\documentclass[nosectionnum,nogerman,heavyfont,noheader]{tudbeamer}
%german | nogerman, heavyfont, navbar, nodin, noheader,nosectionnum, serifmath, ddc | ddcfooter


\usepackage{amsmath,amssymb,mathabx,fdsymbol}
\usepackage{tikz}


\newcommand{\lcm}{{\operatorname{lcm}}}
\newcommand{\He}{{\operatorname{H}}}
\newcommand{\PP}{{\operatorname{PP}}}
\newcommand{\edges}[1]{\stackrel{#1}{\to}}
\newcommand{\smGraphs}{{\mathfrak S}}

\newcommand{\bA}{\ensuremath{\mathfrak{A}}}
\newcommand{\bB}{\ensuremath{\mathfrak{B}}}
\newcommand{\bC}{\ensuremath{\mathfrak{C}}}
\newcommand{\bD}{\ensuremath{\mathfrak{D}}}

\usetikzlibrary{arrows,arrows.meta}
\usetikzlibrary{bending,calc}


\tikzset{
  every overlay node/.style={
    anchor=north west,
  },
}
% Usage:
% \tikzoverlay at (-1cm,-5cm) {content};
% or
% \tikzoverlay[text width=5cm] at (-1cm,-5cm) {content};
\def\tikzoverlay{%
   \tikz[baseline,overlay]\node[every overlay node]
}%

\newcommand{\cycle}[2]{
\def \n {#1}
\def \radius {{#1*0.035+0.15}}
\def \margin {3.8/(#1*0.035+0.15)} % margin in angles, depends on the radius
\node[scale={0.8+#1/10}] at #2 {#1};

\foreach \s in {1,...,\n}
{
%  \node[draw, circle] at ({360/\n * (\s - 1)}:\radius) {};
  \draw #2+({360/\n * (\s - 1)}:\radius) circle (1pt);
  \draw[>=stealth',arrows=-{>[bend]}] #2+({360/\n * (\s - 1)+\margin}:\radius) 
    arc ({360/\n * (\s - 1)+\margin}:{360/\n * (\s)-\margin}:\radius);
}}

\newcommand{\cyclerot}[4]{
\def \n {#1}
\def \radius {{#1*0.035+0.15}}
\def \margin {3.8/(#1*0.035+0.15)} % margin in angles, depends on the radius
\node[scale={0.8+#1/10}] at #2 {#4};

\foreach \s in {1,...,\n}
{
%  \node[draw, circle] at ({360/\n * (\s - 1)}:\radius) {};
  \draw #2+({360/\n * (\s - 1)+#3}:\radius) circle (1pt);
  \draw[>=stealth',arrows=-{>[bend]}] #2+({360/\n * (\s - 1)+\margin+#3}:\radius) 
    arc ({360/\n * (\s - 1)+\margin+#3}:{360/\n * (\s)-\margin+#3}:\radius);
}}

\newcommand{\cycletext}[4]{
\def \n {#1}
\def \radius {{#1*0.035+0.15}}
\def \margin {3.8/(#1*0.035+0.15)} % margin in angles, depends on the radius
\node[scale={0.8+#4/10}] at #2 {#3};

\foreach \s in {1,...,\n}
{
%  \node[draw, circle] at ({360/\n * (\s - 1)}:\radius) {};
  \draw #2+({360/\n * (\s - 1)}:\radius) circle (1pt);
  \draw[>=stealth',arrows=-{>[bend]}] #2+({360/\n * (\s - 1)+\margin}:\radius) 
    arc ({360/\n * (\s - 1)+\margin}:{360/\n * (\s)-\margin}:\radius);
}}

\newcommand{\cycletextrad}[5]{
\def \n {#1}
\def \radius {{#5*0.035+0.15}}
\def \margin {3.8/(#5*0.035+0.15)} % margin in angles, depends on the radius
\node[scale={0.8+#4/10}] at #2 {#3};

\foreach \s in {1,...,\n}
{
%  \node[draw, circle] at ({360/\n * (\s - 1)}:\radius) {};
  \draw #2+({360/\n * (\s - 1)}:\radius) circle (1pt);
  \draw[>=stealth',arrows=-{>[bend]}] #2+({360/\n * (\s - 1)+\margin}:\radius) 
    arc ({360/\n * (\s - 1)+\margin}:{360/\n * (\s)-\margin}:\radius);
}}

\newcommand{\smallcyclenotext}[3]{
\def \n {#1}
\def \radius {{#2*0.035+0.15}}
\def \margin {3.8/(#1*0.035+0.15)} % margin in angles, depends on the radius

\foreach \s in {1,...,\n}
{
%  \node[draw, circle] at ({360/\n * (\s - 1)}:\radius) {};
  \draw #3+({360/\n * (\s - 1)}:\radius) circle (1pt);
  \draw[>=stealth',arrows=-{>[bend]}] #3+({360/\n * (\s - 1)+\margin}:\radius) 
%\draw[arrows=-{Stealth[bend,scale=0.7]}] #2+({360/\n * (\s - 1)+\margin}:\radius) 
    arc ({360/\n * (\s - 1)+\margin}:{360/\n * (\s)-\margin}:\radius);
}}

\newcommand{\smallcycle}[3]{
\def \n {#1}
\def \radius {{#2*0.035+0.15}}
\def \margin {3.8/(#1*0.035+0.15)} % margin in angles, depends on the radius
\node[scale={0.8+#1/10}] at #3 {#1};

\foreach \s in {1,...,\n}
{
%  \node[draw, circle] at ({360/\n * (\s - 1)}:\radius) {};
  \draw #3+({360/\n * (\s - 1)}:\radius) circle (1pt);
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    arc ({360/\n * (\s - 1)+\margin}:{360/\n * (\s)-\margin}:\radius);
}}


%#2 = #1+(0,-0.05)
\newcommand{\sigg}[2]{
\def \n {3}
\def \radius {{3*0.035+0.15}}
\def \margin {3.8/(3*0.035+0.15)}

\foreach \s in {1,...,\n}
{
%  \node[draw, circle] at ({360/\n * (\s - 1)}:\radius) {};
  \draw #1+({360/\n * (\s - 1)}:\radius) circle (1pt);
  \draw[>=stealth',arrows=-{>[bend]}] #1+({360/\n * (\s - 1)+\margin}:\radius) 
    arc ({360/\n * (\s - 1)+\margin}:{360/\n * (\s)-\margin}:\radius);
}
  \draw[>=stealth',arrows=-{>[bend]}] #2+({360/\n * 2+\margin}:\radius) 
    arc ({360/\n * 1+\margin +180}:{360/\n * (2)+180-\margin}:\radius);

}

\newcommand{\cycletwosteps}[1]{
\def \n {#1}
\def \radius {{2*#1*0.035+0.15}}

\foreach \s in {1,...,\n}
{
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}
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\newcommand{\cyclenoedges}[2]{
\def \n {#1}
\def \radius {{#1*0.035+0.15}}

\foreach \s in {1,...,\n}
{
  \draw #2+({360/\n * (\s - 1)}:\radius) circle (1pt);
}
}

\title{Smooth digraphs modulo pp-constructability}

\author{Florian Starke}
\begin{document}

\datecity{}
\date{}

\maketitle

\section{bla}

\begin{frame}
\frametitle{Definitions}
\framesubtitle{}
Let $\bA$ be a structure.
\vspace{5mm}
\begin{columns}
\begin{column}{0.7\textwidth}<1->
    A structure $\bB$ is in $\He(\bA)$ if there are homomorphisms $f\colon\bA\to\bB$ and $g\colon\bB\to\bA$.
\end{column}
\begin{column}{0.3\textwidth}<2->
    \begin{center}
    $\bA^2, \bA\cupdot\bA\in\He(\bA)$
    \end{center}
\end{column}
\end{columns}
\vspace{5mm}
\begin{columns}
\begin{column}{0.7\textwidth}<3->
A structure $\bB$ is in $\PP(\bA)$ if there is a $d$ such that $B=A^d$ and every $k$-ary relation in $\bB$ is, as a $k\cdot d$-ary relation in $\bA$, pp-definable.
\end{column}
\begin{column}{0.3\textwidth}<4->
\begin{tikzpicture}
\smallcyclenotext{1}{-2}{(0,0)}
\smallcyclenotext{1}{-2}{(-0.3,0.15)}
\smallcyclenotext{1}{-2}{(-0.1,0.3)}
\smallcyclenotext{1}{-2}{(-0.15,-0.2)}
\node at (0.85,0) {$\in\PP(\bA)$};
\uncover<4->{\node at (0.35,-0.5) {$\Phi_E(x,y)=``x=y"$};}
\end{tikzpicture}
\end{column}
\end{columns}
\vspace{5mm}
\begin{columns}
\begin{column}{0.7\textwidth}<5->
A structure $\bB$ is \textbf{pp-constructable} from $\bA$ if $\bB\in\He(\PP(\bA))$. In this case we say $\bB\geq\bA$.
\end{column}
\begin{column}{0.3\textwidth}<6->
\begin{tikzpicture}
\cycle{1}{(0,0)}
\node at (1.05,0) {$\in\He(\PP(\bA))$};
\end{tikzpicture}
\end{column}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Definitions}
\begin{columns}
\begin{column}{0.7\textwidth}
A directed graph $G$ is a \textbf{smooth digraphs} if every vertex has in-degree and out-degree at least 1. 
\vspace{5mm}

\uncover<3->{Let $\smGraphs$ be the poset induced by the quasi order $\geq$ on all finite smooth digraphs.}
\end{column}
\begin{column}{0.3\textwidth}<2->
\begin{tikzpicture}
\cycletext{5}{(0,0)}{}{5}
\cycletext{4}{({1.5-(6*0.035+0.15)-(4*0.035+0.15)},1)}{}{4}
\cycletext{6}{(1.5,1)}{}{6}
\sigg{(-1,1)}{(-1,1.05)}
\end{tikzpicture}
\end{column}
\end{columns}

\end{frame}


%\begin{frame}
%\frametitle{Bartos Hammer}
%\begin{tikzpicture}
%\node (0) at (0,0) {$[G]\in\smGraphs$};

%\node[text width = 3cm] (10) at (4,2) {$G$ has 4-ary Siggers and its core is a disjoint union of cycles};
%\node[text width = 3cm] (11) at (4,-2) {$G$ can pp-construct every finite structure};
%\node at (2.5,0) {\includegraphics[scale=0.13]{barto.png}};
%\node[text width = 3cm] (20) at (8,2) {$[G]$ contains a disjoint union of cycles};
%\node (21) at (8,-2) {$[G]$ contains};
%\sigg{(9.2,-2)}{(9.2,-1.95)}
%\path
%    (0) edge (10)
%    (0) edge (11)
%    ;
%\end{tikzpicture}
%\end{frame}

\begin{frame}
\frametitle{Dividing the graphs}
\framesubtitle{}
\begin{center}
\begin{tikzpicture}
\node (0) at (0,0) {$[\hspace{-1pt}G]\in\smGraphs$};

\uncover<2->{\node[text width = 3cm] (10) at (3,3) {$G$ has 4-ary Siggers and its core is a disjoint union of cycles};
\node[text width = 3cm] (11) at (-3,3) {$G$ can pp-construct every finite structure};
\node at (0,3) {\includegraphics[scale=0.38]{barto3.png}};}
%\node at (2.5,0) {\includegraphics[scale=0.13]{barto2.png}};


\uncover<3->{\node[text width = 3cm] (20) at (3,5) {$[\hspace{-1pt}G]$ contains a disjoint union of cycles};
}
\uncover<4->{
\node (21) at (-3,5) {$[\hspace{-1pt}G]$ contains};
\sigg{(-3+1.2,5)}{(-3+1.2,5.05)}
}

\uncover<2->{\path
    (0) edge (10)
    (0) edge (11)
    ;
}
\uncover<3->{
\node[rotate=90,scale =2] at (3,4) {$\Rightarrow$};
}
\uncover<4->{
\node[rotate=90,scale =2] at (-3,4) {$\Rightarrow$};
}
\end{tikzpicture}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Poset first glance}
\begin{center}
\begin{tikzpicture}
\cycle{1}{(0,0)}

\uncover<2->{
\draw (0,-3) ellipse (4cm and 2.3cm);
\node[text width=2.3cm] at (0,-3) {disjoint unions of cycles};
\cycle{3}{(-2,1-3)}
\cycle{5}{(3,0-3)}
\cycle{4}{(-2.2,-1-3)}
\cycle{9}{(1,1-3)}
\cycle{2}{(2,-1.4-3)}
\cycle{3}{(2.4,-1-3)}
}
\sigg{(0,-6)}{(0,-5.95)}

\end{tikzpicture}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Multiples}
\framesubtitle{}
\begin{center}
\begin{tikzpicture}
\uncover<4->{
\node at (-0.2,0.1) {$\stackrel{?}{\equiv}$};
\cycle{4}{(-0.8,0)}
}
\cycle{2}{(0.3,0)}
\cycle{4}{(1,0)}
\uncover<2->{
\node at (1.6,0) {$\equiv$};
\cycle{2}{(2.1,0)}
}
\uncover<3->{
\node at (2.6,0) {$\equiv$};
\cycle{2}{(3.1,0.3)}
\cycle{4}{(3.3,-0.3)}
\cycle{6}{(3.9,0.25)}
}
\node at (4.5,0) {};
\end{tikzpicture}
\end{center}
\end{frame}



\begin{frame}
\frametitle{Division}
\begin{columns}
\begin{column}{0.3\textwidth}
\uncover<2->{
\begin{align*}
\uncover<3->{\Phi_E(\only<1-5>{x,y}
)
&=x\edges{\only<1-4>{2}\only<5->{k}}y}
\end{align*}
}

\end{column}
\begin{column}{0.3\textwidth}
    \begin{center}
    \begin{tikzpicture}
        \uncover<2>{\cyclenoedges{10}{(0,0)}}
        \uncover<3>{\cycletwosteps{5}}
        \uncover<4>{\cycle{5}{(0,0)}}
        \uncover<5>{\cycletext{5}{(0,0)}{$a\dotdiv k$}{0}}
        
        \uncover<1-4>{\cycle{10}{(0,-2)}}
        \uncover<5->{\cycletext{10}{(0,-2)}{$a$}{15}}

        \uncover<2-4>{\path (0,0) edge[shorten >=6.5mm,shorten <=7.5mm] (0,-2);}
        \uncover<4->{\path (0,0) edge[shorten >=6.5mm,shorten <=5.5mm] (0,-2);}
    \end{tikzpicture}
    \end{center}
\end{column}
\begin{column}{0.25\textwidth}
\only<5>{
    \[a\dotdiv k=\frac{a}{\gcd(a,k)}\]
}
\end{column}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Multiplication}
\framesubtitle{}
\begin{columns}
\begin{column}{0.35\textwidth}
\uncover<2->{
\tikzoverlay[text width=0.3\textwidth] at (0cm,2cm) {
\begin{align*}
\uncover<3->{
    \Phi_E\begin{pmatrix}
    x_1,x_2,x_3,\\y_1,y_2,y_3
    \end{pmatrix}
        &=x_1\to y_3\\
        &\wedge x_2= y_1\\
        &\wedge x_3=y_{2}
}
\end{align*}
};
}

\end{column}
\begin{column}{0.3\textwidth}
    \begin{center}
    \begin{tikzpicture}
        \uncover<2-4>{
            \cyclenoedges{9}{(0,0)}
        }
        \uncover<2-5>{
            \cyclenoedges{9}{(1,1)}
            \cyclenoedges{9}{(-1,1)}
        }
        \uncover<5-6>{
            \cycle{9}{(0,0)}
        }
        \uncover<6>{
            \cycle{9}{(1,1)}
            \cycle{9}{(-1,1)}
            }
        \uncover<3->{
        \node[rotate=40*0,scale=0.7] at ({40 * 0}:0.7) {$000$};
        }
        \uncover<4->{
        \node[rotate=40*1,scale=0.7] at ({40 * 1}:0.7) {$001$};
        \def \radius {{9*0.035+0.15}}
        \def \margin {3.8/(9*0.035+0.15)} 
        \draw[>=stealth',arrows=-{>[bend]}] ({\margin}:\radius) 
            arc (\margin:{40 -\margin}:\radius);
        }
        \uncover<5->{
        \node[rotate=40*2,scale=0.7] at ({40 * 2}:0.7) {$011$};
        \node[rotate=40*3+180,scale=0.7] at ({40 * 3}:0.7) {$111$};
        \node[rotate=40*4+180,scale=0.7] at ({40 * 4}:0.7) {$211$};
        \node[rotate=40*5+180,scale=0.7] at ({40 * 5}:0.7) {$221$};
        \node[rotate=40*6+180,scale=0.7] at ({40 * 6}:0.7) {$222$};
        \node[rotate=40*7,scale=0.7] at ({40 * 7}:0.7) {$220$};
        \node[rotate=40*8,scale=0.7] at ({40 * 8}:0.7) {$200$};
        }
        \uncover<1->{
        \node[rotate=120*0,scale=0.7] at ($(0,-2)+({120 * 0}:0.4)$) {$0$};
        \node[rotate=120*1+180,scale=0.7] at ($(0,-2)+({120 * 1}:0.4)$) {$1$};
        \node[rotate=120*2+180,scale=0.7] at ($(0,-2)+({120 * 2}:0.4)$) {$2$};
        }
        
        \uncover<1-6>{\cycle{3}{(0,-2)}}

        \uncover<2-6>{\path (0,0) edge[shorten >=5mm,shorten <=6mm] (0,-2);}
    \end{tikzpicture}
    \end{center}
\end{column}
\begin{column}{0.25\textwidth}
\end{column}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Multiplication}
\framesubtitle{}
\begin{columns}
\begin{column}{0.35\textwidth}
\tikzoverlay[text width=0.3\textwidth] at (0cm,2cm) {
\begin{align*}
    \Phi_E\begin{pmatrix}
    x_1,x_2,\\y_1,y_2
    \end{pmatrix}
        &=x_1\to y_2\\
        &\wedge x_2= y_1
\end{align*}
};

\end{column}
\begin{column}{0.3\textwidth}
    \begin{center}
    \begin{tikzpicture}
        \uncover<2-3>{
            \cyclenoedges{3}{(0,0)}
        }
        \uncover<2>{
            \cyclenoedges{6}{(0.7,1)}
        }
        \uncover<4->{
            \cycle{3}{(0,0)}
        }
        \uncover<3->{
            \cycle{6}{(0.7,1)}
            }
        \uncover<4->{
        \node[rotate=120*0,scale=0.7] at ({120 * 0}:0.5) {$02$};
        \node[rotate=120*1+180,scale=0.7] at ({120 * 1}:0.5) {$12$};
        \node[rotate=120*2+180,scale=0.7] at ({120 * 2}:0.5) {$01$};
        }
        \uncover<1->{
        \node[rotate=120*0,scale=0.7] at ($(0,-2)+({120 * 0}:0.4)$) {$0$};
        \node[rotate=120*1+180,scale=0.7] at ($(0,-2)+({120 * 1}:0.4)$) {$1$};
        \node[rotate=120*2+180,scale=0.7] at ($(0,-2)+({120 * 2}:0.4)$) {$2$};
        }
        
        \uncover<1->{\cycle{3}{(0,-2)}}

        \uncover<2->{\path (0,0) edge[shorten >=5mm,shorten <=6mm] (0,-2);}
    \end{tikzpicture}
    \end{center}
\end{column}
\begin{column}{0.25\textwidth}
\end{column}
\end{columns}
\end{frame}


\begin{frame}
\frametitle{Multiplication}
\framesubtitle{}
\begin{columns}
\begin{column}{0.3\textwidth}
\tikzoverlay[text width=0.3\textwidth] at (0cm,2cm) {
\begin{align*}
\Phi_E\begin{pmatrix}
    x_1,\dots,x_k,\\y_1,\dots,y_k
    \end{pmatrix}
        &=x_1\to y_k\\
        &\wedge x_2= y_1\\
        &\hspace{8mm}\vdots\\
        &\wedge x_k=y_{k-1}
\end{align*}
};
\end{column}%\vrule
\begin{column}{0.3\textwidth}
    \begin{center}
    \begin{tikzpicture}
        \uncover<2-4>{
            \cycletext{9}{(0,0)}{$a\ltimes k$}{2}
        }
        
        \cycletext{3}{(0,-2)}{$a$}{3}
        \uncover<2->{\path (0,0) edge[shorten >=5mm,shorten <=6mm] (0,-2);}
    \end{tikzpicture}
    \end{center}
\end{column}
\begin{column}{0.25\textwidth}
\begin{align*}
\uncover<3->{
a\ltimes k&=\prod_{\alpha_i\neq 0} p_i^{\alpha_i+\kappa_i}\\
}
\uncover<4->{
3\ltimes 3&=9\\
3\ltimes 2&=3\\
3\ltimes 6&=9\\
2\ltimes 12&=8\\
}
\end{align*}
\end{column}
\end{columns}
\end{frame}


\begin{frame}
\frametitle{Normal form}
\framesubtitle{}
\begin{center}
\begin{tikzpicture}
\cycletext{12}{(0,-0.5)}{$2^2\cdot3$}{2}
\cycletext{16}{(1,0.9)}{$2^3\cdot5$}{2}
\node at (2.5,0) {$\equiv$};
\cycletext{6}{(4,-0.5)}{$2\cdot3$}{2}
\cycletext{8}{(4.5,0.5)}{$2^2\cdot5$}{2}
\end{tikzpicture}
\end{center}
\uncover<2->{
$G$ is in \textbf{normal form} if 
\begin{itemize}
\item<3-> for all $a,a'\in G$ we have $a\mid a'$ implies $a=a'$ and
\item<4-> if for an $a\in G$ we have $p\mid a$, then there is an $a'\in G$ with $p\mid a'$ but $p^2\nmid a'$.
\end{itemize} 
}
\end{frame}



\begin{frame}
\frametitle{Poset second glance}
\begin{center}
\begin{tikzpicture}
\cycle{1}{(0,0)}

\def\xs{-1}
\cycle{2}{(-0.9+\xs,-1)}
\cycle{3}{(0+\xs,-1)}
\cycle{5}{(1+\xs,-1)}
\cycle{7}{(2.1+\xs,-1)}
\node at (3.1+\xs,-1) {$\dots$};

\def\xs{-1}
\cycle{6}{(-0.6+\xs,-2.25)}
\smallcycle{15}{9}{(0.8+\xs,-2.3)}

\uncover<2->{
\def\xs{-1}
\cycle{2}{(-2-1.85+\xs,-3.45)}
\cycle{3}{(-2-1.3+\xs,-3.6)}
}
\smallcycle{30}{15}{(0+\xs,-3.7)}


\sigg{(0,-6)}{(0,-5.95)}

\path
    (0,0) edge[shorten >=4mm,shorten <=3mm] (-0.9+\xs,-1)
    (0,0) edge[shorten >=4mm,shorten <=3mm] (0+\xs,-1)
    (0,0) edge[shorten >=4mm,shorten <=3mm] (1+\xs,-1)
    (0,0) edge[shorten >=5mm,shorten <=3mm] (2.1+\xs,-1)
    (0,0) edge[shorten >=4mm,shorten <=3mm] (3.4+\xs,-1)
    
    (-0.9+\xs,-1) edge[shorten >=5mm,shorten <=4mm] (-0.6+\xs,-2.25)
    (0+\xs,-1) edge[shorten >=5mm,shorten <=4mm] (-0.6+\xs,-2.25)
    (0+\xs,-1) edge[shorten >=6mm,shorten <=4mm] (0.8+\xs,-2.3)
    (1+\xs,-1) edge[shorten >=6mm,shorten <=5mm] (0.8+\xs,-2.3)
    
    %(-0.6+\xs,-2.25) edge[shorten >=3mm,shorten <=5mm] (-1.75+\xs,-3.45)
    (-0.6+\xs,-2.25) edge[shorten >=8mm,shorten <=5mm] (0+\xs,-3.7)
    (0.8+\xs,-2.3) edge[shorten >=8mm,shorten <=6mm] (0+\xs,-3.7)
    ;
\node at (4.5,0) {};
\end{tikzpicture}
\end{center}
\end{frame}

\begin{frame}
\frametitle{Combination}
\begin{columns}
\begin{column}{0.35\textwidth}
\uncover<2->{
\tikzoverlay[text width=0.3\textwidth] at (0cm,2cm) {
\begin{align*}
\uncover<2->{
    \Phi_E\begin{pmatrix}
    x_1,x_2,\\y_1,y_2
    \end{pmatrix}
        &=x_1\to y_1\\
        &\wedge x_2\to y_2\\
        \uncover<3->{
        &\wedge x_1\only<2-4>{\edges{2}}\only<5->{\edges{a}}x_1\phantom{\edges{2}}\\
        &\wedge x_2\only<2-4>{\edges{3}}\only<5->{\edges{b}}x_2\\}
}
\end{align*}
};
}

\end{column}
\begin{column}{0.3\textwidth}
    \begin{center}
    \begin{tikzpicture}
        \uncover<2-4>{        
            \cycle{6}{(0,0)}
        }
        \uncover<2>{
            \cycle{6}{(0,1)}
            
            \cycle{2}{(-0.8,0.8)}
            \cycle{2}{(-1.2,1.2)}
            
            \cycle{3}{(1.3-0.25,1-0.15)}
            \cycle{3}{(1.9-0.2,0.7-0.2)}
            \cycle{3}{(1.9-0.2,1.4-0.2)}
        }
        \uncover<3>{
            \cyclenoedges{6}{(0,1)}
            
            \cyclenoedges{2}{(-0.8,0.8)}
            \cyclenoedges{2}{(-1.2,1.2)}
            
            \cyclenoedges{3}{(1.3-0.25,1-0.15)}
            \cyclenoedges{3}{(1.9-0.2,0.7-0.2)}
            \cyclenoedges{3}{(1.9-0.2,1.4-0.2)}
        }
        
        \uncover<5->{
            \cycletext{6}{(0,0)}{$a\vee b$}{1}
        }
        
        \uncover<2-3>{
        \node[rotate=60*0,scale=0.7] at ({60 * 0}:0.55) {$a0$};
        \node[rotate=60*1,scale=0.7] at ({60 * 1}:0.55) {$b1$};
        \node[rotate=60*2+180,scale=0.7] at ({60 * 2}:0.55) {$2a$};
        \node[rotate=60*3+180,scale=0.7] at ({60 * 3}:0.55) {$0b$};
        \node[rotate=60*4+180,scale=0.7] at ({60 * 4}:0.55) {$1a$};
        \node[rotate=60*5,scale=0.7] at ({60 * 5}:0.55) {$b2$};

        \node[rotate=0,scale=0.7] at ($(-0.8,0.8)+({180}:0.4)$) {$aa$};
        \node[rotate=0,scale=0.7] at ($(-1.2,1.2)+({180}:0.4)$) {$ba$};
        
        \node[rotate=0,scale=0.7] at ($(0,1)+({0}:0.55)$) {$0a$};
        
        \node[rotate=120,scale=0.7] at ($(1.3-0.25,1-0.15)+({120}:0.45)$) {$00$};
        \node[rotate=0,scale=0.7] at ($(1.9-0.2,0.7-0.2)+({0}:0.45)$) {$01$};
        \node[rotate=0,scale=0.7] at ($(1.9-0.2,1.4-0.2)+({0}:0.45)$) {$02$};
 
 
        }
        \uncover<1-3>{
        \node[rotate=120*0+40,scale=0.7] at ($(0.4,-2)+({120 * 0+40}:0.4)$) {$0$};
        \node[rotate=120*1+180+40,scale=0.7] at ($(0.4,-2)+({120 * 1+40}:0.4)$) {$1$};
        \node[rotate=120*2+40,scale=0.7] at ($(0.4,-2)+({120 * 2+40}:0.4)$) {$2$};
        
        \node[rotate=180*0-35,scale=0.6] at ($(0,-2.4)+({180 * 0-35}:0.35)$) {$a$};
        \node[rotate=180*1+180-35,scale=0.6] at ($(0,-2.4)+({180 * 1-35}:0.35)$) {$b$};

        }
        
        \uncover<1-4>{
            \cyclerot{2}{(0+0,-2+-0.4)}{-35}{2}
            \cyclerot{3}{(0+0.4,-2+0)}{40}{3}
        }
        \uncover<5->{
            \cyclerot{2}{(0+0,-2+-0.4)}{-35}{$a$}
            \cyclerot{3}{(0+0.4,-2+0)}{40}{$b$}
        }

        \uncover<2->{\path (0,0) edge[shorten >=3mm,shorten <=5mm] (0,-2);}
    \end{tikzpicture}
    \end{center}
\end{column}
\begin{column}{0.25\textwidth}
\uncover<5->{
    $a\nmid b, b\nmid a$
    
    $a\vee b=\lcm(a,b)$
}
\end{column}
\end{columns}
\end{frame}

\begin{frame}
\frametitle{Poset final glance}
\uncover<3->{
\begin{center}
\begin{tikzpicture}[opacity = 0.3]
    \cycletextrad{50}{(0,0)}{Thank You}{30}{100}
%    \clip (0,0) rectangle (0.1,0.1);
\end{tikzpicture}
\end{center}
}
\vspace{-7.5cm}

\begin{center}
\begin{tikzpicture}
\cycle{1}{(0,0)}

\def\xs{-1}
\cycle{2}{(-0.9+\xs,-1)}
\cycle{3}{(0+\xs,-1)}
\cycle{5}{(1+\xs,-1)}
\cycle{7}{(2.1+\xs,-1)}
\node at (3.1+\xs,-1) {$\dots$};

\def\xs{-1}
\cycle{6}{(-0.6+\xs,-2.25)}
\smallcycle{15}{9}{(0.8+\xs,-2.3)}


\def\xs{-1}
\cycle{2}{(-1.85+\xs,-3.45-0.9)}
\cycle{3}{(-1.3+\xs,-3.6-0.9)}
\smallcycle{30}{15}{(0+\xs,-3.7)}

\uncover<2->{
\cycletext{8}{(-1.7+\xs,-3.5)}{$2^3$}{2}
\cycletext{9}{(-2.6+\xs,-3.2)}{$3^3$}{3}
\cycle{6}{(-1.9+\xs,-2.6)}
\path
    (-0.6+\xs,-2.25) edge[shorten >=3mm,shorten <=5mm] (-1.75+\xs,-2.6)
    (-1.7+\xs,-3.5) edge[shorten >=3mm,shorten <=5mm] (-1.75+\xs,-3.45-0.9)
    ;

}
\node at (4.1+\xs,0) {};

\sigg{(0,-6)}{(0,-5.95)}

\path
    (0,0) edge[shorten >=4mm,shorten <=3mm] (-0.9+\xs,-1)
    (0,0) edge[shorten >=4mm,shorten <=3mm] (0+\xs,-1)
    (0,0) edge[shorten >=4mm,shorten <=3mm] (1+\xs,-1)
    (0,0) edge[shorten >=5mm,shorten <=3mm] (2.1+\xs,-1)
    (0,0) edge[shorten >=4mm,shorten <=3mm] (3.4+\xs,-1)
    
    (-0.9+\xs,-1) edge[shorten >=5mm,shorten <=4mm] (-0.6+\xs,-2.25)
    (0+\xs,-1) edge[shorten >=5mm,shorten <=4mm] (-0.6+\xs,-2.25)
    (0+\xs,-1) edge[shorten >=6mm,shorten <=4mm] (0.8+\xs,-2.3)
    (1+\xs,-1) edge[shorten >=6mm,shorten <=5mm] (0.8+\xs,-2.3)
    
    (-0.6+\xs,-2.25) edge[shorten >=8mm,shorten <=5mm] (0+\xs,-3.7)
    (0.8+\xs,-2.3) edge[shorten >=8mm,shorten <=6mm] (0+\xs,-3.7)
    ;
\uncover<1>{
\path
    (-0.6+\xs,-2.25) edge[shorten >=3mm,shorten <=5mm] (-1.75+\xs,-3.45-0.9);
}


\end{tikzpicture}
\end{center}
\end{frame}


\begin{frame}
\frametitle{Summary}
\framesubtitle{}
\uncover<9->{
\begin{center}
\begin{tikzpicture}[opacity = 0.3]
    \cycletextrad{50}{(0,0)}{Thank You}{30}{100}
%    \clip (0,0) rectangle (0.1,0.1);
\end{tikzpicture}
\end{center}
}
\vspace{-7.5cm}

\begin{align*}
a\vee b=\lcm(a,b)&&
a\dotdiv k=\frac{a}{\gcd(a,k)}&&
a\ltimes k=\prod_{\alpha_i\neq 0} p_i^{\alpha_i+\kappa_i}
\end{align*}
\uncover<2->{
%\[A\dotdiv\ltimes((k_1,\ell_1),\dots,(k_d,\ell_d))=\{(a_1\dotdiv k_1)\ltimes \ell_1\vee\dots\vee (a_d\dotdiv k_d)\ltimes \ell_d\mid a_1,\dots,a_d\in A\}\]
\begin{align*}
G\dotdiv (k_1,\dots,k_d)&=\{(a_1\dotdiv k1)\vee\dots\vee (a_d\dotdiv k_d)\mid a_1,\dots,a_d\in G\}\\
G\ltimes (k_1,\dots,k_d)&=\{(a_1\ltimes k1)\vee\dots\vee (a_d\ltimes k_d)\mid a_1,\dots,a_d\in G\}
\end{align*}
}
\uncover<7->{
\[G\, f\, (t_1,\dots,t_d)\text{, where $f\in\{\dotdiv,\ltimes\}^n$ and $t_i\in\{1,2,\dots\}^{n}$}\]}

%If $A$ is in normal form, then all singletons are definable.
\begin{tikzpicture}
%\node at (-2,2) {$A\dotdiv\ltimes((k_1,k'_1),\dots,(k_d,k'_d))$};
\uncover<8->{
\node (a) at (0,1.5+0.4) {$\left(G_1\,f_1\,T_1\right)\vee\dots\vee \left(G_n\,f_n\,T_n\right)$};
\node (b) at (0,0+0.4) {$G$};
\node at (0,-0.5+0.4) {For all $a\in G\setminus G_i$ we have $a\nmid\lcm(G_i)$.};
\draw (a) edge (b);
}

\uncover<3->{
\node (c) at (4,0) {$2\cdot 3,5\cdot 7$};
\node (d) at (4,1.5) {$2\cdot 3,5\cdot 7\dotdiv(2\cdot5,3\cdot7)$};
\draw (c) edge (d);
}
\uncover<4->{
\node at (4,2) {$2\cdot 3,5\cdot 7,2\cdot 7,3\cdot5$};
\node[rotate=90] at (4,1.75) {$=$};
}

\uncover<5->{
\node (e) at (7,0) {$2,3$};
\node (f) at (7,1.5) {$2, 3\ltimes(4,9)$};
\draw (e) edge (f);
}
\uncover<6->{
\node at (7,2) {$2^3, 2\cdot 3,3^3$};
\node[rotate=90] at (7,1.75) {$=$};
}

\end{tikzpicture}
\end{frame}

\end{document}