Partially ordered sets
Abbreviation: Pos
Definition
A partially ordered set (also called ordered set or 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: $x\leq y$, $y\leq x\Longrightarrow x=y$.
Definition
A 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.
Morphisms
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:
$x\leq y\Longrightarrow f(x)\leq f(y)$
Examples
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.
Basic results
Properties
Classtype | Universal Horn class |
---|---|
Universal theory | Decidable |
First-order theory | Undecidable |
Amalgamation property | |
Strong amalgamation property | |
Epimorphisms are surjective |
Finite members
$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &5\\ f(4)= &16\\ f(5)= &63\\ f(6)= &318\\ f(7)= &2045\\ f(8)= &16999\\ f(9)= &183231\\ f(10)= &2567284\\ f(11)= &46749427\\ f(12)= &1104891746\\ f(13)= &33823827452\\ f(14)= &1338193159771\\ f(15)= &68275077901156\\ f(16)= &4483130665195087 \end{array}$
Subclasses
Superclasses
References
Trace: » partially_ordered_sets