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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 A = 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

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Ternary equivalence relation

Main article: Ternary equivalence relation

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Congruence relation

Main article: Congruence modulo m

The ordinary congruence of arithmetics

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 Γ ⊢ e:σ indicates that e is a term of type σ in context Γ, 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 (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 and a complement relation A. Using these involutions, Augustus De Morgan and Ernst Schröder showed that (A, B, C) is equivalent to (C, B, A) and also equivalent to (A, C, B). The mutual equivalences of these forms, constructed from the ternary relation (A, B, C), are called the Schröder rules.

References

Schmidt, Gunther; Ströhlein, Thomas (1993), Relations and Graphs, Springer books, pp. 15–19