Ternary equivalence relation

In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation. A ternary equivalence relation is symmetric, reflexive, and transitive. The classic example is the relation of collinearity among three points in Euclidean space. In an abstract set, a ternary equivalence relation determines a collection of equivalence classes or pencils that form a linear space in the sense of incidence geometry. In the same way, a binary equivalence relation on a set determines a partition.

Definition

A ternary equivalence relation on a set X is a relation E ⊂ X3, written [a, b, c], that satisfies the following axioms:

  1. Symmetry: If [a, b, c] then [b, c, a] and [c, b, a]. (Therefore also [a, c, b], [b, a, c], and [c, a, b].)
  2. Reflexivity: [a, b, b]. Equivalently, if a, b, and c are not all distinct, then [a, b, c].
  3. Transitivity: If a ≠ b and [a, b, c] and [a, b, d] then [b, c, d]. (Therefore also [a, c, d].)

References
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