%Updated 2/01/00
%\undefine \eth
\input amstex
\NoBlackBoxes
\documentstyle{amsppt}
\settabs\+\hskip.1in&\hskip.55in& \cr
\def\psc#1#2{{\tabalign & #1 & \vtop{\hsize=4.325in\parindent 0pt #2\strut }&\cr}\vskip2pt}
\def\ps{\hangindent .3in\hangafter 1}
\def\iput#1{}
\def\icopy#1{#1}
\font \sc cmcsc10
\font\Large=cmr8 scaled \magstep2
\magnification=\magstep1
\loadmsbm
\loadbold
\loadeusb
\def\ac#1{#1}
\pageheight{8.75truein}
\pagewidth{6.5truein}
\centerline{Additions to Part IV: Notes}
\bigskip
In the table of contents for the notes add:
\bigskip
\head 151. Definitions for forms [14 DA]--[14 DF]\endhead
New note:
\head{NOTE 151}\endhead In this note we give the definition of
a lattice ordered group and related notions for forms [14 DA]--
[14 DF].
\definition{Definition}
\item{1.} If $(G,+)$ is a group and $\le$ is a partial ordering on
$G$ compatible with $+$ such that each pair of elements in $G$ have
a least upper bound and a greatest lower bound, then $(G,+)$ is
called a {\it lattice ordered group} or an {\it $l$-group}. (The
partial order $\le$ is {\it compatible} with $+$ if for all
$a,b,c,$ and $d$ in $G$, if $a\le b$, and $c\le d$, then
$a + c\le b + d$. \iput{lattice ordered group} \iput{l-group}
\item{2.} If $(G,+)$ is an $l$-group, $G$ is called {\it archimedian}
if for all pairs of strictly positive elements $g,h\in G$, there is
an $n\in\Bbb N$ such that $g^n\le h$ does not hold. (An element
$g\in G$ is {\it positive} if $e\leq g$ and {\it strictly positive}
if $e\le g$, where $e$ is the unit.) \iput{archimedian l-group}
\item{3.} If $(G,+)$ is an $l$-group, $G$ is called
{\it hyperarchimedian} if all its homomorphic images are
archimedian. \iput{hyperarchimedian l-group}
\item{4.} An $l$-group is called {\it simple} if it does not contain
any proper $l$-ideals. \iput{simple l-group}
\item{5.} Two elements $g$ and $h$ in an $l$-group $G$ are called
{\it orthogonal} if the meet of their absolute values is $e$, the
unit. (The {\it absolute value} of an element $g\in G$ is $g$
if $e\leq g$ and $-g$ otherwise, where $-g$ is the inverse of $g$.)
\item{6.} A {\it weak unit} in an $l$-group is a positive element
$u$ which is orthogonal only to $e$. \iput{weak unit in an
$l$-group}
\item{7.} A topological space is called {\it Boolean} if it is
compact, Hausdorf, and the family of compact open subsets form
a base for the topology. \iput{Boolean topological space}
\item{8.} $A$ is called a {\it Boolean product} of the family
$\Cal A = \{A_i: i\in I\}$ if
\itemitem{(i)} $A$ is a subdirect product of $\Cal A$.
\itemitem{(ii)} $I$ admits a Boolean space topology such that
\itemitem{(a)} For any atomic formula $\phi(v_1,v_2,\cdots,v_n)$
and for all $a_1, a_2, \cdots, a_n \in A$, the subset
$$\{i\in I: A_i\models\phi(a_1(i),a_2(i),\cdots, a_n(i)\}$$
is clopen in $I$.
\itemitem{(b)} For all $a,b\in A$ and for all clopen $J\subseteq I$,
the element $(a\restriction J)\cup (b\restriction (I\setminus J)$
belongs to $A$. \iput{Boolean product}\enddefinition
\medskip
In the table of contents for the notes add:
\bigskip
\head 152. A proof that 409 implies 62 \endhead
\head{NOTE 152}\endhead A proof that form 409 implies form 62.
Let $X$ be a set of disjoint finite sets and construct the graph
$(G,\Gamma) $ where $G=\bigcup X$ and for $t\in G$, if $t\in y\in X$ then
$\Gamma(t) = y\setminus \{t\}$. (So if $y\in X$ there is an edge from
every element of $y$ to every other element of $y$.) Let $K=\{0,1\}$
and define the function $T:\Cal P(K) \to \Cal P(K)$ by $T(\{0\}) =
\{1\}$, $T(\{1\}) = \{0\}$, $T(\emptyset) = K$ and $T(K) =
K$. The hypotheses of 409 are satisfied for
$(G,\Gamma)$ and $K$. Here is the argument:
If $A$ is a finite subgraph of $G$
(I.e., $A\subseteq G$), the $A$ can be written as the disjoint
union $A = (A\cap y_1)\cup \cdots (A\cap y_n)$ where $y_i\in X$
and $y_i\cap A\ne \emptyset$, for $i = 1,\ldots, n$. Choose
one element $t_i$ in each of the sets $A\cap y_i$ for $i=1,\ldots
n$ and define $\phi_A:A\to K$ by $\phi_A(t_i) = 0$ for $i = 1,\ldots,n$
and $\phi_A(t) = 1$ for other elements $t\in A$. For the
proof that $\phi_A(t)\in T(\phi_A(\Gamma_A(t)))$ for all $t\in A$,
($\Gamma_A$ is $\Gamma$ restricted to $A$.)
assume that $t\in A\cap y_i$ and consider two cases\par
Case 1. $|A\cap y_1| = 1$. In this case $t=t_i$ so $\phi_A(t) =
\phi_A(t_i) = 0$ Also in this case $\Gamma_A(t) = \emptyset$ so
$T(\phi_A(\Gamma_A(t))) = T(\phi_A(\emptyset)) = T(\emptyset)
= K = \{ 0,1\}$ so $\phi_A(t)\in T(\phi_A(\Gamma_A(t)))$.\par
Case 2. $|A\cap y_i| >1$. In this case let $s$ be an element
of $A\cap y_i$ different from $t_i$. If $t=t_i$ then $s\in
\Gamma_A(t_i)=\Gamma_A(t)$ so $\phi_A(s)\in \phi_A(\Gamma_A(t))$
so $1\in \phi_A(\Gamma_A(t))$. By the definition of $T$ it
follows that $0\in T(\phi_A(\Gamma_A(t)))$. But $\phi_A(t) =
\phi_A(t_i) = 0$. On the other hand if $t\ne t_i$ then
$t_i\in \Gamma_A(t)$ so $\phi_A(t_i) = 0\in
\phi_A(\Gamma_A(t))$. Using the definition of $T$ again, this
means that $1\in T(\phi_A(\Gamma_A(t)))$. But $\phi_A(t) = 1$.
Therefore by 409, there is a
function $\phi_A : G\to K$ such that for all $t\in G$,
$\phi(t) \in T(\phi(\Gamma(t)))$. We now argue that for each
$y\in X$, $\phi$ restricted to $y$ cannot be constant. (By
contradiction.) Assume $\forall t\in y$, $\phi(t) = 0$. Then
for any $t\in y$, $T(\phi(\Gamma(t))) = T(\{0\}) = \{1\}$.
But then $\phi(t)= 0 \notin T(\phi(\Gamma(t)))$. A similar
argument shows that $\phi(t) = 1$ for all $t\in y$ is
impossible. We can now get a Kinna-Wagner function for $X$ by
defining $f(y) = \{ t\in y : \phi(t) = 0\}$ for each $y\in X$.
\bye