Category:Definitions/Strict Orderings
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This category contains definitions related to Strict Orderings.
Related results can be found in Category:Strict Orderings.
Definition 1
Let $\RR$ be a relation on a set $S$.
Then $\RR$ is a strict ordering (on $S$) if and only if $\RR$ satisfies the strict ordering axioms:
\((1)\) | $:$ | Asymmetry | \(\ds \forall a, b \in S:\) | \(\ds a \mathrel \RR b \) | \(\ds \implies \) | \(\ds \neg \paren {b \mathrel \RR a} \) | |||
\((2)\) | $:$ | Transitivity | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \) | \(\ds \implies \) | \(\ds a \mathrel \RR c \) |
Definition 2
Let $\RR$ be a relation on a set $S$.
Then $\RR$ is a strict ordering (on $S$) if and only if $\RR$ satisfies the strict ordering axioms:
\((1)\) | $:$ | Antireflexivity | \(\ds \forall a \in S:\) | \(\ds \neg \paren {a \mathrel \RR a} \) | |||||
\((2)\) | $:$ | Transitivity | \(\ds \forall a, b, c \in S:\) | \(\ds \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \implies a \mathrel \RR c \) |
Subcategories
This category has only the following subcategory.
O
Pages in category "Definitions/Strict Orderings"
The following 14 pages are in this category, out of 14 total.
S
- Definition:Strict Ordering
- Definition:Strict Ordering on Integers
- Definition:Strict Ordering/Antireflexive and Transitive
- Definition:Strict Ordering/Asymmetric and Transitive
- Definition:Strict Ordering/Notation
- Definition:Strict Ordering/Partial vs. Total
- Definition:Strict Partial Ordering
- Definition:Strict Weak Ordering
- Definition:Strictly Ordered Set
- Definition:Strictly Partially Ordered Set
- Definition:Strictly Totally Ordered Set
- Definition:Strictly Well-Ordered Set