# 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

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

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 *A*^{3} = *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.

### Schröder rules

Given homogeneous relations *A*, *B*, and *C* on a set, a ternary relation 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 Using these involutions, Augustus De Morgan and Ernst Schröder showed that is equivalent to and also equivalent to The mutual equivalences of these forms, constructed from the ternary relation (*A, B, C*), are called the Schröder rules.[1]

## References

- 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,**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