\documentclass[numbers=enddot,12pt,final,onecolumn,notitlepage]{scrartcl}% \usepackage[headsepline,footsepline,manualmark]{scrlayer-scrpage} \usepackage[all,cmtip]{xy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{framed} \usepackage{amsmath} \usepackage{comment} \usepackage{color} \usepackage{hyperref} \usepackage[sc]{mathpazo} \usepackage[T1]{fontenc} \usepackage{amsthm} %TCIDATA{OutputFilter=latex2.dll} %TCIDATA{Version=5.50.0.2960} %TCIDATA{LastRevised=Sunday, January 10, 2016 23:50:11} %TCIDATA{SuppressPackageManagement} %TCIDATA{} %TCIDATA{} %TCIDATA{BibliographyScheme=Manual} %BeginMSIPreambleData \providecommand{\U}{\protect\rule{.1in}{.1in}} %EndMSIPreambleData \theoremstyle{definition} \newtheorem{theo}{Theorem}[section] \newenvironment{theorem}[] {\begin{theo}[#1]\begin{leftbar}} {\end{leftbar}\end{theo}} \newtheorem{lem}[theo]{Lemma} \newenvironment{lemma}[] {\begin{lem}[#1]\begin{leftbar}} {\end{leftbar}\end{lem}} \newtheorem{prop}[theo]{Proposition} \newenvironment{proposition}[] {\begin{prop}[#1]\begin{leftbar}} {\end{leftbar}\end{prop}} \newtheorem{defi}[theo]{Definition} \newenvironment{definition}[] {\begin{defi}[#1]\begin{leftbar}} {\end{leftbar}\end{defi}} \newtheorem{remk}[theo]{Remark} \newenvironment{remark}[] {\begin{remk}[#1]\begin{leftbar}} {\end{leftbar}\end{remk}} \newtheorem{coro}[theo]{Corollary} \newenvironment{corollary}[] {\begin{coro}[#1]\begin{leftbar}} {\end{leftbar}\end{coro}} \newtheorem{conv}[theo]{Convention} \newenvironment{condition}[] {\begin{conv}[#1]\begin{leftbar}} {\end{leftbar}\end{conv}} \newtheorem{quest}[theo]{Question} \newenvironment{algorithm}[] {\begin{quest}[#1]\begin{leftbar}} {\end{leftbar}\end{quest}} \newtheorem{warn}[theo]{Warning} \newenvironment{conclusion}[] {\begin{warn}[#1]\begin{leftbar}} {\end{leftbar}\end{warn}} \newtheorem{conj}[theo]{Conjecture} \newenvironment{conjecture}[] {\begin{conj}[#1]\begin{leftbar}} {\end{leftbar}\end{conj}} \newtheorem{exmp}[theo]{Example} \newenvironment{example}[] {\begin{exmp}[#1]\begin{leftbar}} {\end{leftbar}\end{exmp}} \iffalse \newenvironment{proof}[Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \fi \newenvironment{verlong}{}{} \newenvironment{vershort}{}{} \newenvironment{noncompile}{}{} \excludecomment{verlong} \includecomment{vershort} \excludecomment{noncompile} \newcommand{\kk}{\mathbf{k}} \newcommand{\id}{\operatorname{id}} \newcommand{\ev}{\operatorname{ev}} \newcommand{\Comp}{\operatorname{Comp}} \newcommand{\bk}{\mathbf{k}} \newcommand{\Nplus}{\mathbb{N}_{+}} \newcommand{\NN}{\mathbb{N}} \let\sumnonlimits\sum \let\prodnonlimits\prod \renewcommand{\sum}{\sumnonlimits\limits} \renewcommand{\prod}{\prodnonlimits\limits} \setlength\textheight{22.5cm} \setlength\textwidth{15cm} \ihead{Errata to Semigroup and Ring ...''} \ohead{\today} \begin{document} \begin{center} \textbf{Semigroup and Ring Theoretical Methods in Probability} \textit{Kenneth S. Brown} \texttt{\url{http://www.math.cornell.edu/~kbrown/papers/toronto.pdf}} version of 21 April 2003 \textbf{Errata and addenda by Darij Grinberg} \bigskip \end{center} I will refer to the results appearing in the paper \textquotedblleft Semigroup and Ring Theoretical Methods in Probability\textquotedblright\ by the numbers under which they appear in this paper (specifically, in its version of 21 April 2003). \setcounter{section}{30} \section*{Errata} \begin{itemize} \item \textbf{Page 9, Remark 3.4:} Replace \textquotedblleft below.)\textquotedblright\ by \textquotedblleft below).\textquotedblright. \item \textbf{Page 20, Example A.13:} The notation $L_{\geq X}$ should be defined. (It stands for the sublattice $\left\{ Y\in L\ \mid\ Y\geq X\right\}$ of $L$.) \item \textbf{Page 21, Appendix B:} \textquotedblleft semilattice of supports\textquotedblright\ should be \textquotedblleft support semilattice\textquotedblright\ (in order to use terminology that you have defined). \item \textbf{Page 21, \S B.1:} \textquotedblleft by sending $X$% \textquotedblright\ should be \textquotedblleft by sending $X\in L$\textquotedblright. \item \textbf{Page 22, proof of Theorem B.1:} The case when $L=\varnothing$ should be treated separately. (Trivial as it is, it requires at least explicit mention, because your proof does not work in this case; namely, $\widehat{1}$ does not exist when $L=\varnothing$.) \item \textbf{Page 22, proof of Theorem B.1:} Replace \textquotedblleft the subsemigroup $\left\{ \operatorname*{supp}x\leq Y\right\}$% \textquotedblright\ by \textquotedblleft the subsemigroup $\left\{ x\in S\ \mid\ \operatorname*{supp}x\leq Y\right\}$ of $S$\textquotedblright. Furthermore, after this sentence, I would add another, which justifies the next sentence: \textquotedblleft It is straightforward to see that $S_{\leq Y}$ is itself a band, and its support semilattice is the interval $\left[ \widehat{0},Y\right]$ of $L$.\textquotedblright. \end{itemize} \end{document}