Ternary relation

In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place.

Just as a binary relation is formally defined as a set of pairs, i.e. a subset of the Cartesian product A × B of some sets A and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product A × B × C of three sets A, B and C.

An example of a ternary relation in elementary geometry is the collinearity of points.

Examples

Binary functions

A function ƒ: A × BC in two variables, taking values in two sets A and B, respectively, is formally a function that associates to every pair (a,b) in A × B an element ƒ(a, b) in C. Therefore, its graph consists of pairs of the form ((a, b), ƒ(a, b)). Such pairs in which the first element is itself a pair are often identified with triples. This makes the graph of ƒ a ternary relation between A, B and C, consisting of all triples (a, b, ƒ(a, b)), for all a in A and b in B.

Cyclic orders

Given any set A whose elements are arranged on a circle, one can define a ternary relation R on A, i.e. a subset of A3 = A × A × A, by stipulating that R(a, b, c) holds if and only if the elements a, b and c are pairwise different and when going from a to c in a clockwise direction one passes through b. For example, if A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 } represents the hours on a clock face, then R(8, 12, 4) holds and R(12, 8, 4) does not hold.

Betweenness relations

Congruence relation

The ordinary congruence of arithmetics

which holds for three integers a, b, and m if and only if m divides a  b, formally may be considered as a ternary relation. However, usually, this instead is considered as a family of binary relations between the a and the b, indexed by the modulus m. For each fixed m, indeed this binary relation has some natural properties, like being an equivalence relation; while the combined ternary relation in general is not studied as one relation.

Typing relation

A typing relation indicates that is a term of type in context , and is thus a ternary relation between contexts, terms and types.

Further reading

  • Myers, Dale (1997), "An interpretive isomorphism between binary and ternary relations", in Mycielski, Jan; Rozenberg, Grzegorz; Salomaa, Arto, Structures in Logic and Computer Science, Lecture Notes in Computer Science, 1261, Springer, pp. 84–105, ISBN 3-540-63246-8, doi:10.1007/3-540-63246-8_6 
  • Novák, Vítězslav (1996), "Ternary structures and partial semigroups", Czechoslovak Mathematical Journal, 46 (1): 111–120, hdl:10338.dmlcz/127275 
  • Novák, Vítězslav; Novotný, Miroslav (1989), "Transitive ternary relations and quasiorderings", Archivum Mathematicum, 25 (1–2): 5–12, hdl:10338.dmlcz/107333 
  • Novák, Vítězslav; Novotný, Miroslav (1992), "Binary and ternary relations", Mathematica Bohemica, 117 (3): 283–292, hdl:10338.dmlcz/126278 
  • Novotný, Miroslav (1991), "Ternary structures and groupoids", Czechoslovak Mathematical Journal, 41 (1): 90–98, hdl:10338.dmlcz/102437 
  • Šlapal, Josef (1993), "Relations and topologies", Czechoslovak Mathematical Journal, 43 (1): 141–150, hdl:10338.dmlcz/128381 
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