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, where those terms are meant in the sense defined below. 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, in the presence of symmetry, 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

• Araújo, João; Konieczny, Janusz (2007), "A method of finding automorphism groups of endomorphism monoids of relational systems", Discrete Mathematics, 307: 1609–1620, doi:10.1016/j.disc.2006.09.029
• Bachmann, Friedrich (1959), Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag
• Karzel, Helmut (2007), "Loops related to geometric structures", Quasigroups and Related Systems, 15: 47–76
• Karzel, Helmut; Pianta, Silvia (2008), "Binary operations derived from symmetric permutation sets and applications to absolute geometry", Discrete Mathematics, 308: 415–421, doi:10.1016/j.disc.2006.11.058
• Karzel, Helmut; Marchi, Mario; Pianta, Silvia (December 2010), "The defect in an invariant reflection structure", Journal of Geometry, 99 (1–2): 67–87, doi:10.1007/s00022-010-0058-7
• Karzel, Helmut; Taherian, Sayed-Ghahreman (2018), "Groups with a ternary equivalence relation", Aequationes Mathematicae, 92: 415–423, doi:10.1007/s00010-018-0543-x
• Lingenberg, Rolf (1979), Metric planes and metric vector spaces, Wiley
• Pickett, H.E. (1966), "A note on generalized equivalence relations", American Mathematical Monthly, 73: 860–861, doi:10.2307/2314183
• Rainich, G.Y. (1952), "Ternary relations in geometry and algebra", Michigan Mathematical Journal, 1 (2): 97–111, doi:10.1307/mmj/1028988890
• Szmielew, Wanda (1981), On n-ary equivalence relations and their application to geometry, Warsaw: Instytut Matematyczny Polskiej Akademi Nauk