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\centerline{Additions to Part IV: Notes}
\bigskip
Change the first part of note 32 to the following:
\smallskip
\head{NOTE 32}\endhead Notes for forms [159 A] and [159 B] from
\ac{Blass} \cite{1983b}.
\par
A {\it similarity type} $\tau$ is a set $J$ of operation symbols and for
each $j \in J$, an arity $I_{j}$ ($I_{j}$ indexes the argument places
of $j$). An {\it algebra of type} $\tau$ is a set $A$ and for each $j\in
J$, an operation $j_{A}$ from $A^{I_{j}}$ into $A$. The following
is a theorem of ZF. (See \ac{Blass} \cite{1983b} and Kerkhoff \cite{1965}
\cite{1969}. Also see notes 50 and 115.)
\iput{similarity type} \iput{algebra of type $\tau$}
\proclaim{Theorem} For any similarity type $\tau$ and set $V$
there is an algebra $F(V)$ of type $\tau$ (called the $\tau$ (or {\it
olutely} free algebra generated by $V$) with the property that $V
\subseteq F(V)$ and for any $A$ of type $\tau$ and any $\eta:
V\rightarrow A$ there is a unique homomorphism $\alpha:
F(V)\rightarrow A$ with $\alpha(v) = v$ for all $v \in V$.
\endproclaim
\smallskip
Add the following as the last sentence of the first paragraph
of note 50.
(Also see notes 32 and 115.)
\smallskip
Change the last sentence of note 115 to:
See also forms [1 AP], [1 AQ], [1 AR],
[67 E] and [67 F] and notes 32 and 50.
\smallskip
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 $y\in X$.
\medskip
Change the beginning of note 32 to the following:
\smallskip
\head{NOTE 32}\endhead Notes for forms [159 A] and [159 B] from
\ac{Blass} \cite{1983b}.
\par
A {\it similarity type} $\tau$ is a set $J$ of operation symbols and for
each $j \in J$, an arity $I_{j}$ ($I_{j}$ indexes the argument places
of $j$). An {\it algebra of type} $\tau$ is a set $A$ and for each $j \in
J$, an operation $j_{A}$ from $A^{I_{j}}$ into $A$. The following
is a theorem of ZF. (See \ac{Blass} \cite{1983b} and Kerkhoff \cite{1965}
\cite{1969}. Also see notes 50 and 115.)
\iput{similarity type} \iput{algebra of type $\tau$}
\proclaim{Theorem} For any similarity type $\tau$ and set $V$
there is an algebra $F(V)$ of type $\tau$ (called the $\tau$ (or {\it
absolutely} free algebra
generated by $V$) with the property that $V \subseteq F(V)$ and for any
$A$ of type $\tau$ and any $\eta : V\rightarrow A$ there is a unique
homomorphism $\alpha : F(V)\rightarrow A$ with $\alpha(v) = v$ for all
$v \in V$.
\endproclaim
\smallskip
At the end of the first paragraph of note 50 add:
(Also see notes 32 and 115.)
\smallskip
Change the last sentence of note 115 to:
Also see forms [1 AP], [1 AQ], [1 AR],
[67 E] and [67 F] and notes 32 and 50.
\smallskip
In the table of contents for the notes change the description
for note 23 to the following:
\smallskip
\head 23. Definitions for [14 Q], [52 E], [52 N] and 410-412:
Weak* topology on the dual of a normed linear space, convex-compact subset,
uniformly convex, weak toplology, reflexive space,
affine subspace of a topological vector space\endhead
%\page{245}\endhead
\smallskip
Revised note 23:
\head{NOTE 23}\endhead Definitions for forms [14 Q], [52 E], [52 N],
and 410-412.
If $E$ is a normed linear space or a {Banach space} (complete,
normed, linear space)
$E^{*}$ (the {\it dual} of $E$) is the space of all bounded (continuous)
linear functionals on $E$. (That is, all $f : E \rightarrow {\Bbb R}$
such that $(\exists M > 0)(\forall x \in E)(|f(x)| \le \Vert M \Vert$).)
$E^{*}$ is a Banach space if we
define $\Vert f\Vert = \sup _{x\neq 0}\left(\frac{|f(x)|}{\Vert
x\Vert} \right)$. Each $x \in E$ can be thought of as
a linear functional on $E^{*}$ if we define $x(f) = f(x)$ for all $f$ in
$E^{*}$. The {\it weak$^{*}$ topology} on $E^{*}$ is the weakest topology that
makes all these linear functionals continuous. (A sequence $\{f_n\}$ in
$E^{*}$ is said to be {\it weakly$^{*}$ convergent} if $\lim_{n\to\infty}
f_n(x)$ exists for every $x\in E$. A sequence $\{x_n\}$ in $E$ is said
to be {\it weakly convergent} if $\lim_{n\to\infty}f(x_n)$ exists
for every $f\in E^{*}$.) We let $E^{**}$ be the dual of $E^{*}$.
If $\phi : E\to E^{**}$ such that for each $x\in E$,
$\phi(x)= f_x$, where for all $g\in E^{*}$, $f_x(g)=g(x)$, then $\phi$
is called the {\it natural embedding} of $E$ into $E^{**}$. If the
natural embedding is onto, the space is called {\it reflexive}.
\iput{weak$^{*}$ topology}\iput {weak topology} Sets of the form
$w_{x,\epsilon } =\{ f \in E^{*}: |f(x)| < \epsilon\}$ form a
basis for the weak$^{*}$ topology. If $X\subseteq E$, $X$ is {\it convex-compact}
if whenever $F_i$, for $i\in I$, are closed convex subsets of $X$ and
$\{X\cap F_i: i\in I\}$ has the finite intersection property,
then $\bigcap_{i\in I}(X\cap F_i)\ne\emptyset$. Note that the definition
of {\it convex-compact} may be given in any topological vector space.
\iput{convex-compact subset of a topological vector space}
$E$ is said to be {\it uniformly convex} if for every $\epsilon>0$ there
is a $\delta>0$ such that for all $x, y\in E$ with $\|x\|=\|y\|
=1$, $\|x-y\|\ge\epsilon$ implies $\frac12\| x+y\|\le 1-\delta$.
\iput{uniformly convex}
(A {\it topological vector space (linear topological space)} is a
vector space (linear space) with a topology in which the
operations of addition and scalar multiplication are continuous.
An {\it affine subspace}, $A$, of a topological vector space $E$ is a
translation of a subspace of $E$, $A =v+S = \{v + w: w\in S\}$, where
$S$ is a subspace of $E$ and $v\in E$.)
(The main reason that form [14 Q] implies 410 is that if the space $E$
is reflexive, then there is a mapping of $E$ onto $E^{**}$ and
the weak topology on $E$ corresponds to the weak$^{*}$ on $E^{**}$.)
medskip
Add to the end of note 10
\definition{Definition} If $(f_n)_{n\in\omega}$ is a sequence of
continuous functions from a topological space $X$ to a topological
space $Y$ and $f: X\to Y$, then $(f_n)_{n\in\omega}$ {\it converges
continuously} to $f$ provided $\forall x\in X$, and for all
sequences $(x_n)_{n\in\omega}$ of elements of $X$ such that
$(x_n)_{n\in\omega}\to x$, $(f(x_n))_{n\in\omega} \to f(x)$.
\enddefinition \iput{continuous convergence}
\medskip
Add to the table of contents of the notes:
\head 153. Definitions for Determinateness Axioms (form [94 R])
\endhead
\smallskip
Add note 153:
\head{NOTE 153}\endhead Definitions for Determinateness Axioms
(form [94 R]).
\iput{determinateness}
\definition{Definition} Assume that $A$ is a set of sequences of
of elements from the set $X$.
\item{1.} The game $G(X^{\Bbb N},A)$ is the game played by two
players I and II who alternately choose elements of $X$ (with
I choosing first) to produce a sequence $x =(x_o, x_1, x_2,
\ldots)$. Player I wins if $x\in A$, otherwise II wins.
\item{2.} A {\it strategy} for the game $G(X^{\Bbb N},A)$ is a
function from finite sequences of elements of $X$ to $X$.
\iput{strategy}
\item{3.} If $s$ is a strategy for the game $G(X^{\Bbb N},A)$ and
$x\in X^{\Bbb N}$, then $s[x]$ is the sequence
$(s(\emptyset),x_0,s(x_0),x_1,s(x_0,x_1),x_2,s(x_0,x_1,x_2),\ldots)$
and $[x]s$ is the sequence
$(x_0,s(x_0),x_1,s(x_0,x_1),x_2,s(x_0,x_1,x_2),\ldots)$. ($s[x]$
is the sequence obtained when player I plays the strategy $s$
and player II plays the sequence $x$. Similarly for $[x]s$.)
\item{4.} If $s$ is a strategy for the game $G(X^{\Bbb N},A)$,
then $s$ is a {\it winning strategy for player I} if for all
$x\in X^{\Bbb N}$, $s[x]\in A$. $s$ is a {\it winning strategy for
player II} if for all $x\in X^{\Bbb N}$, $[x]s\notin A$.
\iput{winning strategy}
\enddefinition
\bye