Orderings

Partial Orders

Definition: Partial Order

Let $A$ be a set.

A partial order of $A$ is a relation $R\subseteq A\times A$ which is reflexive, transitive and asymmetric.

NOTATION

Partial orders are usually denoted by $\le$ instead of $R$.

Definition: Strict Partial Order

Let $A$ be a set.

A strict partial order of $A$ is a relation $R\subseteq A\times A$ which is irreflexive, transitive and antisymmetric.

NOTATION

Partial orders are usually denoted by $<$ instead of $R$.

Total Orders

Definition: Total Order

Let $A$ be a set.

A total order of $A$ is a partial order of $A$ in which any two elements are comparable:

$$a\le b\text{ or }b\le a\forall a,b\in A$$

Definition: Strict Total Order

Let $A$ be a set.

A strict total order of $A$ is a strict partial order of $A$ in which any two elements are comparable:

$$a<b\text{ or }b<a\forall a,b\in A$$