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Reflexive variant

If l, m, n are three distinct lines, then

In this case, parallelism is a transitive relation. However, in case l = n, the superimposed lines are not considered parallel in Euclidean geometry. The binary relation between parallel lines is evidently a transitive relation. However, in case l = n, the superimposed lines are not considered parallel in Euclidean geometry. The binary relation between parallel lines is evidently a symmetric relation. According to Euclid's tenets, parallelism is not a reflexive relation and thus fails to be an equivalence relation. Nevertheless, in affine geometry a pencil of parallel lines is taken as an equivalence class in the set of lines where parallelism is an equivalence relation.[15][16][17]

To this end, Emil Artin (1957) adopted a definition of parallelism where two lines are parallel if they have all or none of their points in common.[18] Then a line is parallel to itself so that the reflexive and transitive properties belong to this type of parallelism, creating an equivalence relation on the set of lines. In the study of incidence geometry, this variant of parallelism is used in the affine plane.