Mathematical Structures: History of Partially ordered sets

# History of Partially ordered sets

 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Partially ordered sets} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Partially_ordered_sets}{edit} \abbreviation{Pos} \begin{definition} A \emph{partially ordered set} (also called \emph{ordered set} or \emph{poset} for short) is a structure $\mathbf{P}=\left\langle P,\leq \right\rangle$ such that $P$ is a set and $\leq$ is a binary relation on $P$ that is reflexive: $x\leq x$ transitive: $x\leq y$, $y\leq z\implies x\leq y$ antisymmetric: $x\leq y$, $y\leq x\implies x=y$. \end{definition} \begin{definition} A \emph{strict partial order} is a structure $\left\langle P,<\right\rangle$ such that $P$ is a set and $<$ is a binary relation on $P$ that is irreflexive: $\neg(x  %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Partially ordered sets} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Partially_ordered_sets}{edit} \abbreviation{Pos} \begin{definition} A \emph{partially ordered set} (also called \emph{ordered set} or \emph{poset} for short) is a structure$\mathbf{P}=\left\langle P,\leq \right\rangle $such that$P$is a set and$\leq $is a binary relation on$P$that is reflexive:$x\leq x$transitive:$x\leq y$,$y\leq z\implies x\leq y$antisymmetric:$x\leq y$,$y\leq x\implies x=y$. \end{definition} \begin{definition} A \emph{strict partial order} is a structure$\left\langle P,<\right\rangle $such that$P$is a set and$<$is a binary relation on$P$that is irreflexive:$\neg(x