# 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 ƒ: A × BC 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 ƒ(ab) 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)), satisfying a in A, b in B, and ƒ(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 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.

### Congruence relation

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

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

A ternary relation ${\displaystyle (A,\ B,\ C)}$ can be defined in the category of binary relations using composition of relations AB and inclusion ABC. Within the calculus of relations each relation A has a converse relation AT 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