Euclidean Relation

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."

Definition

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:

:$\forall a, b, c\in X\,(a\,R\, b \land a \,R\, c \to b \,R\, c).$

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'':

:$\forall a, b, c\in X\,(b\,R\, a \land c \,R\, a \to b \,R\, c).$

Properties

# 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, also a non-symmetric relation can 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 range of a right Euclidean relation is always a subset of its domain. 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.
# 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.
# A right Euclidean relation is always quasitransitive, as is a left Euclidean relation.
# A connected right Euclidean relation is always transitive; and so is a connected left Euclidean relation.
# If ''X'' has at least 3 elements, a connected right Euclidean relation ''R'' on ''X'' cannot be antisymmetric, and neither can a connected left Euclidean relation on ''X''. On the 2-element set ''X'' = , e.g. the relation ''xRy'' defined by ''y''=1 is connected, right Euclidean, and antisymmetric, and ''xRy'' defined by ''x''=1 is connected, left Euclidean, and antisymmetric.
# A relation ''R'' on a set ''X'' is right Euclidean if, and only if, the restriction ''R'' := ''R''_ran(''R'') is an equivalence and for each ''x'' in ''X''\ran(''R''), all elements to which ''x'' is related under ''R'' are equivalent under ''R''. Similarly, ''R'' on ''X'' is left Euclidean if, and only if, ''R'' := ''R''_dom(''R'') is an equivalence and for each ''x'' in ''X''\dom(''R''), all elements that are related to ''x'' under ''R'' are equivalent under ''R''.
# A left Euclidean relation is left-unique if, and only if, it is antisymmetric. Similarly, a right Euclidean relation is right unique if, and only if, it is anti-symmetric.
# A left Euclidean and left unique relation is vacuously transitive, and so is a right Euclidean and right unique relation.
# A left Euclidean relation is left quasi-reflexive. For left-unique relations, the converse also holds. Dually, each right Euclidean relation is right quasi-reflexive, and each right unique and right quasi-reflexive relation is right Euclidean. Lemma 44-46.

References

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Binary relations
Relation