\documentclass[12pt]{article}
\usepackage{latexsym}
\usepackage{amssymb}
\newcommand{\ga}{\alpha}
\newcommand{\gb}{\beta}
\renewcommand{\gg}{\gamma}
\newcommand{\gd}{\delta}
\newcommand{\gep}{\epsilon}
\newcommand{\gz}{\zeta}
\newcommand{\gee}{\eta}
\newcommand{\gth}{\theta}
\newcommand{\gi}{\iota}
\newcommand{\gk}{\kappa}
\newcommand{\gl}{\lambda}
\newcommand{\gm}{\mu}
\newcommand{\gn}{\nu}
\newcommand{\gx}{\xi}
\newcommand{\gom}{\omicron}
\newcommand{\gp}{\pi}
\newcommand{\gr}{\rho}
\newcommand{\gs}{\sigma}
\newcommand{\gt}{\tau}
\newcommand{\gu}{\upsilon}
\newcommand{\gph}{\phi}
\newcommand{\gch}{\chi}
\newcommand{\gps}{\psi}
\newcommand{\go}{\omega}
\newcommand{\gA}{A}
\newcommand{\gB}{B}
\newcommand{\gG}{\Gamma}
\newcommand{\gD}{\Delta}
\newcommand{\gEp}{E}
\newcommand{\gZ}{Z}
\newcommand{\gEe}{H}
\newcommand{\gTh}{\Theta}
\newcommand{\gI}{I}
\newcommand{\gK}{K}
\newcommand{\gL}{\Lambda}
\newcommand{\gM}{M}
\newcommand{\gN}{N}
\newcommand{\gX}{\Xi}
\newcommand{\gOm}{O}
\newcommand{\gP}{\Pi}
\newcommand{\gR}{P}
\newcommand{\gS}{\Sigma}
\newcommand{\gT}{T}
\newcommand{\gU}{\Upsilon}
\newcommand{\gPh}{\Phi}
\newcommand{\gCh}{X}
\newcommand{\gPs}{\Psi}
\newcommand{\gO}{\Omega}
\newcommand{\bA}{{\bf A}}
\newcommand{\bB}{{\bf B}}
\newcommand{\bG}{\boldGamma}
\newcommand{\bD}{\boldDelta}
\newcommand{\bEp}{{\bf E}}
\newcommand{\bZ}{{\bf Z}}
\newcommand{\bEe}{{\bf H}}
\newcommand{\bTh}{\boldTheta}
\newcommand{\bI}{{\bf I}}
\newcommand{\bK}{{\bf K}}
\newcommand{\bL}{{\bf L}}
\newcommand{\bM}{{\bf M}}
\newcommand{\bN}{{\bf N}}
\newcommand{\bX}{\boldXi}
\newcommand{\bOm}{{\bf O}}
\newcommand{\bP}{\boldPi}
\newcommand{\bR}{{\bf P}}
\newcommand{\bS}{\boldSigma}
\newcommand{\bT}{{\bf T}}
\newcommand{\bU}{\boldUpsilon}
\newcommand{\bPh}{\boldPhi}
\newcommand{\bCh}{{\bf X}}
\newcommand{\bPs}{\boldPsi}
\newcommand{\bO}{\boldOmega}
\newcommand{\rest}{\restriction}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\ov}{\overline}
\newcommand{\add}{{\rm Add}}
\newcommand{\K}{{\mathfrak K}}
\newcommand{\U}{{\cal U}}
%
% Hebrew letters
%
\newcommand{\ha}{\aleph}
\newcommand{\hb}{\beth}
\newcommand{\hg}{\gimel}
\newcommand{\hd}{\daleth}
%
% basic set theory constructions
%
\newcommand{\setof}[2]{{\{\; #1 \; \vert \; #2 \; \} } }
\newcommand{\seq}[1]{{\langle #1 \rangle} }
\newcommand{\card}[1]{{\vert #1 \vert} }
\newcommand{\ot}[1]{\hbox{o.t.($#1$)}}
\newcommand{\forces}{\Vdash}
\newcommand{\decides}{\parallel}
\newcommand{\ndecides}{\nparallel}
\renewcommand{\models}{\vDash}
\newcommand{\powerset}{{\cal P}}
\newcommand{\bool}{{\bf b} }%
%
% stuff for use inside math formulae
%
\newcommand{\dom}{{\rm dom}}
\newcommand{\rge}{{\rm rge}}
\newcommand{\crit}{{\rm crit}}
\renewcommand{\top}{{\rm top}}
\newcommand{\supp}{{\rm supp}}
\newcommand{\support}{{\rm support}}
\newcommand{\cf}{{\rm cf}}
\newcommand{\lh}{{\rm lh}}
\newcommand{\lp}{{\rm lp}}
\newcommand{\up}{{\rm up}}
\newcommand{\FF}{{\mathbb F}}
\newcommand{\FP}{{\mathbb P}}
\newcommand{\FQ}{{\mathbb Q}}
\newcommand{\FR}{{\mathbb R}}
\newcommand{\FS}{{\mathbb S}}
\newcommand{\FT}{{\mathbb T}}
\newcommand{\implies}{\Longrightarrow}
%\newcommand{\commtriangle}[6]
%{
%\medskip
%\[
%\setlength{\dgARROWLENGTH}{6.0em}
%\begin{diagram}
%\node{#1} \arrow[2]{e,t}{#6} \arrow{se,b}{#4} \node[2]{#3} \\
%\node[2]{#2} \arrow{ne,r}{#5}
%\end{diagram}
%\]
%\medskip
%}
%
% This picture tells you what order to put the arguments in
%
%
%
%
% #6
% #1 --------- #3
% \ /
% \#4 / #5
% \ /
% #2/
%
\newtheorem{theorem}{Theorem}
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}[definition]{Remark}
\newtheorem{fact}[definition]{Fact}
\newtheorem{lemma}[definition]{Lemma}
\newtheorem{claim}[definition]{Claim}
\newtheorem{conjecture}{Conjecture}
\newenvironment{proof}{\noindent{\bf
Proof:}}{\nopagebreak\mbox{}\newline\makebox[\textwidth]{\hfill$\square$}
\par\bigskip}
\newenvironment{sketch}{\noindent{\bf
Sketch of Proof:}}{\nopagebreak\mbox{}\newline
\makebox[\textwidth]{\hfill$\square$}\par\bigskip}
\newenvironment{pf}{\indent{${}$}}{\nopagebreak\mbox{}\newline
\makebox[\textwidth]{\hfill$\square$}\par\bigskip}
\newcommand{\lra}{\longrightarrow}
\setlength{\topmargin}{-0.62in}
\setlength{\textheight}{9.10in}
\setlength{\oddsidemargin}{-0.15in}
\setlength{\textwidth}{6.95in}
\setlength{\parindent}{1.5em}
%\setcounter{section}{-1}
%\setcounter{theorem}{-1}
% IndWC.tex
% The following macros are a selection from Joel's general math
% macros used in the document below
%
\def\tlt{\triangleleft}
\def\k{\kappa}
\def\a{\alpha}
\def\b{\beta}
\def\d{\delta}
\def\s{\sigma}
\def\t{\tau}
\def\l{\lambda}
\def\lted{{{\leq}\d}}
\def\ltk{{{<}\k}}
%
% Arthur, in the next two macro definitions, use blackboard bold for
% \bm. Latex uses a different name I think.
%
\def\P{{\mathbb P}}
\def\Q{{\mathbb Q}}
\def\Qdot{\dot\Q}
\def\Pforces{\forces_{\P}}
\def\of{{\subseteq}}
%\def\card#1{\left|#1\right|}
\def\boolval#1{\mathopen{\lbrack\!\lbrack}\,#1\,\mathclose{\rbrack\!
\rbrack}}
\def\restrict{\mathbin{\mathchoice{\hbox{\am\char'26}}{\hbox{\am\char'
26}}{\hbox{\eightam\char'26}}{\hbox{\sixam\char'26}}}}
\def\st{\mid}
\def\set#1{\{\,{#1}\,\}}
\def\th{{\hbox{\fiverm th}}}
\def\muchgt{>>}
\def\cof{\mathop{\rm cof}\nolimits}
\def\iff{\mathrel{\leftrightarrow}}
\def\intersect{\cap}
\def\minus{\setminus}
\def\Union{\bigcup}
\def\union{\bigcup}
\def\and{\mathrel{\kern1pt\&\kern1pt}}
\def\image{\mathbin{\hbox{\tt\char'42}}}
\def\elesub{\prec}
\def\iso{\cong}
\def\<#1>{\langle\,#1\,\rangle}
\def\ot{\mathop{\rm ot}\nolimits}
%
% ------------------------------------------------------------------------------
%
\title{Removing Laver Functions from
Supercompactness Arguments
%Forcing Con(PFA) and Con(SPFA) via
%Lottery Sums
\thanks{2000 Mathematics Subject Classifications:
03E35, 03E55}
\thanks{Keywords: Supercompact cardinal, proper
forcing axiom, semiproper forcing axiom,
lottery sum, strong cardinal}}
\author{Arthur W.~Apter\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
%http://math.baruch.cuny.edu/$\sim$apter\\
http://faculty.baruch.cuny.edu/apter\\
awabb@cunyvm.cuny.edu}
\date{April 15, 2004}
\begin{document}
\maketitle
\begin{abstract}
We show how the use of
a Laver function in the
proof of the consistency,
relative to the existence
of a supercompact cardinal, of
both the Proper Forcing Axiom and the
Semiproper Forcing Axiom can
be eliminated via the use
of lottery sums of the
appropriate partial orderings.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
In \cite{H4}, Hamkins introduced a
general method for forcing indestructibility
he calls the lottery preparation.
The advantage of Hamkins' method is
that it avoids the need for Laver functions
\cite{L}, although as presented in
\cite{H4}, it still requires
the use of a non-canonical function called
a fast function (whose definition can
be found in both \cite{H4} and \cite{A03}).
For the case of supercompactness,
this was rectified in \cite{A03},
in which it was shown how to apply the
methods of \cite{H4} to force
indestructibility for a supercompact
cardinal via a
lottery preparation but without
employing either a Laver function
or a fast function.
%a non-canonical function.
The purpose of this note is to demonstrate
that the ideas of \cite{A03}
can be extended to other situations
in which a Laver function was
originally found. In particular,
we will show that
it is possible to remove the
use of a Laver function from
the standard arguments due to Baumgartner
(see \cite{J}) and Foreman, Magidor,
and Shelah (see \cite{FMS})
for obtaining the consistency of
both the Proper Forcing Axiom (PFA) and
the Semiproper Forcing Axiom (SPFA)
relative to the existence of a
supercompact cardinal.
Specifically, we will prove the
following theorems.
\begin{theorem}\label{t1}
Suppose
$V \models ``$ZFC + $\gk$ is
a supercompact cardinal''.
There is then a partial ordering
$\FP \in V$ defined without the
use of a Laver function
such that
$V^\FP \models ``$ZFC + PFA''.
\end{theorem}
\begin{theorem}\label{t2}
Suppose
$V \models ``$ZFC + $\gk$ is
a supercompact cardinal''.
There is then a partial ordering
$\FP \in V$ defined without the
use of a Laver function
such that
$V^\FP \models ``$ZFC + SPFA''.
\end{theorem}
\noindent In fact, as our methods
of proof will show, our partial
orderings will be defined completely
canonically, i.e., without the use
of a fast function as well.
Before beginning the proofs of
Theorems \ref{t1} and \ref{t2},
we briefly mention some preliminary
information. First,
we recall for the benefit of readers Hamkins'
definition from Section 3 of \cite{H4} of the lottery sum
of a collection of partial orderings.
If ${\mathfrak A}$ is a collection of partial orderings, then
the lottery sum is the partial ordering
$\oplus {\mathfrak A} =
\{\la \FP, p \ra : \FP \in {\mathfrak A}$
and $p \in \FP\} \cup \{0\}$, ordered with
$0$ below (i.e., weaker than) everything and
$\la \FP, p \ra \le \la \FP', p' \ra$ iff
$\FP = \FP'$ and $p \le p'$.
Intuitively, if $G$ is $V$-generic over
$\oplus {\mathfrak A}$, then $G$
first selects an element of
${\mathfrak A}$ (or as Hamkins says in \cite{H4},
``holds a lottery among the posets in
${\mathfrak A}$'') and
then forces with it.\footnote{The
terminology ``lottery sum'' is due
to Hamkins, although the concept
of the lottery sum of partial
orderings has been around for quite
some time and has been referred to
at different junctures via the names
``disjoint sum of partial orderings'',
``side-by-side forcing'', and
``choosing which partial ordering to
force with generically''.}
We assume some familiarity with
the notions of properness and
semiproperness, as found, e.g.,
in Part III of \cite{J}
(or more extensively, in \cite{Sh}).
Although there are a number of
equivalent definitions of these
properties, we present the ones
which will be most useful in
our context.
Suppose $\FP$ is a partial
ordering, and consider
the following game $\mathfrak G$.
Player I begins by selecting
a condition $p \in \FP$
and an ordinal name $\check \ga_0$
in the forcing language with
respect to $\FP$.
Player II responds by choosing an
ordinal $\gb_0$.
At the $n^{\rm th}$ move for $n > 0$,
I plays an ordinal name $\check \ga_n$
in the forcing language with
respect to $\FP$, and II responds by
choosing an ordinal $\gb_n$.
II wins the game iff there is some
$q$ extending $p$ such that for
every natural number $n$,
$q \forces ``$There is a natural
number $k$ such that
$\check \ga_n = \gb_k$''.
$\FP$ is then said to be proper
if II has a winning strategy
for this game.
If the game is changed such that
instead of each $\check \ga_i$
being a name for an arbitrary ordinal,
each $\check \ga_i$ is a name for
a countable ordinal, then
$\FP$ is said to be semiproper
if II has a winning strategy for the game.
From these definitions, it is
easy to see that the lottery sum
of a collection of either proper
or semiproper partial orderings
remains either proper or semiproper.
Also, using these definitions, we
may now give both PFA and SPFA,
which state that if $\FP$ is
proper (semiproper) and
${\mathfrak D} = \la D_\ga : \ga < \ha_1 \ra$ is
a collection of dense open subsets
of $\FP$, then there is a
$\mathfrak D$-generic filter for $\FP$.
The remainder of
our terminology and notation are
standard.
We note only that for any
ordinal $\gd$, $\gd'$ is
the least strong cardinal above $\gd$.
For anything left unexplained,
we refer readers to \cite{A03}.
\section{The Proofs of Theorems \ref{t1}
and \ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t1}.
\begin{proof}
Suppose that
$\ov V$ is any model of
ZFC in which there is a proper
partial ordering for which PFA fails,
i.e., for which
there is a collection of $\ha_1$ many
dense open sets for which there is
no generic filter.
We begin by noting that if
$\gg$ is any $\ov V$-strong
cardinal, then there is a proper
partial ordering having rank below
$\gg$ witnessing the failure of PFA.
To see this, let $\FQ$ be a proper
partial ordering having rank
$\gz \ge \gg$ for which PFA fails. Let
$\eta > \gz$ be sufficiently large with
$j : V \to M$ an elementary embedding
witnessing the $\eta$ strongness of $\gg$
such that
$M \models ``\FQ$ is a proper partial
ordering having rank
$\gz$ for which PFA fails''.
By reflection, there will be
unboundedly in $\gg$ many cardinals
$\gz'$ below $\gg$ and partial
orderings having rank $\gz'$ for
which PFA fails.
Let
$V \models ``$ZFC + $\gk$ is supercompact''.
As in Baumgartner's original proof, our
partial ordering $\FP =
\la \la \FP_\ga, \dot \FQ_\ga \ra :
\ga < \gk \ra$
used in the proof of Theorem \ref{t1}
is a countable support iteration having
length $\gk$.
In analogy to the iteration given in
the proof of Theorem 2 of \cite{A03},
$\FP$ will be non-trivial only at
those stages which are in $V$
regular limits of strong cardinals.
At such a stage $\gd$, we let
$\mathfrak A$ be the collection of
all proper partial orderings in
$V^{\FP_\gd}$ of minimal rank
for which PFA fails.
We then let
$\FP_{\gd + 1} = \FP_\gd \ast \dot \FQ_\gd$,
where $\dot \FQ_\gd$ is a term for the
lottery sum $\oplus {\mathfrak A}$.
Since inductively
$\card{\FP_\gd} < {(\gd')}^V$,
by the work found in the paper by
Hamkins and Woodin \cite{HW},
${(\gd')}^V = {(\gd')}^{V^{\FP_\gd}}$.
Hence, the results of the preceding paragraph
tell us that
$\forces_{\FP_\gd} ``\dot \FQ_\gd$ has
rank below ${(\gd')}^V$''.
%$\gd'$''.
Suppose now
$V^\FP \models
``$PFA fails as witnessed
by the proper partial ordering $\FQ$ having
minimal rank and the sequence of
dense open sets
${\mathfrak D} = \la D_\ga : \ga
< \ha_1 \ra$''. Let
$\gl > \card{{\rm TC}(\dot \FQ)}$ be a
sufficiently large cardinal such that
for any elementary embedding
$j : V \to M$ witnessing the $\gl$
supercompactness of $\gk$,
$M^\FP \models
``$PFA fails as witnessed
by the proper partial ordering $\FQ$ having
minimal rank and the sequence of
dense open sets
${\mathfrak D} = \la D_\ga : \ga
< \ha_1 \ra$''. If $j$ is chosen
to be such that
$M \models ``\gk$ isn't $\gl$ supercompact'',
then since as in the proof of Theorem 2
of \cite{A03},
$M \models ``\gk$ is a regular limit of
strong cardinals and no cardinal $\gd$ with
$\gk < \gd \le \gl$ is strong'',
it is possible to opt for $\FQ$ in the
stage $\gk$ lottery held in $M^\FP$
in the definition of $j(\FP)$. Hence,
above a condition opting for $\FQ$,
%in the stage $\gk$ lottery held in $M$,
$j(\FP)$ is forcing equivalent in $M$ to
$\FP \ast \dot \FQ \ast \dot \FR$ for
$\dot \FR$ a term for the
portion of $j(\FP)$ defined between
stages $\gk + 1$ and $j(\gk)$.
%appropriate partial ordering.
We conclude the proof of Theorem \ref{t1}
by following the reasoning found in
Baumgartner's original argument. Let
$G \ast H \ast K$ be $V$-generic for
$\FP \ast \dot \FQ \ast \dot \FR$. In
$V[G][H][K]$, since
$j '' G \subseteq G \ast H \ast K$,
$j$ lifts to an embedding which,
with a slight abuse of notation, we denote by
$j : V[G] \to M[G][H][K]$.
Define now
in $V[G][H][K]$ the filter
${\cal F}$ generated by $\{j(p) : p \in H\}$.
Because $j '' \gl \in M$,
the choice of $\gl$ yields that
$j '' \FQ \in M[G][H][K]$.
This allows us to infer that
${\cal F} \in M[G][H][K]$.
Therefore, as all iterations with which
we are dealing are countable support
iterations of proper partial orderings
and hence preserve
${(\ha_1)}^V = {(\ha_1)}^M < \gk$,
$j(\la D_\ga : \ga < \ha_1 \ra) = j({\mathfrak D}) =
\la j(D_\ga) : \ga < \ha_1 \ra$. Hence,
$M[G][H][K] \models ``$There is a
$j({\mathfrak D})$-generic filter for
the proper partial ordering $j(\FQ)$'',
so by elementarity,
$V[G] \models ``$There is a
${\mathfrak D}$-generic filter for
the proper partial ordering $\FQ$''.
This contradiction completes the proof of
Theorem \ref{t1}.
\end{proof}
\begin{pf}
The proof of Theorem \ref{t2}
proceeds in much the same
way as the proof of Theorem \ref{t1},
with the appropriate modifications.
In particular, instead of taking a
countable support iteration, we take
a revised countable support iteration,
in the sense of \cite{FMS} and \cite{Sh}.
Also, when $\dot \FQ_\gd$ is a term
for a forcing done at a non-trivial
stage $\gd$ of the iteration, instead
of $\dot \FQ_\gd$ only being a term for
the lottery sum $\oplus {\mathfrak A}$
of the collection of all partial orderings
of minimal rank for which SPFA fails, as in
\cite{FMS}, $\dot \FQ_\gd$ is actually a term for
$\oplus {\mathfrak A} \ast
\dot {\rm Coll}(\ha_1, 2^{\card{\FP_\gd \ast
\oplus \dot {\mathfrak A}}})$.\footnote{We
remark that in the original proof found in
\cite{FMS}, there are non-trivial stages
in the definition of the iteration at
which one forces only with
${\rm Coll}(\ha_1, \gd)$ for the
appropriate value of $\gd$.
This is unnecessary for our purposes,
since this partial ordering is only
used to obtain that the conditions
``preservation of stationary subsets
of $\ha_1$'' and ``semiproper'' are
equivalent.}
With these changes, and using properties
a) - f) of revised countable support
iterations as given in the proof of
Theorem 5 of \cite{FMS},
the proof of Theorem \ref{t2} is
now finished as in the proof
of Theorem \ref{t1} above.
This completes the proof of Theorem \ref{t2}.
\end{pf}
We note that the original arguments
used in the proofs of Con(PFA) and
Con(SPFA) from a supercompact cardinal
will show that in both Theorems \ref{t1}
and \ref{t2}, $2^{\ha_0} =
\gk$ and $\gk$ is collapsed to $\ha_2$.
Also, since PFA is equivalent to
Martin's Maximum (MM) (see \cite{J} or \cite{Sh}),
the model constructed for Theorem \ref{t2}
is actually a model for MM.
Finally, slight modifications of our techniques,
which we leave to the readers of
this paper, will show that our methods of
proof can be used to construct models for
PFA${}^+$, SPFA${}^+$, and MM${}^+$
(the definitions for which can be found in
\cite{FMS}, \cite{J}, and \cite{Sh}).
\begin{thebibliography}{99}
\bibitem{A03} A.~Apter, ``Some Remarks on Indestructibility
and Hamkins' Lottery Preparation'',
{\it Archive for Mathematical Logic 42}, 2003, 717--735.
\bibitem{FMS} M.~Foreman, M.~Magidor, and
S.~Shelah, ``Martin's Maximum, Saturated
Ideals, and Non-Regular Ultrafilters:
Part I'', {\it Annals of Mathematics 127},
1988, 1--47.
\bibitem{H4} J.~D.~Hamkins, ``The Lottery Preparation'',
{\it Annals of Pure and Applied Logic 101},
2000, 103--146.
\bibitem{HW} J.~D.~Hamkins, W.~H.~Woodin,
``Small Forcing Creates neither Strong nor
Woodin Cardinals'', {\it Proceedings of the
American Mathematical Society 128},
2000, 3025--3029.
\bibitem{J} T.~Jech, {\it Multiple Forcing},
Cambridge University Press, Cambridge, London,
and New York, 1986.
\bibitem{L} R.~Laver, ``Making the Supercompactness
of $\gk$ Indestructible under $\gk$-Directed
Closed Forcing'', {\it Israel Journal of
Mathematics 29}, 1978, 385--388.
%\bibitem{K} A.~Kanamori, {\it The
%Higher Infinite}, Springer-Verlag,
%Berlin and New York, 1994.
\bibitem{Sh} S.~Shelah, {\it Proper
and Improper Forcing}, Springer-Verlag,
Berlin and New York, 1997.
\end{thebibliography}
\end{document}
The partial ordering $\FP$
will be said to be proper
if for some $\gl$ such that
$\wp(\FP) \in V_\gl$,
there is a club subset
$C \subseteq {[V_\gl]}^\go$ of
countable elementary submodels
$M \prec \la V_\gl, \in, \FP \ra$
with the property that for all
$p \in M$ and every name
$\check \ga \in M$ for an ordinal,
there is some $q \in \FP$ extending
$p$ such that
$q \forces ``$For some $\gb \in M$,
$\check \ga = \gb$''.
%if, for every uncountable set $A$,
%every stationary subset of
%${[A]}^\go$ remains stationary
%after forcing with $\FP$.
$\FP$ will be said to be semiproper
if for some $\gl$ such that
$\wp(\FP) \in V_\gl$,
there is a club subset
$C \subseteq {[V_\gl]}^\go$ of
countable elementary submodels
$M \prec \la V_\gl, \in, \FP \ra$
with the property that for all
$p \in M$ and every name
$\check \ga \in M$ for a countable ordinal,
there is some $q \in \FP$ extending
$p$ such that
$q \forces ``$For some $\gb \in M$,
$\check \ga = \gb$''.
It is shown in both \cite{J} and
\cite{Sh} that in the preceding
definitions, the conditions
``some $\gl$ such that
$\wp(\FP) \in V_\gl$''
and
``all $\gl$ such that
$\wp(\FP) \in V_\gl$''
are equivalent. From this,
it easily follows that
the lottery sum
of a collection of either proper
or semiproper partial orderings
remains either proper or semiproper.