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 {{nowrap|A × B}} of some sets A and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product {{nowrap|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 {{nowrap|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 {{nowrap|A × B}} an element f(a, b) in C. Therefore, its graph consists of pairs of the form {{nowrap|((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 {{nowrap|(a, b, f(a, b))}}, satisfying {{nowrap|a in A}}, {{nowrap|b in B}}, and {{nowrap|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 {{nowrap|1=A³ = A × A × A}}, by stipulating that {{nowrap|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 {{nowrap|1=A = {{mset| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }}}} represents the hours on a clock face, then {{nowrap|R(8, 12, 4)}} holds and {{nowrap|R(12, 8, 4)}} does not hold.

= Betweenness relations =

= Ternary equivalence relation =

= Congruence relation =

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 {{nowrap|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 {{nowrap|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 {{nowrap|(A, B, C)}} can be defined using composition of relations AB and inclusion {{nowrap|AB ⊆ C}}. Within the calculus of relations each relation A has a converse relation A^T and a complement relation {{overline|A}}. Using these involutions, Augustus De Morgan and Ernst Schröder showed that {{nowrap|(A, B, C)}} is equivalent to {{nowrap|({{overline|C}}, B^T, {{overline|A}})}} and also equivalent to {{nowrap|(A^T, {{overline|C}}, {{overline|B}})}}. The mutual equivalences of these forms, constructed from the ternary relation {{nowrap|(A, B, C),}} are called the Schröder rules.{{citation |last1=Schmidt |first1=Gunther |author-link1=Gunther Schmidt |last2=Ströhlein |first2=Thomas |year=1993 |title=Relations and Graphs |pages=15–19 |publisher=Springer books }}