Category:Orderings
Jump to navigation
Jump to search
This category contains results about Orderings.
Definitions specific to this category can be found in Definitions/Orderings.
Let $S$ be a set.
Definition 1
$\RR$ is an ordering on $S$ if and only if $\RR$ satisfies the ordering axioms:
| \((1)\) | $:$ | $\RR$ is reflexive | \(\ds \forall a \in S:\) | \(\ds a \mathrel \RR a \) | |||||
| \((2)\) | $:$ | $\RR$ is transitive | \(\ds \forall a, b, c \in S:\) | \(\ds a \mathrel \RR b \land b \mathrel \RR c \implies a \mathrel \RR c \) | |||||
| \((3)\) | $:$ | $\RR$ is antisymmetric | \(\ds \forall a, b \in S:\) | \(\ds a \mathrel \RR b \land b \mathrel \RR a \implies a = b \) |
Definition 2
$\RR$ is an ordering on $S$ if and only if $\RR$ satisfies the ordering axioms:
| \((1)\) | $:$ | \(\ds \RR \circ \RR \) | |||||||
| \((2)\) | $:$ | \(\ds \RR \cap \RR^{-1} = \Delta_S \) |
where:
- $\circ$ denotes relation composition
- $\RR^{-1}$ denotes the inverse of $\RR$
- $\Delta_S$ denotes the diagonal relation on $S$.
Subcategories
This category has the following 2 subcategories, out of 2 total.
O
S
- Subset Relation is Ordering (4 P)
Pages in category "Orderings"
The following 20 pages are in this category, out of 20 total.