← Notes on Rieg Theory

Older Versions__2024_09_06_Ver__20240906.tex

\documentclass{amsart}
\usepackage[left=2cm, right=2cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts, amsthm, amssymb, mathtools,etoolbox}
\usepackage{blindtext}
\usepackage[colorlinks=true, urlcolor=blue, linkcolor=blue, citecolor=blue]{hyperref}
\usepackage{tikz,tikz-cd}
\usepackage{cleveref}
\usepackage{framed}
\usepackage{array}
\usepackage[style=alphabetic,sorting=nyt]{biblatex}
\renewbibmacro{in:}{}
% \addbibresource{biblio.bib}
\addbibresource{CommonBiblio20240922.bib}
\tikzset{pullback/.style={minimum size=1.2ex,path picture={
\draw[opacity=1,black,-,#1] (-0.5ex,-0.5ex) -- (0.5ex,-0.5ex) -- (0.5ex,0.5ex);%
}}}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{todo}[theorem]{Todo}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{fact}[theorem]{Fact}

\theoremstyle{definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{question}[theorem]{Question}
\newtheorem{idea}[theorem]{Idea}
\newtheorem{problem}[theorem]{Problem}
\newtheorem*{answer}{Answer}

\newcommand{\dq}[1]{``#1"}
\newcommand{\memo}[1]{\textcolor{red}{memo: #1}}
\newcommand{\invmemo}[1]{\textcolor{blue}{memo: #1}}
\newcommand{\para}[1]{\paragraph{\textbf{#1}}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\op}{\mathrm{op}}
\newcommand{\ob}{\mathrm{ob}}
\newcommand{\Set}{\mathbf{Set}}
\newcommand{\FinSet}{\mathbf{FinSet}}
\newcommand{\PSh}{\mathbf{PSh}}
\newcommand{\Sh}{\mathbf{Sh}}
\newcommand{\Func}[2]{[#1,#2]}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\demph}[1]{\textbf{#1}}
\font\maljapanese=dmjhira at 2.5ex
\newcommand{\yo}{\textrm{\!\maljapanese\char"48}}

\newcommand{\Po}[1]{\mathcal{P}(#1)}
\newcommand{\Top}{\mathrm{Top}}
\newcommand{\HeyAlg}{\mathrm{HeyAlg}}
\newcommand{\Rieg}{\mathbf{Rieg}}
\newcommand{\Ring}{\mathbf{Ring}}
\newcommand{\Rig}{\mathbf{Rig}}
\newcommand{\ti}{\times}
\newcommand{\ex}{\uparrow}
\newcommand{\co}{\mathrm{c}}
\newcommand{\2}{\mathbf{2}}
\newcommand{\1}{\mathbf{1}}
\newcommand{\mult}[1]{\mathrm{mult}(#1)}
\newcommand{\mN}{\mult{\N}}
\newcommand{\di}{\mathrm{dim}}
\newcommand{\rad}{\mathrm{rad}}
\newcommand{\Mr}[2]{\N_{#1,#2}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\Groupoid}{\mathrm{Groupoid}}
\newcommand{\Groupoidfin}{\Groupoid_{\mathrm{fin}}}
\newcommand{\TB}{\star}
\newcommand{\ep}{\varepsilon}
\newcommand{\loga}{log-able}
\newcommand{\ch}{\mathrm{char}}
\newcommand{\TV}{\Omega}
\newcommand{\chz}{(\infty, 0)}
\newcommand{\Tame}{\mathbb{T}\mathrm{ame}}
\newcommand{\Char}{\mathbb{C}\mathrm{har}}
\newcommand{\finChar}{\mathrm{f}\mathbb{C}\mathrm{har}}
\newcommand{\qTame}{\overline{\N\times \N_{>}}}
\newcommand{\K}[1]{#1 / \hspace{-1pt} \scalebox{0.7}{$\cong$}}
\newcommand{\biK}[1]{#1 / \hspace{-1pt} \scalebox{0.7}{$\simeq$}}
\newcommand{\lcm}{\mathrm{lcm}}
\newcommand{\proN}{\hat{\mathbb{N}}}
\newcommand{\proZ}{\hat{\mathbb{Z}}}
\newcommand\binuparrow{\mathbin{\uparrow}}
\newcommand{\powow}{\binuparrow \binuparrow}

\title{Notes on Rieg theory: semiring with exponentials in logic, number theory, enumerative combinatorics and category theory}
\author{Ryuya Hora}
\thanks{Graduate School of Mathematical Sciences, University of Tokyo. \url{hora@ms.u-tokyo}}
% \date{\today}
\subjclass[2020]{MSC}
\keywords{Keywords}


\begin{document}
\definecolor{shadecolor}{gray}{0.9}
\maketitle
\tableofcontents


\end{document}