Abbreviation: Pos
A \emph{partially ordered set} (also called \emph{ordered set} or \emph{poset} for short) is a structure $\mathbf{P}=\langle P,\leq \rangle $ such that $P$ is a set and $\leq $ is a binary relation on $P$ that is
reflexive: $x\leq x$
transitive: $x\leq y$, $y\leq z\Longrightarrow x\leq y$
antisymmetric: $x\leq y$, $y\leq x\Longrightarrow x=y$.
A \emph{strict partial order} is a structure $\langle P,<\rangle $ such that $P$ is a set and $<$ is a binary relation on $P$ that is
irreflexive: $\neg(x<x)$
transitive: $x<y$, $y<z\Longrightarrow x<y$
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 }=\langle P,\geq \rangle $ by $x\geq y\Longleftrightarrow y\leq x$. Then $\mathbf{P}^{\partial }$ is also a partially ordered set.
Let $\mathbf{P}$ and $\mathbf{Q}$ be posets. A morphism from $\mathbf{P}$ to $\mathbf{Q}$ is a function $f:P\to Q$ that is order-preserving:
$x\leq y\Longrightarrow f(x)\leq f(y)$
Example 1: $\langle \mathbb{R},\leq \rangle $, the real numbers with the standard order.
Example 2: $\langle P(S),\subseteq \rangle $, the collection of subsets of a sets $S$, ordered by inclusion.
Example 3: Any poset is order-isomorphic to a poset of subsets of some set, ordered by inclusion.
| Classtype | Universal Horn class |
|---|---|
| Universal theory | Decidable |
| First-order theory | Undecidable |
| Amalgamation property | |
| Strong amalgamation property | |
| Epimorphisms are surjective |
$\begin{array}{lr}
f(1)= &1
f(2)= &2
f(3)= &5
f(4)= &16
f(5)= &63
f(6)= &318
f(7)= &2045
f(8)= &16999
f(9)= &183231
f(10)= &2567284
f(11)= &46749427
f(12)= &1104891746
f(13)= &33823827452
f(14)= &1338193159771
f(15)= &68275077901156
f(16)= &4483130665195087
\end{array}$