Ternary_equivalence_relation Knowpia

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 edit

A ternary equivalence relation on a set $X$ is a relation $E \subset X 3$ , written $[a, b, c]$ , that satisfies the following axioms:

Symmetry: If $[a, b, c]$ then $[b, c, a]$ and $[c, b, a]$ . (Therefore also $[a, c, b]$ , $[b, a, c]$ , and $[c, a, b]$ .)
Reflexivity: $[a, b, b]$ . Equivalently, in the presence of symmetry, if $a$ , $b$ , and $c$ are not all distinct, then $[a, b, c]$ .
Transitivity: If $a \neq b$ and $[a, b, c]$ and $[a, b, d]$ then $[b, c, d]$ . (Therefore also $[a, c, d]$ .)

References edit

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

Summary

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