Editing Ternary operation (section)

==Examples==
[[File:Volledige vierhoek.PNG|thumb|right|Given ''A'', ''B'' and point ''P'', geometric construction yields ''V'', the projective harmonic conjugate of ''P'' with respect to ''A'' and ''B''.]]

The [[function (mathematics)|function]] <math>T(a, b, c) = ab + c</math> is an example of a ternary operation on the [[integer]]s (or on any structure where <math>+</math> and <math>\times</math> are both defined). Properties of this ternary operation have been used to define [[planar ternary ring]]s in the foundations of [[projective geometry]].

In the [[Euclidean plane]] with points ''a'', ''b'', ''c'' referred to an origin, the ternary operation <math>[a, b, c] = a - b + c</math> has been used to define [[free vector]]s.<ref>Jeremiah Certaine (1943) [https://www.ams.org/journals/bull/1943-49-12/S0002-9904-1943-08042-1/S0002-9904-1943-08042-1.pdf The ternary operation (abc) = a b<sup>−1</sup>c of a group], [[Bulletin of the American Mathematical Society]] 49: 868–77 {{MR|id=0009953}}</ref> Since (''abc'') = ''d'' implies ''b'' – ''a'' = ''c'' – ''d'', the [[directed line segment]]s ''b'' – ''a'' and ''c'' – ''d'' are [[equipollence (geometry)|equipollent]] and are associated with the same free vector. Any three points in the plane ''a, b, c'' thus determine a [[parallelogram]] with ''d'' at the fourth vertex.

In [[projective geometry]], the process of finding a [[projective harmonic conjugate]] is a ternary operation on three points. In the diagram, points ''A'', ''B'' and ''P'' determine point ''V'', the harmonic conjugate of ''P'' with respect to ''A'' and ''B''. Point ''R'' and the line through ''P'' can be selected arbitrarily, determining ''C'' and ''D''. Drawing ''AC'' and ''BD'' produces the intersection ''Q'', and ''RQ'' then yields ''V''.

Suppose ''A'' and ''B'' are given sets and <math>\mathcal{B}(A, B)</math> is the collection of [[binary relation]]s between ''A'' and ''B''. [[Composition of relations]] is always defined when ''A'' = ''B'', but otherwise a ternary composition can be defined by <math>[p, q, r] = p q^T r</math> where <math>q^T</math> is the [[converse relation]] of ''q''. Properties of this ternary relation have been used to set the axioms for a [[heap (mathematics)|heap]].<ref>Christopher Hollings (2014) ''Mathematics across the Iron Curtain: a history of the algebraic theory of semigroups'', page 264, History of Mathematics 41, [[American Mathematical Society]] {{ISBN|978-1-4704-1493-1}}</ref>

In [[Boolean algebra]], <math>T(A,B,C) = AC+(1-A)B</math> defines the formula <math>(A \lor B) \land (\lnot A \lor C)</math>.