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((more citations needed|date=December 2009))
In [[mathematics]], a '''ternary relation''' or '''triadic relation''' is a [[finitary relation]] in which the number of places in the relation is three.  [[wikt:ternary|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 [[Collinearity|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 [[Incidence (geometry)|incident]] with) the line.

== Examples ==

=== Binary functions ===
((further|Graph of a function|Binary function))

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'',&nbsp;''b'') in&nbsp;''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 ===
((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 ((nowrap|1=''A''<sup>3</sup> = ''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 ===
((Main article|Betweenness relation))
((Expand section|date=May 2011))

=== Ternary equivalence relation ===
((Main article|Ternary equivalence relation))
((Expand section|date=August 2020))

=== Congruence relation ===
((Main article|Congruence modulo m))

The ordinary congruence of arithmetics
: <math> a \equiv b \pmod{m}</math>
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 relation]]s between the ''a'' and the ''b'', indexed by the [[Modular arithmetic|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'' ((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 relation]]s ''A'', ''B'', and ''C'' on a set, a ternary relation ((nowrap|(''A'', ''B'', ''C''))) can be defined using [[composition of relations]] ''AB'' and [[inclusion (set theory)|inclusion]] ((nowrap|''AB'' ⊆ ''C'')). Within the [[calculus of relations]] each relation ''A'' has a [[converse relation]] ''A''<sup>T</sup> and a complement relation ((overline|''A'')). Using these [[involution (mathematics)|involution]]s, [[Augustus De Morgan]] and [[Ernst Schröder (mathematician)|Ernst Schröder]] showed that ((nowrap|(''A'', ''B'', ''C''))) is equivalent to ((nowrap|(((overline|''C'')), ''B''<sup>T</sup>, ((overline|''A''))))) and also equivalent to ((nowrap|(''A''<sup>T</sup>, ((overline|''C'')), ((overline|''B''))))). The mutual equivalences of these forms, constructed from the ternary relation ((nowrap|(''A'', ''B'', ''C''),)) are called the [[composition of relations#Schröder rules|Schröder rules]].<ref>((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]] ))</ref>

== References ==
((reflist))

== Further reading ==
((refbegin))
* ((citation
  |last1=Myers |first1=Dale
  |year=1997
  |title=Structures in Logic and Computer Science
  |chapter=An interpretive isomorphism between binary and ternary relations
  |editor1-last=Mycielski |editor1-first=Jan
  |editor2-last=Rozenberg |editor2-first=Grzegorz
  |editor3-last=Salomaa |editor3-first=Arto 
  |series=Lecture Notes in Computer Science |volume=1261
  |pages=84–105
  |publisher=Springer
  |isbn=3-540-63246-8 |doi=10.1007/3-540-63246-8_6
  ))
* ((citation
  |last1=Novák |first1=Vítězslav
  |year=1996
  |title=Ternary structures and partial semigroups
  |journal=Czechoslovak Mathematical Journal
  |volume=46 |issue=1
  |pages=111–120
  |hdl=10338.dmlcz/127275
  ))
* ((citation
  |last1=Novák |first1=Vítězslav
  |last2=Novotný |first2=Miroslav
  |year=1989
  |title=Transitive ternary relations and quasiorderings
  |journal=Archivum Mathematicum
  |volume=25 |issue=1–2
  |pages=5–12
  |hdl=10338.dmlcz/107333
  ))
* ((citation
  |last1=Novák |first1=Vítězslav
  |last2=Novotný |first2=Miroslav
  |year=1992
  |title=Binary and ternary relations
  |journal=Mathematica Bohemica
  |volume=117 |issue=3
  |pages=283–292
  |hdl=10338.dmlcz/126278
  ))
* ((citation
  |last1=Novotný |first1=Miroslav
  |year=1991
  |title=Ternary structures and groupoids
  |journal=Czechoslovak Mathematical Journal
  |volume=41 |issue=1
  |pages=90–98
  |hdl=10338.dmlcz/102437
  ))
* ((citation
  |last1=Šlapal |first1=Josef
  |year=1993
  |title=Relations and topologies
  |journal=Czechoslovak Mathematical Journal
  |volume=43 |issue=1
  |pages=141–150
  |hdl=10338.dmlcz/128381
  ))
((refend))

((DEFAULTSORT:Ternary Relation))
[[Category:Mathematical relations]]

[[ru:Тернарное отношение]]

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