Definition:Euclidean Relation
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Definition
Let $\RR \subseteq S \times S$ be a relation in $S$.
Left-Euclidean
$\RR$ is left-Euclidean if and only if:
- $\tuple {x, z} \in \RR \land \tuple {y, z} \in \RR \implies \tuple {x, y} \in \RR$
Right-Euclidean
$\RR$ is right-Euclidean if and only if:
- $\tuple {x, y} \in \RR \land \tuple {x, z} \in \RR \implies \tuple {y, z} \in \RR$
Euclidean
$\RR$ is Euclidean if and only if it is both left-Euclidean and right-Euclidean.
The concept of a Euclidean relation was named for Euclid.
It derives ultimately from the first of Euclid's common notions.
In the words of Euclid:
- Things which are equal to the same thing are also equal to each other.
(The Elements: Book $\text{I}$: Common Notions: Common Notion $1$)
However, Euclid did not delve deeply into the field of relation theory.
The concept of equivalence relations was a much later development.
Also see
- Results about Euclidean relations can be found here.
Source of Name
This entry was named for Euclid.