\documentclass[nosectionnum,nogerman,heavyfont,noheader,14pt]{tudbeamer}
%german | nogerman, heavyfont, navbar, nodin, noheader,nosectionnum, serifmath, ddc | ddcfooter

\usepackage{hyperref}
\usepackage{amsmath,amssymb,mathabx,fdsymbol}
\usepackage{tikz,array}
\newcolumntype{C}[1]{>{\centering\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}

\newcommand{\PL}{\ensuremath{\operatorname{PL}}}
\newcommand{\SDPoset}{{\mathfrak P}_{\operatorname{{SD}}}}
\newcommand{\FP}{\ensuremath{\operatorname{FP}}}
\newcommand{\UP}{\ensuremath{\big\downarrow}}

\newcommand{\myBox}[1]{\vbox to 3mm{\vfill\hbox to 3mm{\hfill$#1$\hfill}}}
\newcommand{\Cyc}[1]{\mathfrak{C}_{#1}}
\newcommand{\lcm}{{\operatorname{lcm}}}
\newcommand{\Pol}{{\operatorname{Pol}}}
\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,
  },
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% 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};

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{
%  \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};

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{
%  \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

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%  \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);
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\def \n {#1}
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\node[scale={0.8+#1/10}] at #3 {#1};

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    arc ({360/\n * (\s - 1)+\margin}:{360/\n * (\s)-\margin}:\radius);
}}


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\def \n {3}
\def \radius {{3*0.035+0.15}}
\def \margin {3.8/(3*0.035+0.15)}

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{
%  \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);

}

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\def \n {#1}
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{
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  \draw ({360/\n * (\s - 1)+180/\n}:\radius) edge[cyan,>=stealth',arrows=-{>[bend]},bend right = 80,looseness=1.7,shorten >=1mm,shorten <=1mm] ({360/\n * (\s)+180/\n}:\radius);
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{
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}
}

\title{The return of the smooth digraphs {\small (modulo pp-constructability)}}

\author{Florian Starke}
\begin{document}

\datecity{}
\date{}

\maketitle

\section{bla}
\title{The return of the smooth digraphs modulo pp-constructability}

\setbeamercolor{background canvas}{bg=black}
\setbeamercolor{normal text}{fg=white}\usebeamercolor*{normal text}

\begin{frame}
\frametitle{Definitions}
%h1 conditions
\begin{columns}
\begin{column}{0.5\textwidth}<1->
\end{column}
\begin{column}{0.55\textwidth}<2->
    \begin{center}
    \textbf{Examples}
    \end{center}
\end{column}
\end{columns}
\vspace{5mm}
\begin{columns}
\begin{column}{0.5\textwidth}<1->
    \textbf{h1-identity}: $f(x_1,\dots,x_n)\approx g(y_1,\dots,y_m)$ ($x_i,y_j$ not necessarily distinct)
\end{column}
\begin{column}{0.55\textwidth}<2->
    h1: $f(x)\approx f(y), f(x,y,z)\approx g(x)$\\%f(x_0,x_1,x_2)\approx f(x_1,x_2,x_0)
    \textbf{not} h1: $f(x,x)\approx x, f(f(x,y),z)\approx f(x,f(y,z))$
\end{column}
\end{columns}
\vspace{5mm}
\begin{columns}
\begin{column}{0.5\textwidth}<3->
    \textbf{h1-condition}: finite set of h1-identities 
\end{column}
\begin{column}{0.55\textwidth}<4->
    $f(x,x,y)\approx f(y,x,x)\approx f(y,y,y)$
\end{column}
\end{columns}
\vspace{5mm}
\begin{columns}
\begin{column}{1.05\textwidth}<5->
$\tilde f\colon A\to A$ \textbf{satisfies} $f(x)\approx f(y)$ if $\tilde f(a)= \tilde f(b)$ for all $a,b\in A$.
\end{column}
\end{columns}
\vspace{5mm}
\begin{columns}
\begin{column}{1.05\textwidth}<6->
A set of functions \textbf{satisfies} a h1-condition $\Sigma$ if it contains functions that satisfy $\Sigma$.
\end{column}
\end{columns}

\end{frame}


\begin{frame}
\frametitle{Define the Order}
\framesubtitle{}
Let $\bA$, $\bB$ be finite relational structures.

%$\lambda\colon \Pol(\bB)\to\Pol(\bA)$ is \textbf{minor-preserving} iff for all h1-conditions $\Sigma$ we have that $f_1,\dots,f_n\in\Pol(\bB)$ satisfy $\Sigma$ implies $\lambda(f_1),\dots,\lambda(f_n)$ satisfy $\Sigma$.
    
\vspace{7mm}
\centering
\begin{tikzpicture}
\node (A) at (0,3) {$\bA$};
\node (B) at (0,2) {$\bB$};
\path (B) edge (A);

%\node at (1.7,0.5) {iff};\node at (1.7,0.8) {\tiny definition};
\node at (1.7,2.5) {:iff};
\node (A) at (3.4,3) {$\Pol(\bA)\models\Sigma$};
\node (B) at (3.4,2) {$\Pol(\bB)\models\Sigma$};
\node[rotate = 90] at (3.4,2.5) {$\Rightarrow$};



\uncover<2->{
\node at (-1.7,0.5) {iff};
\node (A) at (0,1) {$\Pol(\bA)$};
\node (B) at (0,0) {$\Pol(\bB)$};
\path[->,>=latex'] (B) edge node[above,rotate=90] {\tiny minor} (A);
}
\uncover<3->{
\node at (1.7,0.5) {iff};%TODO \node at (5.2,0.8) {\tiny for finite structures};
\node at (3.5,0.5) {$\bB$ pp-constructs $\bA$};
}
\uncover<4->{
\node[orange] at (2.0,3.6) {\textbf{Central definition of this Talk}};
\draw[very thick,orange] (-0.3,3.3) rectangle (4.3,1.7);
}
\node at (-1.7+4*1.7+0.5,0.5) {};

\end{tikzpicture}

\end{frame}




% ------------------ introduce Pfin and PSD and Barto -----------------------------
\begin{frame}
\frametitle{Poset-Introduction}
\alt<1-4>{Goal: understand finite structures ordered by pp-constructability}
{goal: understand finite smooth digraphs ordered by pp-constructability}
\vspace{-22mm}
\begin{center}
\begin{tikzpicture}
\draw (0,-3) ellipse (4cm and 2.3cm);
\uncover<1-4>{\node at (0,-3) {finite relational structures};}
\uncover<5-5>{\node at (0,-3) {finite smooth digraphs};}
\uncover<6->{
%\node at (-4.35,0) {\includegraphics[scale=0.18]{boda3.png}};
\node at (-3.65,0.5) {\includegraphics[scale=0.18]{boda4.png}};
}


%smooth digraphs
\only<6>{\tikzset{every path/.style={opacity =0.2}}}
\cycletext{4}{({-0.5-(6*0.035+0.15)-(4*0.035+0.15)},-4.5)}{}{4}
\cycletext{6}{(-0.5,-4.5)}{}{6}

\cycletext{2}{({-0.5-(2*0.035+0.15)-(2*0.035+0.15)},-2.3)}{}{2}
\cycletext{2}{(-0.5,-2.3)}{}{2}
\only<6>{\tikzset{every path/.style={}}}


% finite structures
\only<5->{\tikzset{every path/.style={opacity =0.2}}}
%tree
\node[circle, draw, scale=0.3] at (1,-3.5) (t0) {};
\node[circle, draw, scale=0.3] at (1,-3.85) (t1) {};
\node[circle, draw, scale=0.3] at (0.75,-4.1) (t2) {};
\node[circle, draw, scale=0.3] at (1.25,-4.1) (t3) {};
\path[->,>=latex'] 
    (t0) edge (t1) 
    (t1) edge (t2) 
    (t1) edge (t3)
    ;

%other thing
\node[circle, draw, scale=0.3] at ($(2.5,-2)+(0:0.2)$) {};
\node[circle, draw, scale=0.3] at ($(2.5,-2)+(120:0.2)$) {};
\node[circle, draw, scale=0.3] at ($(2.5,-2)+(240:0.2)$) {};
\draw[rotate around ={30:(2.5,-2)}] (2.5,-2.1) ellipse (0.3cm and 0.081cm);
\draw[rotate around ={150:(2.5,-2)}] (2.5,-2.1) ellipse (0.3cm and 0.081cm);
\draw[rotate around ={270:(2.5,-2)}] (2.5,-2.1) ellipse (0.3cm and 0.081cm);
\only<5-6>{\tikzset{every path/.style={}}}



\only<1>{
\sigg{(-1,-1.5)}{(-1,-1.45)}
}
\uncover<2->{\sigg{(0,-6)}{(0,-5.95)}}
\only<1-2>{//loop inside
\node[circle, draw, scale=0.3] at (0,-4) (0) {};
\path (0) edge[out=120, in=60, looseness=20,->,>=latex'] (0);
}
\only<1-3>{%other trivial ones inside
\node at ($(-30:0.2)+(-3,-3)$) [circle, draw, scale=0.3] (0) {};
\node at ($(90:0.2)+(-3,-3)$) [circle, draw, scale=0.3] (1) {};
\node at ($(210:0.2)+(-3,-3)$) [circle, draw, scale=0.3] (2) {};
\path 
    (1) edge[out=120, in=60, looseness=20,->,>=latex'] (1)
    (0) edge (1)
    (1) edge (2)
    (2) edge (0);

\node[circle, draw, scale=0.3] at (1,-4.5) (0) {};

}

\node at (0,0) {$
\uncover<3->{\tikz{
\node[circle, draw, scale=0.3] at (0,0) (0) {};
\path (0) edge[out=120, in=60, looseness=20,->,>=latex'] (0);
}}
\uncover<4->{=\tikz{
\only<6->{\tikzset{every path/.style={opacity =0.2}}}

\node at (-30:0.2) [circle, draw, scale=0.3] (0) {};
\node at (90:0.2) [circle, draw, scale=0.3] (1) {};
\node at (210:0.2) [circle, draw, scale=0.3] (2) {};
\path 
    (1) edge[out=120, in=60, looseness=20,->,>=latex'] (1)
    (0) edge (1)
    (1) edge (2)
    (2) edge (0);
\only<6->{\tikzset{every path/.style={}}}
}}
\uncover<4->{=
\tikz{
\only<5->{\tikzset{every path/.style={opacity =0.2}}}
\node[circle, draw, scale=0.3] at (0,0) (0) {};
\only<5->{\tikzset{every path/.style={}}}
}=}$};
\only<5->{\tikzset{every path/.style={opacity =0.2}}}
\uncover<4->{\node[text width = 3.5cm] at (2.7,-0.25) {any finite structure with a constant polymorphism};}
\only<5->{\tikzset{every path/.style={}}}

\uncover<6->{
\node[text width=2.3cm] at (0,-3) {finite 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)}

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



% ------------------ cyclic loop conditions examples ------------------------------

\begin{frame}
\frametitle{Cyclic loop conditions}
\centering
% 3 notmodels 3, 2 notmodels 4, general f(A' < A^n) = B nohomo A, 6,20,15 notmodels 2,3
% introduce Sigma_2 notation
$\Pol\left(
\begin{tikzpicture}[baseline=-1mm]
\cycle{3}{(0,0)}
\end{tikzpicture}\right)\not\models \uncover<8->{\hbox to 0mm{\large $\Sigma_3$}}
\begin{tikzpicture}[baseline=1mm]
\uncover<1-7>{
\node at (0,0.6) {$f(\alt<2>{\myBox0,\myBox1,\myBox2}{\myBox{x_0},\myBox{x_1},\myBox{x_2}})$};
\node at (0,-0.1) {$f(\alt<2>{\myBox1,\myBox2,\myBox0}{\myBox{x_1},\myBox{x_2},\myBox{x_0}})$};
\node[rotate=90] at (-0.7,0.25) {$\approx$}; 
\uncover<2>{
\path[->,>=latex'] 
    (-0.35,0.4) edge (-0.35,0.1)
    (0.05,0.4) edge (0.05,0.1)
    (0.48,0.4) edge (0.48,0.1)
    ;}
}
\end{tikzpicture}$
\hspace{5mm}
\uncover<3->{
$\Pol\left(
\begin{tikzpicture}[baseline=-1mm]
\cycle{2}{(0,0)}
\end{tikzpicture}\right)\not\models\uncover<9->{\hbox to 0mm{\large $\Sigma_4$}}
\begin{tikzpicture}[baseline=1mm]
\uncover<3-8>{
\node at (0,0.6) {$f(\alt<4>{\myBox0,\myBox1,\myBox0,\myBox1}{\myBox{x_0},\myBox{x_1},\myBox{x_2},\myBox{x_3}})$};
\node at (0,-0.1) {$f(\alt<4>{\myBox1,\myBox0,\myBox1,\myBox0}{\myBox{x_1},\myBox{x_2},\myBox{x_3},\myBox{x_0}})$};
\node[rotate=90] at (-0.87,0.25) {$\approx$}; 
\uncover<4>{
\path[->,>=latex'] 
    (-0.56,0.4) edge (-0.56,0.1)
    (-0.15,0.4) edge (-0.15,0.1)
    (0.35,0.4) edge (0.35,0.1)
    (0.7,0.4) edge (0.7,0.1)
    ;}
}    
\end{tikzpicture}$
}

\vspace{5mm}
\uncover<5->{
$\Pol\left(
\Cyc{6}\cupdot\Cyc{20}\cupdot\Cyc{15}\right)\not\models \uncover<9->{\hbox to 0mm{\large $\Sigma_{2,3}$}}
\begin{tikzpicture}[baseline=1mm]
\uncover<5-8>{
\node at (0,0.6) {$f(\alt<6>{\myBox{0'},\myBox{10'},\myBox{0},\myBox{4},\myBox{2}}{\myBox{x_0},\myBox{x_1},\myBox{y_0},\myBox{y_1},\myBox{y_2}})$};
\node at (0,-0.1) {$f(\alt<6>{\myBox{10'},\myBox{0'},\myBox4,\myBox2,\myBox0}{\myBox{x_1},\myBox{x_0},\myBox{y_1},\myBox{y_2},\myBox{y_0}})$};
\node[rotate=90] at (-1.15,0.25) {$\approx$}; 
\uncover<6>{
\path[->,>=latex'] 
    (-0.82,0.4) edge node[above,rotate=90] {\tiny $10$} (-0.82,0.1)
    (-0.82+1*1.7/4,0.4) edge node[above,rotate=90] {\tiny $10$} (-0.82+1*1.7/4,0.1)
    (-0.82+2*1.7/4,0.4) edge node[above,rotate=90] {\tiny $10$} (-0.82+2*1.7/4,0.1)
    (-0.82+3*1.7/4,0.4) edge node[above,rotate=90] {\tiny $10$} (-0.82+3*1.7/4,0.1)
    (-0.82+4*1.7/4,0.4) edge node[above,rotate=90] {\tiny $10$} (-0.82+4*1.7/4,0.1)
    ;}
}
\end{tikzpicture}$
}
\vspace{3mm}

\uncover<10->{
\begin{align*}
&\Pol(\bA)\models\Sigma_{b_1,\dots,b_n}&&\text{iff}&&\text{for all $\Cyc{a_1},\dots,\Cyc{a_n}\hookrightarrow\bA$ exists $\Cyc a\hookrightarrow\bA$ such that}\\
&&&&&\text{$a$ divides $\lcm(a_1\dotdiv b_1,\dots,a_n\dotdiv b_n)$}
\end{align*}
}
\end{frame}


% --------------- Jacub + Thm 3.5 + Thm 3.9 + Example {6,20,15} -------------------
\begin{frame}
\frametitle{Order}
\centering
%\begin{tabular*}{\textwidth}{c @{\extracolsep{\fill}} ccccc}
\vspace{-19mm}
\begin{tabular}{C{1.5cm} C{0.5cm}C{3cm} C{0.5cm}C{3cm}}
%B<A  &  Thm 3.5  &  Thm 3.9
\begin{tikzpicture}[baseline=-29.5mm]
    \node at (0,-2.75) {};
    \node at (0,3.6) {};
    \node (A) at (0,0.7) {$\Pol(\bA)\models \Sigma$};
    \node (B) {$\Pol(\bB)\models \Sigma$};
    \node[rotate=90] at (0,0.35) {$\Rightarrow$};
\end{tikzpicture} & 
\uncover<3->{iff &}
\uncover<3->{
\begin{tikzpicture}[baseline=1.5mm]
    \node (A) at (0,0.7) {$\Pol(\bA)\models \Sigma_{A\dotdiv c}$};
    \node (B) {$\Pol(\bB)\models \Sigma_{A\dotdiv c}$};
    \node[rotate=90] at (0,0.35) {$\Rightarrow$};
\end{tikzpicture} &}
\uncover<7->{iff & 
\begin{tikzpicture}[baseline=1.5mm]
    \node (A) at (0,0.7) {$\Pol(\bA)\models \Sigma_{P}$};
    \node (B) {$\Pol(\bB)\models \Sigma_{P}$};
    \node[rotate=90] at (0,0.35) {$\Rightarrow$};
\end{tikzpicture}}
\only<2-6> {
\hspace*{-1.5cm}\includegraphics[scale=0.25]{jakub_Obi2.png}
}
%Example:
\\\hline
\begin{tikzpicture}[baseline=-5.5mm]
    \node at (0,1.3) {};
    \node[scale = 0.7] (A) at (0,0.7) {$\Cyc{6}\cupdot\Cyc{20}\cupdot\Cyc{15}$};
    \node (B) {$\bB$};
    \path (A) edge (B);
\end{tikzpicture} &
\uncover<3->{iff &}
\only<3>{
$\Pol(\bB)\not\models
\Sigma_{6,20,15}$, $\Sigma_{2,20,5}$, $\Sigma_{3,10,15}$, $\Sigma_{6,4,3}$, $\Sigma_{3,5,15}$, $\Sigma_{3,2,3}$
 &}
\only<4->{
\begin{tikzpicture}[scale=0.6]
    \node[scale=0.8] at (-1.8,0) {$\Pol(\bB)\not\models$};
    \node[scale=0.8] (21) at (1.5,0) {$\Sigma_{3,2\only<-5>{,3}}$};
    \node[scale=0.8] (10) at (-1.5,-1) {$\Sigma_{2,\only<-5>{20,}5}$};
    \node[scale=0.8] (20) at (0,0) {$\Sigma_{3,5\only<-5>{,15}}$};

\only<5->{\tikzset{every path/.style={opacity = 0.2}}}
    
    \node[scale=0.8] (00) at (0,-2) {$\Sigma_{6,20,15}$};    
    
    \node[scale=0.8] (11) at (0,-1) {$\Sigma_{3,10,15}$};
    \node[scale=0.8] (12) at (1.5,-1) {$\Sigma_{6,4,3}$};
    
    \path 
        (00) edge (10)
        (00) edge (11)
        (00) edge (12)
        (11) edge (20)
        (11) edge (21)
        (12) edge (21)
        ;
\end{tikzpicture} &}
\uncover<7->{iff & 
\begin{tikzpicture}[scale = 0.6]

    \node[scale=0.8] at (0,1) {$\Pol(\bB)\not\models$};
    \node[opacity=0,scale = 0.8] (00) at (0,1) {$\Sigma_A$};
        
    \node[scale=0.8] (10) at (-1.5,0) {$\Sigma_{2,5}$};
    \node[scale=0.8] (11) at (0,0) {$\Sigma_{3,5}$};
    \node[scale=0.8] (12) at (1.5,0) {$\Sigma_{2,3}$};


\tikzset{every path/.style={opacity = 0.2}}
    \node[scale=0.8] (00) at (-1.5,-1) {$\Sigma_{5}$};
    \node[scale=0.8] (01) at (0,-1) {$\Sigma_{2}$};
    \node[scale=0.8] (02) at (1.5,-1) {$\Sigma_{3}$};
    
    \path 
        (00) edge (10)
        (00) edge (11)
        (01) edge (10)
        (01) edge (12)
        (02) edge (12)
        (02) edge (11)
        ;
\end{tikzpicture}}
\end{tabular}
\end{frame}


% -------------- final poset free dist step by set + Thm 3.11 ---------------------
\begin{frame}
\frametitle{Understanding Smooth Digraphs}
\centering
% PSD(2,3,5)  FP(2,3,5)
% _____________________
%       Thm 3.11

\begin{tikzpicture}[scale=0.4]
\def\myScale{0.6}
\only<12>{\node[scale=\myScale,orange] (0) at (2,-2)  {$\Cyc2\cupdot\Cyc3\cupdot\Cyc5$};}
\uncover<13->{\node[scale=\myScale] (0) at (2,-2)  {$\Cyc2\cupdot\Cyc3\cupdot\Cyc5$};}

\only<11>{\node[scale=\myScale,orange] (00) at (2,0)  {$\Cyc{15}\cupdot\Cyc{10}\cupdot\Cyc6$};}
\uncover<12->{\node[scale=\myScale] (00) at (2,0)  {$\Cyc{15}\cupdot\Cyc{10}\cupdot\Cyc6$};}

\uncover<10->{%-----------------------------------10------------------------------------
\node[scale=\myScale] (12) at (2,2) {$\Cyc{10}\cupdot\Cyc{3}$};
\node[scale=\myScale] (13) at (4,2) {$\Cyc6\cupdot\Cyc5$};
}
\uncover<8->{%-----------------------------------8------------------------------------
\node[scale=\myScale] (22) at (2,4) {$\Cyc6\cupdot\Cyc{15}$};
\node[scale=\myScale] (23) at (4,4) {$\Cyc6\cupdot\Cyc{10}$};
}
\only<9-10>{\node[scale=\myScale,orange] (11) at (0,2)  {$\Cyc2\cupdot\Cyc{15}$};}
\uncover<11->{\node[scale=\myScale] (11) at (0,2)  {$\Cyc2\cupdot\Cyc{15}$};}

\only<7-8>{\node[scale=\myScale,orange] (21) at (0,4) {$\Cyc{10}\cupdot\Cyc{15}$};}
\uncover<9->{\node[scale=\myScale] (21) at (0,4) {$\Cyc{10}\cupdot\Cyc{15}$};}

\uncover<6->{%-----------------------------------6------------------------------------
\node[scale=\myScale] (33) at (4,6) {$\Cyc2\cupdot\Cyc{5}$};
\node[scale=\myScale] (34) at (6,6) {$\Cyc3\cupdot\Cyc{5}$};
\node[scale=\myScale] (42) at (2,8) {$\Cyc{10}$};
\node[scale=\myScale] (43) at (4,8) {$\Cyc{15}$};
\node[scale=\myScale] (32) at (2,6) {$\Cyc{30}$};
\node[scale=\myScale] (53) at (4,10) {$\Cyc{5}$};
}

\only<5-6>{\node[scale=\myScale,orange] (31) at (-2,6) {$\Cyc2\cupdot\Cyc{3}$};}
\uncover<7->{\node[scale=\myScale] (31) at (-2,6) {$\Cyc2\cupdot\Cyc{3}$};}

\only<4>{\node[scale=\myScale,orange] (41) at (0,8) {$\Cyc{6}$};}
\uncover<5->{\node[scale=\myScale] (41) at (0,8) {$\Cyc{6}$};}


\only<2>{\node[scale=\myScale,orange] (51) at (0,10) {$\Cyc{2}$};}
\uncover<3->{\node[scale=\myScale] (51) at (0,10) {$\Cyc{2}$};}

\only<3>{\node[scale=\myScale,orange] (52) at (2,10) {$\Cyc{3}$};}
\uncover<4->{\node[scale=\myScale] (52) at (2,10) {$\Cyc{3}$};}



\node[scale=\myScale] (60) at (2,12) {$\Cyc{1}$};
\uncover<6->{
\path 
    (32) edge (41)
    (32) edge (42)
    (32) edge (43)
    (33) edge (42)
    (34) edge (43)
    (42) edge (51)
    (42) edge (53)
    (43) edge (52)
    (43) edge (53)
    
    (53) edge (60)
    ;
}
\uncover<8->{
\path 
    (21) edge (32)
    (22) edge (32)
    (22) edge (33)
    (23) edge (32)
    (23) edge (34)
    ;
}

\uncover<10->{
\path 
    (12) edge (21)
    (12) edge (23)
    (13) edge (22)
    (13) edge (23)
    ;
}    

\uncover<11->{
\path     
    (00) edge (11)
    (00) edge (12)
    (00) edge (13)
    ;
}    

\uncover<2->{\path (51) edge (60);}
\uncover<3->{\path (52) edge (60);}
\uncover<4->{\path (41) edge (51)
                   (41) edge (52);}
\uncover<5->{\path (31) edge (41);}    
\uncover<7->{\path (21) edge (31)
                   (21) edge (32);} 
\uncover<9->{\path (11) edge (21)
                   (11) edge (22);}  

\uncover<12->{\path (0)  edge (00);}                                                   
\end{tikzpicture}
\hspace{10mm}
\begin{tikzpicture}


\alt<12>{\node[orange] (20) at (0,1) {$\Sigma_{2,3,5}$};}{\node (20) at (0,1) {$\Sigma_{2,3,5}$};} 


\alt<2,4,5-12>{\node[orange] (00) at (-1.5,-1) {$\Sigma_{2}$};}{\node (00) at (-1.5,-1) {$\Sigma_{2}$};}
\alt<3-12>{\node[orange] (01) at (0,-1) {$\Sigma_{3}$};}{\node (01) at (0,-1) {$\Sigma_{3}$};}
\alt<7-12>{\node[orange] (02) at (1.5,-1) {$\Sigma_{5}$};}{\node (02) at (1.5,-1) {$\Sigma_{5}$};}
   
\alt<5-12>{\node[orange] (10) at (-1.5,0) {$\Sigma_{2,3}$};}{\node (10) at (-1.5,0) {$\Sigma_{2,3}$};}  
\alt<9-12>{\node[orange] (11) at (0,0) {$\Sigma_{2,5}$};}{\node (11) at (0,0) {$\Sigma_{2,5}$};}  
\alt<11-12>{\node[orange] (12) at (1.5,0) {$\Sigma_{3,5}$};}{\node (12) at (1.5,0) {$\Sigma_{3,5}$};}  
    
\alt<12>{
    \path 
        (20) edge[orange] (10)
        (20) edge[orange] (12)
        (20) edge[orange] (11);}{
    \path 
        (20) edge (10)
        (20) edge (12)
        (20) edge (11);}
        
\alt<5-12>{\path (00) edge[orange] (10)
                 (01) edge[orange] (10);}
          {\path (00) edge (10)
                 (01) edge (10);}        
\alt<9-12>{\path (02) edge[orange] (11)
                 (00) edge[orange] (11);}
          {\path (02) edge (11)
                 (00) edge (11);}  
                 
\alt<11-12>{\path (02) edge[orange] (12)
                  (01) edge[orange] (12);}
           {\path (02) edge (12)
                  (01) edge (12);}        

                                        
\end{tikzpicture}

\end{frame}


\begin{frame}
\frametitle{}
\centering
\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{theorem}
The following map is an isomorphism of posets where the order on the image set is reverse inclusion:
\begin{align*}
    \text{Smooth Digraphs}&\to\UP_\text{fin}\{\text{prime cyclic loop conditions}\}\cupdot\{\text{all pclc}\}\\
    [\bA]&\mapsto\{\Sigma_P\mid \Pol(\bA)\not\models\Sigma_P\}
\end{align*}
\end{theorem}

\uncover<2->{
\begin{tikzpicture}[scale=0.6]
\node (0) at (2,-2)  {$\Cyc2\cupdot\Cyc3$};
\node (00) at (2,0)  {$\Cyc6$};
\node (11) at (0,2)  {$\Cyc2$};
\node (13) at (4,2) {$\Cyc3$};
\node (22) at (2,4) {$\Cyc1$};

\node at (6.2,1) {$\mapsto$};
\path 
    (0)  edge (00)
    (00) edge (11)
    (00) edge (13)
    (11) edge (22)
    (13) edge (22)
    ;
\end{tikzpicture}
\hspace{5mm}
\begin{tikzpicture}[scale=0.6]


\node[scale=0.8] (0) at (2,-2)  {$\left\{\Sigma_{2,3},\Sigma_{3},\Sigma_{2}\right\}$};
\node[scale=0.8] (00) at (2,0)  {$\left\{\Sigma_{3},\Sigma_{2}\right\}$};
\node[scale=0.8] (11) at (0,2)  {$\left\{\Sigma_{2}\right\}$};
\node[scale=0.8] (13) at (4,2) {$\left\{\Sigma_{3}\right\}$};
\node[scale=0.8] (22) at (2,4) {$\emptyset$};

\path 
    (0)  edge (00)
    (00) edge (11)
    (00) edge (13)
    (11) edge (22)
    (13) edge (22)
    ;
\end{tikzpicture}
}
\end{frame}

\addtocounter{framenumber}{-1}
\begin{frame}
\centering\large
Paper: \href{https://arxiv.org/abs/1906.05699}{https://arxiv.org/abs/1906.05699}
\end{frame}

\end{document}