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.

Properties

Classtype Universal Horn class Decidable Undecidable

Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &5\\ f(4)= &16\\ f(5)= &63\\ f(6)= &318\\ f(7)= &2045\\ f(8)= &16999\\ f(9)= &183231\\ f(10)= &2567284\\ f(11)= &46749427\\ f(12)= &1104891746\\ f(13)= &33823827452\\ f(14)= &1338193159771\\ f(15)= &68275077901156\\ f(16)= &4483130665195087 \end{array}$