# Category:Strict Orderings

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This category contains results about Strict Orderings.

Definitions specific to this category can be found in Definitions/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 the following two conditions hold:

\((1)\) | $:$ | Asymmetry | \(\displaystyle \forall a, b \in S:\) | \(\displaystyle a \mathrel \RR b \) | \(\displaystyle \implies \) | \(\displaystyle \neg \paren {b \mathrel \RR a} \) | ||

\((2)\) | $:$ | Transitivity | \(\displaystyle \forall a, b, c \in S:\) | \(\displaystyle \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \) | \(\displaystyle \implies \) | \(\displaystyle a \mathrel \RR c \) |

### Definition 2

Let $\RR$ be a relation on a set $S$.

Then $\RR$ is a **strict ordering (on $S$)** if and only if the following two conditions hold:

\((1)\) | $:$ | Antireflexivity | \(\displaystyle \forall a \in S:\) | \(\displaystyle \neg \paren {a \mathrel \RR a} \) | ||||

\((2)\) | $:$ | Transitivity | \(\displaystyle \forall a, b, c \in S:\) | \(\displaystyle \paren {a \mathrel \RR b} \land \paren {b \mathrel \RR c} \implies a \mathrel \RR c \) |

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Strict Orderings"

The following 11 pages are in this category, out of 11 total.