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 can be given on triples of points, where a triple is in the relation if the three points are collinear. Another geometric example can be obtained by considering triples consisting of two points and a line, where a triple is in the ternary relation if the two points determine (are incident with) the line.

Examples

Binary functions

Further information: Graph of a function and Binary function

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

Cyclic orders

Main article: Cyclic order

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

Main article: Betweenness relation

Ternary equivalence relation

Main article: Ternary equivalence relation

Congruence relation

Main article: Congruence modulo m

The ordinary congruence of arithmetics

${\displaystyle a\equiv b{\pmod {m}}}$ 

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

Main article: Simply typed lambda calculus § Typing rules

A typing relation ${\displaystyle \Gamma \vdash e\!:\!\sigma }$  indicates that ${\displaystyle e}$  is a term of type ${\displaystyle \sigma }$  in context ${\displaystyle \Gamma }$ , and is thus a ternary relation between contexts, terms and types.

Schröder rules

Given homogeneous relations A, B, and C on a set, a ternary relation ${\displaystyle (A,\ B,\ C)}$  can be defined using composition of relations AB and inclusion AB ⊆ C. Within the calculus of relations each relation A has a converse relation A^T and a complement relation ${\displaystyle {\bar {A}}.}$  Using these involutions, Augustus De Morgan and Ernst Schröder showed that ${\displaystyle (A,\ B,\ C)}$ is equivalent to ${\displaystyle ({\bar {C}},B^{T},{\bar {A}})}$  and also equivalent to ${\displaystyle (A^{T},\ {\bar {C}},\ {\bar {B}}).}$  The mutual equivalences of these forms, constructed from the ternary relation (A, B, C), are called the Schröder rules.^[1]

References

1. Gunther Schmidt & Thomas Ströhlein (1993) Relations and Graphs, pages 15–19, Springer books

Further reading

• Myers, Dale (1997), "An interpretive isomorphism between binary and ternary relations", in Mycielski, Jan; Rozenberg, Grzegorz; Salomaa, Arto (eds.), Structures in Logic and Computer Science, Lecture Notes in Computer Science, vol. 1261, Springer, pp. 84–105, doi:10.1007/3-540-63246-8_6, ISBN 3-540-63246-8
• 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