m

invariantL.tex

@@ -151,15 +151,34 @@ \section{Lascar strong types}
 
 
 \begin{proposition}
-  The relation $\equivL_A$ is the finest equivalence relation with $<\kappa$ classes that is invariant over $A$.\QED
+  The relation $\equivL_A$ is the finest equivalence relation 
+  with $<\kappa$ classes that is invariant over $A$.\QED
 \end{proposition}
 
 \begin{proof}
-  Clearly $\equivL_A$ is an equivalence relation invariant over $A$. Each equivalence class is Lascar invariant over $A$, hence the number of equivalence classes is bounded by the number of Lascar invariant sets over $A$. To see that $\equivL_A$ is the finest of such equivalences. Suppose $\grD$ is an equivalence class of an $A$-invariant equivalence relation with $<\kappa$ classes. Then $\oorbit(\grD/A)$ has also cardinality $<\kappa$. Then $\grD$ is Lascar invariant and as such it is union of classes of the relation $\equivL_A$. 
+  Clearly $\equivL_A$ is an equivalence relation invariant over $A$.
+  %
+  Each equivalence class is Lascar invariant over $A$, 
+  hence the number of equivalence classes is bounded by 
+  the number of Lascar invariant sets over $A$.
+  %
+  To see that $\equivL_A$ is the finest of such equivalences.
+  %
+  Suppose $\grD$ is an equivalence class of an $A$-invariant 
+  equivalence relation with $<\kappa$ classes.
+  %
+  Then $\oorbit(\grD/A)$ has also cardinality $<\kappa$.
+  Then $\grD$ is Lascar invariant and as such it is union of 
+  classes of the relation $\equivL_A$. 
 \end{proof}
 
-
-Let $p({\mr x})\in S(\U)$ be global type; we say that $p$ is \emph{Lascar invariant\/} over $A$ if the sets ${\gr\D_{p,\phi}}$ for $\phi({\mr x}\,;{\gr z})\in L$ all Lascar invariant over $A$. (The sets ${\gr\D_{p,\phi}}$ are defined in Section~\hyperref[invariant_sets]{~\ref*{invariant}.\ref*{invariant_sets}}.)
+Let $p({\mr x})\in S(\U)$ be global type; 
+we say that $p$ is \emph{Lascar invariant\/} over $A$ if 
+the sets ${\gr\D_{p,\phi}}$ for $\phi({\mr x}\,;{\gr z})\in L$ are all 
+Lascar invariant over $A$. 
+%
+(The sets ${\gr\D_{p,\phi}}$ are defined in 
+Section~\hyperref[invariant_sets]{~\ref*{invariant}.\ref*{invariant_sets}}.)
 
 \begin{proposition}\label{prop_Lascar_indiscernibles}
   Let $p({\mr x})\in S(\U)$ be a global type. Then the following are equivalent
@@ -170,9 +189,15 @@ \section{Lascar strong types}
   \end{itemize}
   For convenience all tuples ${\gr c_i}$ have length $|{\gr z}|=\omega$.
 \end{proposition}
-\begin{proof} We prove \ssf{1}$\IFF$\ssf{3}. Equivalence  \ssf{1}$\IFF$\ssf{2} can be proved similarly because the Lascar invariance of $p({\mr x})$ easily implies the Lascar invariance of the sets ${\gr\D_{p,\phi}}$ for all $\phi({\mr x}\,;{\gr z})\in L(A)$.
 
-  \def\ceq#1#2#3{\parbox[t]{30ex}{$\displaystyle #1$}\parbox[t]{5ex}{$\displaystyle\hfil #2$}{$\displaystyle #3$}}
+\def\ceq#1#2#3{\parbox[t]{30ex}{$\displaystyle #1$}\parbox[t]{5ex}{$\displaystyle\hfil #2$}{$\displaystyle #3$}}
+\begin{proof} 
+  We prove \ssf{1}$\IFF$\ssf{3}.
+  %
+  Equivalence  \ssf{1}$\IFF$\ssf{2} can be proved similarly because 
+  the Lascar invariance of $p({\mr x})$ easily implies the Lascar invariance of 
+  the sets ${\gr\D_{p,\phi}}$ for all $\phi({\mr x}\,;{\gr z})\in L(A)$.
+
   \ssf{3}$\IMP$\ssf{1}\quad If $p({\mr x})$ is not Lascar invariant over $A$ then ${\gr c_0}\in{\gr\D_{p,\phi}}\niff {\gr c_1}\in{\gr\D_{p,\phi}}$ for some $A$-indiscernible sequence ${\gr\bar c}=\<{\gr c_i}:i<\omega\>$ and some $\phi({\mr x}\,;{\gr z})\in L$. Then $p({\mr x})$ contains the formula $\phi({\mr x}\,;{\gr c_0})\niff \phi({\mr x}\,;{\gr c_1})$. Hence, ${\gr\bar c}$ is not indiscernible over any realization of $p({\mr x})_{\restriction{\gr c_0},{\gr c_1}}$ 
 
   \ssf{1}$\IMP$\ssf{3}\quad Assume \ssf{1} and fix an $A$-indiscernible sequence ${\gr\bar c}=\<{\gr c_i}:i<\omega\>$ and some ${\mr a}\models p_{\restriction{\gr\bar c}}$. We need to prove that for every formula $\phi({\mr x}\,;{\gr z'})\in L$, where ${\gr z'}={\gr z_1},\dots,{\gr z_n}$,

ramsey.tex

@@ -755,7 +755,6 @@ \section{The Hales-Jewett Theorem}
 Let $\mrG$ be a semigroup.
 A \emph{nice subsemigroup\/} of $\mrG$ is a subsemigroup $\mrC$ with the property that if ${\mr a}{\cdot}{\mr b}\in\mrC$ then both ${\mr a},{\mr b}\in \mrC$.
 
-
 \theoremstyle{mio}
 \newtheorem{HalesJewett}[thm]{Hales-Jewett Theorem}
 \begin{HalesJewett}[(Koppelberg's version)]\label{thm_abstract_HJ}
@@ -779,19 +778,28 @@ \section{The Hales-Jewett Theorem}
 By Proposition~\ref{prop_minimal_existence1}, there is a left-minimal ${\mr c}\in\mrC$.
 As $\mrC\cdot_M{\mr c}$ is clearly idempotent, we can further require that ${\mr c}$ has idempotent orbit.
 
-By nicety, $\mrA=\mrG\sm\mrC$ satisfy the assumptions of Proposition~\ref{prop_minimal_existence2}.
-Hence there is some idempotent ${\mr g}\in{\mr c}\cdot_G(\mrG\sm\mrC)\cdot_G{\mr c}$.
+By nicety, $\mrG\sm\mrC$ satisfy the assumptions of 
+Proposition~\ref{prop_minimal_existence2}.
+%
+Hence, by the first claim of that proposition, there is an idempotent
+${\mr g}\in{\mr c}\cdot_G(\mrG\sm\mrC)\cdot_G{\mr c}$.
+%
 In particular, ${\mr g}\in\mrG\sm\mrC$.
-From Proposition~\ref{prop_HJ_tecnical} we obtain
-$\sigma\,{\mr g}\in{\mr c}\cdot_G\mrC\cdot_G{\mr c}$
+%
+Now apply the second claim of Proposition~\ref{prop_HJ_tecnical}, 
+with $\mrC$ for $\mrA$ to  obtain 
+$\sigma\,{\mr g}\in{\mr c}\cdot_G\mrC\cdot_G{\mr c}$ 
 for all $\sigma\in\Sigma$.
-As $\sigma\,{\mr g}$ is also idempotent, 
-we apply Proposition~\ref{prop_minimal_existence2} to conclude that 
+%
+As $\sigma\,{\mr g}$ is also idempotent, we apply 
+Proposition~\ref{prop_minimal_existence2} to conclude that 
 $\sigma\,{\mr g}\equiv_G{\mr c}$.
+%
 In particular the set $\{\sigma\,{\mr g}:\sigma\in\Sigma\}$ is monochromatic.
 
 Though the element ${\mr g}$ above need not belong to $G\sm C$,
-by elementarity $G\sm C$ contains some $a$ with the same property and this proves the theorem.
+by elementarity $G\sm C$ contains some $a$ with the same property and 
+this proves the theorem.
 \end{proof}
 
 Finally we show how the classical Hales-Jewett theorem follows from its abstract version.
@@ -837,15 +845,16 @@ \section{The Hales-Jewett Theorem}
 In fact, this extension is unique: the elements of $G$ that occur in a word are replaced by their image under $\sigma$, finally the elements in the resulting sequence are multiplied.
 This extension is denoted by the same symbol $\sigma$.
 
-Apply Theorem~\ref{thm_abstract_HJ} to obtain some $w\in G*C$ such that $\{\sigma \,w:\sigma\in\Sigma\}$ is monochromatic. Suppose $w=c_0\cdot g_0\cdots\cdots c_n\cdot g_n$ for some $g_i\in G$ and $c_i\in C$, where one or both of $c_0$ or $g_n$ could be absent. Pick some $h_i\in\Sigma^{-1}[c_i]$ and let $g=h_0\cdot g_0\cdots\cdots h_n\cdot g_n$. Then $\{\sigma\,g:\sigma\in\Sigma\}$ is monochromatic as required to complete the proof.
+Apply Theorem~\ref{thm_abstract_HJ} to obtain some $w\in G*C$ such that 
+$\{\sigma \,w:\sigma\in\Sigma\}$ is monochromatic.
+%
+Suppose $w=c_0\cdot g_0\cdots\cdots c_n\cdot g_n$ for some $g_i\in G$ and $c_i\in C$, 
+where one or both of $c_0$ or $g_n$ could be absent. Pick some $h_i\in\Sigma^{-1}[c_i]$ and 
+let $g=h_0\cdot g_0\cdots\cdots h_n\cdot g_n$.
+%
+Then $\{\sigma\,g:\sigma\in\Sigma\}$ is monochromatic as required to complete the proof.H
 \end{proof}
 
-
-
-
-
-
-
 \section{Notes and references}
 
 The first application of the algebraic structure of $\beta G$