History of Partially ordered sets

 
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 \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:  $\neg(x<x)$
 
 
 transitive:  $x<y$, $y<z\implies x<y$
 
 
 Remark: 
 The above definitions are related via: $x\leq y\Longleftrightarrow x<y \mbox{or} x=y$ and 
 $x<y\Longleftrightarrow x\leq y$, $x\neq y$.
 
 For a partially ordered set $\mathbf{P}$, define the dual $\mathbf{P}^{\partial }=\left\langle P,\geq \right\rangle $ by $x\geq
 y\Longleftrightarrow y\leq x$. Then $\mathbf{P}^{\partial }$ is also a
 partially ordered set.
 
 
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{P}$ and $\mathbf{Q}$ be posets. A morphism from $\mathbf{P}$ to 
 $\mathbf{Q}$ is a function $f:P\rightarrow Q$ that is order-preserving: 
 
 $x\leq y\implies f(x)\leq f(y)$
 
 \end{morphisms}
 \begin{basic_results}
 \end{basic_results}
 \begin{examples}
 \begin{example}
 $\left\langle \mathbb{R},\leq \right\rangle $, the real numbers with the standard order.
 
 \end{example}
 \begin{example}
 $\left\langle P(S),\subseteq \right\rangle $, the collection of subsets of a
 sets $S$, ordered by inclusion.
 
 \end{example}
 \begin{example}
 Any poset is order-isomorphic to a poset of subsets of some set, ordered by
 inclusion.
 
 \end{example}
 \end{examples}
 \begin{table}[h]
 \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})
 
 \begin{tabular}{|ll|}\hline
 Classtype & Universal Horn class\\\hline
 Universal theory & Decidable\\\hline
 First-order theory & Undecidable\\\hline
 Amalgamation property & \\\hline
 Strong amalgamation property & \\\hline
 Epimorphisms are surjective & \\\hline
 \end{tabular}
 \end{properties}
 \end{table}
 \begin{finite_members} $f(n)=$ number of members of size $n$.
 
 $\begin{array}{lr}
 f(1)= &1\\
 f(2)= &2\\
 f(3)= &5\\
 f(4)= &16\\
 f(5)= &63\\
 f(6)= &318\\
 f(7)= &2045\\
 f(8)= &16999\\
 f(9)= &183231\\
 f(10)= &2567284\\
 f(11)= &46749427\\
 f(12)= &1104891746\\
 f(13)= &33823827452\\
 f(14)= &1338193159771\\
 \end{array}$
 \end{finite_members}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Connected_partial_orders.pdf}{Connected partial orders} 
 
 \href{Complete_partial_orders.pdf}{Complete partial orders} 
 
 \href{Directed_partial_orders.pdf}{Directed partial orders} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Preordered_sets.pdf}{Preordered sets} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
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