii) If $e_{\alpha , \beta}$ is injective for all $\alpha \precc \beta$ in $I$ then $\ordercontn{\left(\ind{\cal{D}}\right)} \cong \proj{\ordercontn{\cal{D}}}$ in $\nvlc$; that is, there exists a lattice isomorphism $S: \ordercontn{\vlat{E}} \to \vlat{G}$ such that the following diagram commutes for all $\alpha\in I$.

\begin{eqnarray}\label{EQ:  Thm-Order continuous dual of ind sys in NRIP is proj of duals in NRiesz diagram}
\begin{tikzcd}[cramped]
\ordercontn{\vlat{E}} \arrow[rd, "e^\sim_\alpha"'] \arrow[rr, "S"] & & \vlat{G} \arrow[dl, "p_\alpha"]\\
& \ordercontn{(\vlat{E}_\alpha)}
\end{tikzcd}
\end{eqnarray}

The proof proceeds in a similar fashion to that of Theorem <ref>. That (i) is true follows from Proposition <ref> and Theorem <ref> (ii).

For the proof of (ii), assume that $e_{\alpha , \beta}$ is injective for all $\alpha \precc \beta$ in $I$.  By Proposition <ref>, $\ordercontn{\cal{S}}$ is a compatible system of $\ordercontn{\cal{D}}$ in $\nvlic$, hence in $\nvlc$.  Therefore there exits a unique normal lattice homomorphism $S:\ordercontn{\vlat{E}}\to \vlat{G}$ so that the diagram (<ref>) commutes.

It follows by exactly the same reasoning as employed in the proof of Theorem <ref> that $S$ is injective.  It remains to verify that $S$ maps $\ordercontn{\vlat{E}}$ onto $\vlat{G}$.  Let $(\varphi_\alpha)\in\vlat{G}^+$.  As in the proof of Theorem <ref> we define a positive functional $\varphi \in \vlat{E}^\sim$ by setting, for each $u\in \vlat{E}$,

\[
\varphi(u)\defeq \varphi_\alpha (u_\alpha) \text{ if } u = e_\alpha (u_\alpha).
\]

We claim that $\varphi$ is order continuous.  To see that this is so, let $A\downarrow 0$ in $\vlat{E}$.  Without loss of generality, we may assume that $A$ is bounded above by some $0\leq w\in \vlat{E}$.  By Remark <ref> (ii) there exist an $\alpha\in I$ and a $0\leq w_\alpha\in \vlat{E}_\alpha$ so that $e_\alpha(w_\alpha)=w$, and, by Remark <ref> (iii), $e_\alpha$ is injective.  Because $e_\alpha$ is also interval preserving, there exists for every $u\in A$ a unique $0\leq u_\alpha \leq w_\alpha$ in $\vlat{E}_\alpha$ so that $e_\alpha (u_\alpha)=u$.  Let $A_\alpha \defeq \{u_\alpha ~:~ u\in A\}$.  Then $A_\alpha \downarrow 0$ in $\vlat{E}_\alpha$. Indeed, let $0\leq v\in\vlat{E}_\alpha$ be a lower bound for $A_\alpha$.  Then $0\leq e_\alpha(v) \leq e_\alpha(u_\alpha)=u$ for all $u\in A$.  Because $A\downarrow 0$ in $\vlat{E}$ it follows that $e_\alpha (v)=0$, hence $v=0$.  By definition of $\varphi$ and the order continuity of $\varphi_\alpha$ we now have $\varphi[A]=\varphi_\alpha[A_\alpha]\downarrow 0$.  Hence $\varphi\in \ordercontn{\vlat{E}}$.

By definition of $\varphi$ and the commutativity of the diagram (<ref>) it follows that $S(\varphi)=(\varphi_\alpha)$.  Therefore $S$ is surjective.

Let $\cal{D}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ be a direct system in $\vlic$, and let $\cal{S} \defeq \left( \vlat{E},(e_\alpha)_{\alpha\in I}\right)$ be the direct limit of $\cal{D}$ in $\vlic$.  In general, it does not follow from $\preann{\orderdual{(\vlat{E}_\alpha)}}=\{0\}$ for all $\alpha\in I$ that $\preann{\orderdual{\vlat{E}}}=\{0\}$, even if all the $\vlat{E}_\alpha$ are non-trivial and the $e_\alpha$ injective.  Indeed, it is well known that $\vlat{L}^0[0,1]$, the space of Lebesgue measurable functions on the unit interval $[0,1]$, has trivial order dual, see for instance <cit.>.  However, by Example <ref>, $\vlat{L}^0[0,1]$ can be expressed as the direct limit of its principal ideals, each of which has a separating order dual.

In view of the above remark, the following proposition is of interest.

Let $\cal{D}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha, \beta})_{\alpha\precc\beta}\right)$ be a direct system in $\vlic$, and let $\cal{S} \defeq \left( \vlat{E},(e_\alpha)_{\alpha\in I}\right)$ be the direct limit of $\cal{D}$ in $\vlic$.  Assume that for every $\alpha\in I$, $e_\alpha$ is injective and $e_\alpha[\vlat{E}_\alpha]$ is a projection band in $\vlat{E}$.  The following statements are true.

(i) If $\preann{\orderdual{(\vlat{E}_\alpha)}}=\{0\}$ for every $\alpha\in I$ then $\preann{\orderdual{\vlat{E}}}=\{0\}$.

(ii) If $\preann{\ordercontn{(\vlat{E}_\alpha)}}=\{0\}$ for every $\alpha\in I$ then $\preann{\ordercontn{\vlat{E}}}=\{0\}$.

The proofs of (i) and (ii) are identical, except that for (ii) we note that for all $\alpha\in I$, $e_\alpha$ and $e_\alpha^{-1}$ are order continuous by Proposition <ref> (i).  We therefore omit the proof of (ii).

Assume that $\preann{\orderdual{(\vlat{E}_\alpha)}}=\{0\}$ for every $\alpha\in I$.  Let $u\in\vlat{E}$ be non-zero.  Then there exist $\alpha\in I$ and a non-zero $u_\alpha\in \vlat{E}_\alpha$ so that $e_\alpha(u_\alpha)=u$, see Remark <ref>.  By assumption there exists $\varphi_\alpha \in\orderdual{\vlat{E}_\alpha}$ so that $\varphi_\alpha(u_\alpha)\neq 0$.  Denote by $P_\alpha:\vlat{E}\to e_\alpha[\vlat{E}_\alpha]$ the projection onto $e_\alpha[\vlat{E}_\alpha]$.  We note that $e_\alpha$ is an isomorphism onto $e_\alpha[\vlat{E}_\alpha]$.  Let $\varphi \defeq (e_\alpha^{-1} \circ P_\alpha)^\sim(\varphi_\alpha)$.  Then $\varphi\in\orderdual{\vlat{E}}$ and, because $u \in e_\alpha[\vlat{E}_\alpha]$, $\varphi(u) = \varphi_\alpha(e_\alpha^{-1}(P_\alpha(u)))=\varphi_\alpha(u_\alpha)\neq 0$.  Hence $\preann{\orderdual{\vlat{E}}}=\{0\}$.

§.§ Duals of inverse limits

We now turn to duals of inverse limits. For inverse systems over $\N$, we prove results analogous to those of Theorems <ref> and <ref>. We identify the main obstacle to more general results for inverse systems over arbitrary index sets: Positive (order continuous) functionals defined on a proper sublattice of a vector lattice $\vlat{E}$ do not necessarily extend to $\vlat{E}$.

Let $\cal{I}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta\scc \alpha}\right)$ be an inverse system in $\vlic$. The dual system of $\cal{I}$ is the pair $\cal{I}^\sim \defeq \left( (\vlat{E}^\sim_\alpha)_{\alpha\in I}, (p^\sim_{\beta , \alpha})_{\beta\scc \alpha}\right)$.

If $\cal{I}$ is an inverse system in $\nvlc$, define the order continuous dual system of $\cal{I}$ as the pair $\ordercontn{\cal{I}} \defeq \left( (\ordercontn{(\vlat{E}_\alpha)})_{\alpha\in I}, (p^\sim_{\beta , \alpha})_{\beta\scc \alpha}\right)$ with $p^\sim_{\beta , \alpha}: \ordercontn{(E_\alpha)} \to \ordercontn{(\vlat{E}_\beta)}$.

The following preliminary results, analogous to Propositions <ref> and <ref>, are proven in the same way as the corresponding results for direct limits.  As such, we omit the proofs.

Let $\cal{I}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta\scc \alpha}\right)$ be an inverse system in $\vlc$.  The following statements are true.

(i) If $\cal{I}$ is an inverse system in $\vlic$ then the dual system $\cal{I}^\sim$ is a direct system in $\nvlic$.

(ii) If $\cal{I}$ is an inverse system in $\nvlic$ then the order continuous dual system $\ordercontn{\cal{I}}$ is a direct system in $\nvlic$.

Let $\cal{I}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta\scc \alpha}\right)$ be an inverse system in $\vlic$ and  $\cal{S}\defeq \left( \vlat{E}, (p_\alpha)_{\alpha\in I} \right)$ a compatible system of $\cal{I}$ in $\vlic$. The following statements are true.

(i) $\cal{S}^\sim \defeq \left( \vlat{E}^\sim, (p^\sim_\alpha)_{\alpha\in I} \right)$ is a compatible system for the direct system $\cal{I}^\sim$ in $\nvlic$.

(ii) If $\cal{I}$ is an inverse system in $\nvlic$ then $\ordercontn{\cal{S}} \defeq \left( \ordercontn{\vlat{E}}, (p^\sim_\alpha)_{\alpha\in I} \right)$ is a compatible system for the direct system $\ordercontn{\cal{I}}$ in $\nvlic$.

Let $\cal{I}\defeq \left( (\vlat{E}_n)_{n\in \N}, (p_{m, n})_{m\geq n}\right)$ be an inverse system in $\vlic$ and let $\cal{S} \defeq \left( \vlat{E}, (p_n)_{n\in \N} \right)$ be the inverse limit of $\cal{I}$ in $\vlc$.  Assume that $p_{m , n}$ is a surjection for all $m \geq n$ in $\N$. Then $p_n$ is surjective and interval preserving for every $n\in\N$. In particular, $\cal{S}$ is a compatible system of $\cal{I}$ in $\vlic$.

Fix $n_0\in \N$. Consider any $u_{n_0}\in \vlat{E}_{n_0}$.  For $n<n_0$ let $u_n = p_{n_0 , n}(u_{n_0})$.  Because $p_{n_0+1 , n_0}$ is a surjection, there exists $u_{n_0+1}\in \vlat{E}_{n_0+1}$ so that $p_{n_0+1 , n_0}(u_{n_0+1}) = u_{n_0}$.  Inductively, for each $n>n_0$ there exists $u_n\in\vlat{E}_n$ so that $p_{n , n-1}(u_n) = u_{n-1}$.

We show that $(u_n)\in\vlat{E}$.  Let $n<m$ in $\N$.  By the definition of an inverse system it follows that $p_{m , n} = p_{n+1 , n} \circ p_{n+2 , n+1}\circ  \cdots \circ p_{m-1 , m-2}\circ p_{m,m-1}$.  It thus follows that $p_{m , n}(u_m) = u_n$ so that $(u_n)\in \vlat{E}$.  We have $p_{n_0}((u_n))=u_{n_0}$ so that $p_{n_0}$ is a surjection.  It follows from Proposition <ref> that $p_{n_0}$ is interval preserving. Since $\cal{S}$ is a compatible system of $\cal{I}$ in $\vlc$ and the $p_n$ are interval preserving, we conclude that $\cal{S}$ is a compatible system of $\cal{I}$ in $\vlic$.

Let $\cal{I}\defeq \left( (\vlat{E}_n)_{n\in \N}, (p_{m, n})_{m\geq n}\right)$ be an inverse system in $\vlic$, and let $\cal{S} \defeq \left( \vlat{E}, (p_n)_{n\in \N} \right)$ be the inverse limit of $\cal{I}$ in $\vlc$. Assume that $p_{m , n}$ is a surjection for all $m \geq n$ in $\N$.  Then the following statements are true.

(i) $\ind{\cal{I}^\sim}\defeq \left(\vlat{F},(e_n)_{n\in \N}\right)$ exists in $\nvlic$.

(ii) $\left(\proj{\cal{I}}\right)^\sim \cong \ind{\cal{I}^\sim}$ in $\nvlic$; that is, there exists a lattice isomorphism $T: \vlat{F} \to \vlat{E}^\sim$ such that the following diagram commutes for all $n\in \N$.

\[
\begin{tikzcd}[cramped]
\vlat{F} \arrow[rr, "T"] & & \vlat{E}^\sim\\
& \vlat{E}_n^\sim \arrow[lu, "e_{n}"] \arrow[ru, "p_n^\sim"']
\end{tikzcd}
\]

By Proposition <ref>, $\cal{I}^\sim$ is a direct system in $\nvlic$.   Because the $p_{m , n}$ are surjections their adjoints are injective.  Thus by Theorem <ref>, $\ind{\cal{I}^\sim}$ exists in $\nvlic$.

We proceed to prove (ii).  Because the $p_{m , n}^\sim: \left(\vlat{E}_n\right)^\sim \to \left(\vlat{E}_m\right)^\sim$ are injective, so are the $e_n:\left(\vlat{E}_n\right)^\sim \to \vlat{F}$, see Remark <ref>.  By Lemma <ref>, each $p_n:\vlat{E}\to\vlat{E}_n$ is surjective and interval preserving, and $\cal{S}$ is a compatible system of $\cal{I}$ in $\vlic$. Therefore $p_n^\sim:\left(\vlat{E}_n\right)^\sim \to\vlat{E}^\sim$ is an injection for every $n$ in $\N$.

By Proposition <ref>, $\cal{S}^\sim = (\vlat{E}^\sim , (p_n^\sim)_{n\in\N})$ is a compatible system of $\cal{I}^\sim$ in $\nvlic$.  Therefore there exists a unique interval preserving normal lattice homomorphism $T : \vlat{F}\to \vlat{E}^\sim$ so that the diagram

\[
\begin{tikzcd}[cramped]
\vlat{F} \arrow[rr, "T"] & & \vlat{E}^\sim\\
& \vlat{E}_n^\sim \arrow[lu, "e_{n}"] \arrow[ru, "p_n^\sim"']
\end{tikzcd}
\]

commutes for all $n\in\N$.  We show that $T$ is a lattice isomorphism.

Our first goal is to establish that $T$ is injective.  Consider $\varphi\in \vlat{F}$ so that $T(\varphi)=0$.  There exist an $n\in\N$ and a unique $\varphi_{n}\in \orderdual{\vlat{E}}_n$ so that $e_{n}(\varphi_n) = \varphi$.  Then $p_n^\sim (\varphi_n) = T (e_n(\varphi_n)) = T(\varphi)=0$.  But $p_n^\sim$ is injective so that $\varphi_n=0$, hence $\varphi=e_n(\varphi_n)=0$.

It remains to show that $T$ maps $\vlat{F}$ onto $\vlat{E}^\sim$.  This follows from

\[
\vlat{E}^\sim = \displaystyle\bigcup p_n^\sim\left[ \left(\vlat{E}_n\right)^\sim \right],
\]

a fact which we now establish.  Suppose that $\vlat{E}^\sim \neq \displaystyle\bigcup p_n^\sim[\vlat{E}^\sim_n]$.  Because $p_n^\sim$ is an interval preserving lattice homomorphism for every $n\in\N$, each $p_n^\sim[\vlat{E}_n^\sim]$ and hence $\displaystyle\bigcup p_n^\sim[\vlat{E}^\sim_n]$ is a solid subset of $\vlat{E}^\sim$.  Therefore, because $\vlat{E}^\sim \neq \displaystyle\bigcup p_n^\sim[\vlat{E}^\sim_n]$, there exists $0\leq \psi\in \vlat{E}^\sim \setminus \displaystyle\bigcup p_n^\sim[\vlat{E}^\sim_n]$.  By Proposition <ref> (i), $p_n^\sim[\vlat{E}^\sim_n]=\ann{\ker(p_n)}$ for every $n\in \N$ so that $\psi\notin \ann{\ker(p_n)}$ for $n\in\N$.  Hence, for every $n\in\N$, there exists $0\leq u^{(n)}\in\ker(p_n)$ so that $\psi(u^{(n)})=1$.  We claim that there exists $w\in\vlat{E}$ so that $w \geq u^{(1)}+\cdots + u^{(n)}$ for all $n\in\N$.  This claim leads to $\psi(w) \geq \psi(u^{(1)}+\cdots + u^{(n)}) = n$ for every $n\in\N$, which is impossible, so that $\vlat{E}^\sim = \displaystyle\bigcup p_n^\sim\left[ \left(\vlat{E}_n\right)^\sim \right]$.

Write $u^{(n)} = (u_m^{(n)})\in\vlat{E} \subseteq \dsprod\vlat{E}_m$.  Fix $m\in \N$.  If $n>m$ then $u_m^{(n)} = p_{n , m}(p_n(u^{(n)})) = 0$ because $u^{(n)}\in\ker(p_n)$.  Let $w_m \defeq u_m^{(1)} + \cdots + u_m^{(m)}$ and $w\defeq (w_m)$.  Then $w \geq u^{(1)}+ \cdots + u^{(n)}$ for every $n\in\N$ because $u_{m}^{(n)}\geq 0$ for all $m,n\in\N$.  To see that $w\in \vlat{E}$ consider $m_1\geq m_0$ in $\N$.  Then

\[
p_{m_1 , m_0}(w_{m_1}) = p_{m_1 , m_0}(u_{m_1}^{(1)}) + \cdots + p_{m_1 , m_0}(u_{m_1}^{(m_1)}).
\]

But $u^{(n)}=(u_{m}^{(n)})\in\vlat{E}$ for all $n\in\N$, so

\[
p_{m_1 , m_0}(w_{m_1}) = u_{m_0}^{(1)} + \cdots + u_{m_0}^{(m_1)}.
\]

Finally, because $u_m^{(n)}=0$ for all $n>m$ in $\N$ we have

\[
p_{m_1 , m_0}(w_{m_1}) = u_{m_0}^{(1)} + \cdots + u_{m_0}^{(m_0)} = w_{m_0}.
\]

Hence $w\in \vlat{E}$, which verifies our claim.  This completes the proof.

Let $\cal{I}\defeq \left( (\vlat{E}_n)_{n\in \N}, (p_{m, n})_{m\geq n}\right)$ be an inverse system  in $\nvlic$, and let $\cal{S} \defeq \left( \vlat{E}, (p_n)_{n\in \N} \right)$ be the inverse limit of $\cal{I}$ in $\vlc$.  Assume that $\vlat{E}_n$ is Archimedean for each $n\in \N$ and that $p_{m , n}$ is a surjection for all $m \geq n$ in $\N$.  The following statements are true.

(i) $\ind{\ordercontn{\cal{I}}}\defeq \left(\vlat{G},(e_n)_{n\in \N}\right)$ exists in $\nvlic$.

(ii) $\ordercontn{\left(\proj{\cal{I}}\right)} \cong \ind{\ordercontn{\cal{I}}}$ in $\nvlic$; that is, there exists a lattice isomorphism $S: \vlat{G} \to \ordercontn{\vlat{E}}$ such that the following diagram commutes for all $n\in \N$.

\[
\begin{tikzcd}[cramped]
\vlat{G} \arrow[rr, "S"] & & \ordercontn{\vlat{E}}\\
& \ordercontn{(\vlat{E}_n)} \arrow[lu, "e_{n}"] \arrow[ru, "p_n^\sim"']
\end{tikzcd}
\]

The existence of $\ind{\ordercontn{\cal{I}}}$ in $\nvlic$ follows by the same reasoning as given in Theorem <ref>.

For (ii), as in the proof of Theorem <ref>, we see that $e_n:\ordercontn{(\vlat{E}_n)} \to \vlat{G}$ and $p_n^\sim:\ordercontn{(\vlat{E}_n)}\to\ordercontn{\vlat{E}}$ are injective interval preserving maps for all $n\in \N$. In addition, $\cal{S}$ is a compatible system for $\cal{I}$ in $\vlic$.

By Proposition <ref>, $\ordercontn{\cal{S}} = (\ordercontn{\vlat{E}} , (p_n^\sim)_{n\in\N})$ is a compatible system of $\ordercontn{\cal{I}}$ in $\nvlic$. Therefore there exists a unique interval preserving normal lattice homomorphism $S : \vlat{G}\to \ordercontn{\vlat{E}}$ so that the diagram

\[
\begin{tikzcd}[cramped]
\vlat{G} \arrow[rr, "S"] & & \ordercontn{\vlat{E}}\\
& \ordercontn{(\vlat{E}_n)} \arrow[lu, "e_{n}"] \arrow[ru, "p_n^\sim"']
\end{tikzcd}
\]

commutes for all $n\in\N$. Exactly the same argument as used in the proof of Theorem <ref> shows that $S$ is a lattice isomorphism, this time making use of Proposition <ref> (ii).

Theorems <ref> and <ref> cannot be generalised to systems over an arbitrary directed set $I$.  Indeed, the assumption that the inverse system $\cal{I}$ is indexed by the natural numbers is used in essential ways to show that the mappings $T$ and $S$ in Theorems <ref> and <ref>, respectively, are both injective and surjective.  The injectivity of $S$ and $T$ follows from the surjectivity of the maps $p_n$, which in turn follows from Lemma <ref> where the total ordering of $\N$ is used explicitly.  We are not aware of any conditions on a general inverse system $\cal{I}$ in $\vlc$, indexed over an arbitrary directed set, which implies that the projections from $\proj{\cal{I}}$ into the component spaces are surjective.  Furthermore, the method of proof for surjectivity of $S$ and $T$ cannot be generalised to systems over arbitrary directed sets.  As we show next, this issue is related to the extension of positive linear functionals.

Let $\cal{I}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta \scc \alpha}\right)$ be an inverse system in $\vlic$ and $\cal{S} \defeq \left( \vlat{E}, (p_\alpha)_{\alpha\in I} \right)$ its inverse limit in $\vlc$.   Assume that $p_{\beta , \alpha}$ and $p_\alpha$ are surjections for all $\beta\scc \alpha$ in $I$.  Then the following statements are true.

(i) $\ind{\cal{I}^\sim}\defeq \left(\vlat{F},(e_\alpha)_{\alpha\in I}\right)$ exists in $\nvlic$.

(ii) There exists an injective interval preserving normal lattice homomorphism $T:\vlat{F}\to \vlat{E}^\sim$ so that the diagram

\[
\begin{tikzcd}[cramped]
\vlat{F} \arrow[rr, "T"] & & \vlat{E}^\sim\\
& \vlat{E}_\alpha^\sim \arrow[lu, "e_{\alpha}"] \arrow[ru, "p_\alpha^\sim"']
\end{tikzcd}
\]

commutes for every $\alpha \in I$.

(iii) If $T$ is a bijection, hence a lattice isomorphism, then every order bounded linear functional on $\vlat{E}$ has a positive linear extension to $\dsprod \vlat{E}_\alpha$.  The converse is true if $I$ has non-measurable cardinal.

That (i) and (ii) are true follow as in the proof of Theorem <ref>.  We verify (iii).

Let $\iota:\vlat{E}\to\dsprod \vlat{E}_\alpha$ be the inclusion map.  The diagram

\[
\begin{tikzcd}[cramped]
\vlat{E} \arrow[rd, "p_\alpha"'] \arrow[rr, "\iota"] & & \dsprod\vlat{E}_\alpha \arrow[dl, "\pi_\alpha"]\\
& \vlat{E}_\alpha
\end{tikzcd}
\]

commutes for every $\alpha\in I$, and therefore the diagram

\[
\begin{tikzcd}[cramped]
\left(\dsprod\vlat{E}_\alpha\right)^\sim  \arrow[rr, "\iota^\sim"] & & \vlat{E}^\sim  \\
& \vlat{E}_\alpha^\sim \arrow[lu, "\pi_{\alpha}^\sim"] \arrow[ru, "p_\alpha^\sim"']
\end{tikzcd}%\label{EQ:  Remark Proj Lim Dual Diagram Dual of proj lim vs dual of product}
\]

also commutes for each $\alpha\in I$.  Hence, for all $\alpha\in I$, the diagram

\[
\begin{tikzcd}[cramped]
\left(\dsprod\vlat{E}_\alpha\right)^\sim  \arrow[rr, "\iota^\sim"] & &  \vlat{E}^\sim &\\
& \vlat{E}^\sim_\alpha \arrow[ul, "\pi_\alpha^\sim"] \arrow[ur, "p_\alpha^\sim"'] \arrow[rr, "e_\alpha"'] & & \vlat{F} \arrow[ul, "T"']
\end{tikzcd}
\]

Assume that $T$ is a lattice isomorphism, and therefore a surjection.  Let $\varphi\in \vlat{E}^\sim$.  There exists a $\psi\in \vlat{F}$ so that $T(\psi)=\varphi$.  By Remark <ref>, there exist $\alpha\in I$ and $\psi_\alpha\in\vlat{E}^\sim_\alpha$ so that $e_\alpha(\psi_\alpha)=\psi$.  Then

\[
\iota^\sim (\pi_\alpha^\sim(\psi_\alpha)) = p_\alpha^\sim (\psi_\alpha) = T(e_\alpha(\psi_\alpha)) = \varphi.
\]

Therefore $\iota^\sim$ is a surjection; that is, every $\varphi\in \vlat{E}^\sim$ has an order bounded linear extension to $\dsprod \vlat{E}_\alpha$.

Assume that $I$ has non-measurable cardinal, and every order bounded linear functional on $\vlat{E}$ extends to an order bounded linear functional on $\dsprod \vlat{E}_\alpha$.  Then $\iota^\sim$, which acts as restriction of functionals on $\dsprod \vlat{E}_\alpha$ to $\vlat{E}$, is a surjection.  Fix $\varphi\in \vlat{E}^\sim$.  There exists $\psi \in \left(\dsprod\vlat{E}_\alpha\right)^\sim$ so that $\varphi = \iota^\sim (\psi)$.  By Theorem <ref> (iv) there exist $\alpha_1,\ldots,\alpha_n\in I$ and $\psi_1\in \vlat{E}_{\alpha_1}^\sim,\ldots,\psi_n\in \vlat{E}_{\alpha_n}^\sim$ so that $\psi = \pi_{\alpha_1}^\sim (\psi_{\alpha_1}) + \ldots + \pi_{\alpha_n}^\sim (\psi_{\alpha_n})$.  Then

\[
\varphi =\iota^\sim \left( \sum_{i=1}^n \pi_{\alpha_i}^\sim(\psi_i)\right) =  \sum_{i=1}^n \iota^\sim(\pi_{\alpha_i}^\sim(\psi_i)) = \sum_{i=1}^n p_{\alpha_i}^\sim(\psi_i) = \sum_{i=1}^n T(e_{\alpha_i}(\psi_i)) = T\left( \sum_{i=1}^n e_{\alpha_i}(\psi_i)\right).
\]

Therefore $T$ is surjective, and hence a lattice isomorphism.

A similar result holds for the order continuous dual of an inverse limit.  We omit the proof of the next theorem, which is virtually identical to that of Theorem <ref>.  Note, however, that unlike in Theorem <ref>, we make no assumption on the cardinality of $I$.

Let $\cal{I}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta \scc \alpha}\right)$ be an inverse system in $\nvlic$ and $\cal{S} \defeq \left( \vlat{E}, (p_\alpha)_{\alpha\in I} \right)$ its inverse limit in $\vlc$.   Assume that $p_{\beta , \alpha}$ and $p_\alpha$ are surjections for all $\beta \scc \alpha$ in $I$.  Then the following statements are true.

(i) $\ind{\left(\ordercontn{\cal{I}}\right)}\defeq \left( \vlat{G}, (e_\alpha)_{\alpha \in I} \right)$ exists in $\nvlic$.

(ii) There exists an injective and interval preserving normal lattice homomorphism $S:\vlat{G}\to \ordercontn{\vlat{E}}$ so that the diagram

\[
\begin{tikzcd}[cramped]
\vlat{G} \arrow[rr, "S"] & & \ordercontn{\vlat{E}}\\
& \ordercontn{(\vlat{E}_\alpha)} \arrow[lu, "e_{\alpha}"] \arrow[ru, "p_\alpha^\sim"']
\end{tikzcd}
\]

commutes for every $\alpha \in I$.

(iii) $S$ is a lattice isomorphism if and only if every order continuous linear functional on $\vlat{E}$ has an order continuous linear extension to $\dsprod \vlat{E}_\alpha$.

The following two results are immediate consequences of Theorems <ref> and <ref>, respectively.

Let $\cal{I}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\alpha\precc \beta}\right)$ be an inverse system in $\vlic$, $\cal{S} \defeq \left( \vlat{E}, (p_\alpha)_{\alpha\in I} \right)$ its inverse limit in $\vlc$ and $(\vlat{F},(e_\alpha)_{\alpha\in I})$ the direct limit of $\cal{I}^\sim$ in $\nvlic$.   Assume that $p_{\beta , \alpha}$ and $p_\alpha$ are surjections for all $\beta\scc\alpha$ in $I$.  If $\vlat{E}$ is majorising in $\dsprod \vlat{E}_\alpha$ then $\left(\proj{\cal{I}}\right)^\sim \cong \ind{\cal{I}^\sim}$ in $\nvlic$; that is, there exists a lattice isomorphism $T: \vlat{F} \to \vlat{E}^\sim$ such that the diagram

\[
\begin{tikzcd}[cramped]
\vlat{F} \arrow[rr, "T"] & & \vlat{E}^\sim\\
& \vlat{E}_\alpha^\sim \arrow[lu, "e_{\alpha}"] \arrow[ru, "p_\alpha^\sim"']
\end{tikzcd}
\]

commutes for all $\alpha\in I$.

This follows immediately from <cit.> and Theorem <ref>.

Let $\cal{I}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\alpha\precc \beta}\right)$ be an inverse system in $\nvlic$, $\cal{S} \defeq \left( \vlat{E}, (p_\alpha)_{\alpha\in I} \right)$ its inverse limit in $\vlc$ and $(\vlat{F},(e_\alpha)_{\alpha\in I})$ the direct limit of $\ordercontn{\cal{I}}$ in $\nvlic$.   Assume that $p_{\beta , \alpha}$ and $p_\alpha$ are surjections for all $\beta\scc\alpha$ in $I$.  If $\vlat{E}$ is majorising and order dense in $\dsprod \vlat{E}_\alpha$ then $\ordercontn{\left(\proj{\cal{I}}\right)} \cong \ind{\ordercontn{\cal{I}}}$ in $\nvlic$; that is, there exists a lattice isomorphism $S: \vlat{F} \to \ordercontn{\vlat{E}}$ such that the diagram

\[
\begin{tikzcd}[cramped]
\vlat{F} \arrow[rr, "S"] & & \ordercontn{\vlat{E}}\\
& \ordercontn{\left(\vlat{E}_\alpha\right)} \arrow[lu, "e_{\alpha}"] \arrow[ru, "p_\alpha^\sim"']
\end{tikzcd}
\]

commutes for all $\alpha\in I$.

This follows immediately from <cit.> and Theorem <ref>.

In contradistinction with direct limits, the inverse limit construction always preserves the property of having a separating order (continuous) dual.

Let $\cal{I}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (p_{\beta, \alpha})_{\beta \scc \alpha}\right)$ be an inverse system in $\vlc$ and $\cal{S} \defeq \left( \vlat{E}, (p_\alpha)_{\alpha\in I} \right)$ its inverse limit in $\vlc$.  Then the following statements are true.

(i) If $\preann{\orderdual{(\vlat{E}_\alpha)}}=\{0\}$ for every $\alpha \in I$ then $\preann{\orderdual{\vlat{E}}}=\{0\}$.

(ii) If $\preann{\ordercontn{(\vlat{E}_\alpha)}}=\{0\}$ and $p_\alpha$ is order continuous for every $\alpha \in I$ then $\preann{\ordercontn{\vlat{E}}}=\{0\}$.

The proofs of (i) and (ii) are identical.  Hence we omit the proof of (ii).

Assume that $\preann{\orderdual{(\vlat{E}_\alpha)}}=\{0\}$ for every $\alpha \in I$.  Let $u\in\vlat{E}$ be non-zero.  Then there exists $\alpha \in I$ so that $p_\alpha(u)\neq 0$.  Since $\preann{\orderdual{(\vlat{E}_\alpha)}}=\{0\}$, there exists $\varphi\in \orderdual{(\vlat{E}_\alpha)}$ so that $\varphi(p_\alpha (u))\neq 0$; that is, $p_\alpha^\sim(\varphi)(u)\neq 0$.  Hence $\preann{\orderdual{\vlat{E}}}=\{0\}$.

§ APPLICATIONS

In this section we apply the duality results for direct and inverse limits obtained in Section <ref>.  In particular, we consider order (continuous) duals of some of the function spaces which are expressed as direct and inverse limits in Sections <ref> and <ref>, respectively.  This is followed by an investigation of perfect spaces. We show that, under certain conditions, the direct and inverse limits of perfect spaces are perfect.  We then specialise these results to the case of $\cont(X)$ and obtain a solution to the decomposition problem mentioned in the introduction.  Finally, we show that an  Archimedean vector lattice has a relatively uniformly complete predual if and only if it can be expressed, in a suitable way, as an inverse limit of spaces of Radon measures on compact Hausdorff spaces.  The following two simple propositions are used repeatedly.  These results are proved in <cit.> in the context of direct and inverse systems of sets.  The arguments in [10] suffice to verify the results in the vector lattice context, so we do not repeat them here.

Let $\cal{D}\defeq \left( (\vlat{E}_\alpha)_{\alpha\in I}, (e_{\alpha , \beta})_{\alpha\precc \beta}\right)$ and $\cal{D}'\defeq \left( (\vlat{E}_\alpha')_{\alpha\in I}, (e_{\alpha , \beta}')_{\alpha\precc \beta}\right)$ be direct systems in $\vlc$ with direct limits $\cal{S} \defeq \left( \vlat{E}, (e_\alpha)_{\alpha\in I} \right)$ and $\cal{S}' \defeq \left( \vlat{E}', (e_\alpha')_{\alpha\in I} \right)$ in $\vlc$.  Assume that for every $\alpha\in I$ there exists a lattice homomorphism ${T_\alpha :\vlat{E}_\alpha \to \vlat{E}_\alpha'}$ so that the diagram

\begin{eqnarray}
\begin{tikzcd}[cramped]
\vlat{E}_\alpha \arrow[rr, "T_\alpha"] \arrow[dd, "e_{\alpha , \beta}"'] & & \vlat{E}_\alpha' \arrow[dd, "e_{\alpha , \beta}' "]\\
& & \\
\vlat{E}_\beta \arrow[rr, "T_\beta"']  & & \vlat{E}_\beta'
\end{tikzcd}\label{EQ:  Prop-Morphisms between inductive limits 1}
\end{eqnarray}

commutes for all $\alpha\precc \beta$ in $I$.  The following statements are true.

(i) There exists a unique lattice homomorphism $T:\vlat{E}\to\vlat{E}'$ so that the diagram

\begin{eqnarray}
\begin{tikzcd}[cramped]
\vlat{E}_\alpha \arrow[rr, "T_\alpha"] \arrow[dd, "e_{\alpha}"'] & & \vlat{E}_\alpha' \arrow[dd, "e_{\alpha}' "]\\
& & \\
\vlat{E} \arrow[rr, "T"']  & & \vlat{E}'
\end{tikzcd}\label{EQ:  Prop-Morphisms between inductive limits 2}
\end{eqnarray}

commutes for every $\alpha\in I$.

(ii) If $T_\alpha$ is a lattice isomorphism for every $\alpha\in I$, then so is $T$.

Let $u\in\vlat{E}$.  There exists $\alpha\in I$ and $u_\alpha\in\vlat{E}_\alpha$ so that $u=e_\alpha(u_\alpha)$.  Define $T(u)\defeq e_\alpha'(T_\alpha (u_\alpha))$.  It follows from Remark <ref> and the commutative diagram (<ref>) that $T(u)$ is well defined.  Indeed, if also $u=e_{\gamma}(u_\gamma)$ for some $\gamma\neq \alpha$ in $I$ then there exists $\beta\scc\alpha,\gamma$ in $I$ so that $e_{\alpha,\beta}(u_\alpha)=e_{\gamma,\beta}(u_\gamma)$.  Then

\[
e_{\alpha,\beta}'(T_\alpha(u_\alpha)) = T_\beta(e_{\alpha,\beta}(u_\alpha)) = T_\beta(e_{\gamma,\beta}(u_\gamma))=e_{\gamma,\beta}'(T_\gamma(u_\gamma)).
\]

Therefore $e_\alpha'(T_\alpha(u_\alpha))=e_\gamma'(T_\gamma(u_\gamma))$.

With $T$ so defined, it is clear that the diagram (<ref>) commutes.  It follows from the definitions of the vector space and lattice operations on $\vlat{E}$ and $\vlat{E}'$ that $T$ is a linear lattice homomorphism.  It follows from the diagram (<ref>) and the fact that $\cal{S}'$ is the direct limit, and hence a compatible system, of $\cal{D}'$ in $\vlc$ that $\left(\vlat{E}',(e_\alpha'\circ T_\alpha)_{\alpha\in I}\right)$ is a compatible system of $\cal{D}$ in $\vlc$.  The uniqueness of $T$ now follows from the definition of a direct limit.

Assume that $T_\alpha$ is a lattice isomorphism for every $\alpha\in I$.  Then, following the same argument as in the proof of (i), there exists a unique lattice homomorphism $S:\vlat{E}'\to\vlat{E}$ so that the diagram

\[
\begin{tikzcd}[cramped]
\vlat{E}_\alpha' \arrow[rr, "T_\alpha^{-1}"] \arrow[dd, "e_{\alpha}'"'] & & \vlat{E}_\alpha \arrow[dd, "e_{\alpha} "]\\
& & \\
\vlat{E}' \arrow[rr, "S"']  & & \vlat{E}
\end{tikzcd}
\]

We have that $S\circ T$ is the identity on $\vlat{E}$ and $T\circ S$ is the identity on $\vlat{E}'$.  Therefore $T$ is a lattice isomorphism onto $\vlat{E}'$.