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Concept
Euclidean relation
Summary
In mathematics, Euclidean relations are a class of binary relations that formalize "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other."
A binary relation R on a set X is Euclidean (sometimes called right Euclidean) if it satisfies the following: for every a, b, c in X, if a is related to b and c, then b is related to c. To write this in predicate logic:
Dually, a relation R on X is left Euclidean if for every a, b, c in X, if b is related to a and c is related to a, then b is related to c:
Due to the commutativity of ∧ in the definition's antecedent, aRb ∧ aRc even implies bRc ∧ cRb when R is right Euclidean. Similarly, bRa ∧ cRa implies bRc ∧ cRb when R is left Euclidean.
The property of being Euclidean is different from transitivity. For example, ≤ is transitive, but not right Euclidean, while xRy defined by 0 ≤ x ≤ y + 1 ≤ 2 is not transitive, but right Euclidean on natural numbers.
For symmetric relations, transitivity, right Euclideanness, and left Euclideanness all coincide. However, a non-symmetric relation can also be both transitive and right Euclidean, for example, xRy defined by y=0.
A relation that is both right Euclidean and reflexive is also symmetric and therefore an equivalence relation. Similarly, each left Euclidean and reflexive relation is an equivalence.
The of a right Euclidean relation is always a subset of its . The restriction of a right Euclidean relation to its range is always reflexive, and therefore an equivalence. Similarly, the domain of a left Euclidean relation is a subset of its range, and the restriction of a left Euclidean relation to its domain is an equivalence. Therefore, a right Euclidean relation on X that is also right total (respectively a left Euclidean relation on X that is also left total) is an equivalence, since its range (respectively its domain) is X.
A relation R is both left and right Euclidean, if, and only if, the domain and the range set of R agree, and R is an equivalence relation on that set.
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