Ternary relation
Template:More citations needed 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 Template:Nowrap of some sets A and B, so a ternary relation is a set of triples, forming a subset of the Cartesian product Template:Nowrap 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 Template:Nowrap in two variables, mapping two values from sets A and B, respectively, to a value in C associates to every pair (a,b) in Template:Nowrap an element f(a, b) in C. Therefore, its graph consists of pairs of the form Template:Nowrap. 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 Template:Nowrap, satisfying Template:Nowrap, Template:Nowrap, and Template:Nowrap
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 Template:Nowrap, by stipulating that Template:Nowrap 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 Template:Nowrap represents the hours on a clock face, then Template:Nowrap holds and Template:Nowrap does not hold.
Betweenness relations
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Ternary equivalence relation
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Congruence relation
The ordinary congruence of arithmetics
which holds for three integers a, b, and m if and only if m divides Template:Nowrap, 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 Template:Nowrap 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 Template:Nowrap can be defined using composition of relations AB and inclusion Template:Nowrap. Within the calculus of relations each relation A has a converse relation AT and a complement relation Template:Overline. Using these involutions, Augustus De Morgan and Ernst Schröder showed that Template:Nowrap is equivalent to Template:Nowrap and also equivalent to Template:Nowrap. The mutual equivalences of these forms, constructed from the ternary relation Template:Nowrap are called the Schröder rules.[1]