Ternary operation

In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A.

In computer science, a ternary operator is an operator that takes three arguments as input and returns one output.

The function ${\displaystyle T(a,b,c)=ab+c}$ is an example of a ternary operation on the integers (or on any structure where ${\displaystyle +}$ and ${\displaystyle \times }$ are both defined). Properties of this ternary operation have been used to define planar ternary rings in the foundations of projective geometry.

In the Euclidean plane with points a, b, c referred to an origin, the ternary operation ${\displaystyle [a,b,c]=a-b+c}$ has been used to define free vectors. Since (abc) = d implies b – a = c – d, the directed line segments b – a and c – d are 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 ${\displaystyle {\mathcal {B}}(A,B)}$ is the collection of binary relations between A and B. Composition of relations is always defined when A = B, but otherwise a ternary composition can be defined by ${\displaystyle [p,q,r]=pq^{T}r}$ where ${\displaystyle q^{T}}$ is the converse relation of q. Properties of this ternary relation have been used to set the axioms for a heap.

In Boolean algebra, ${\displaystyle T(A,B,C)=AC+(1-A)B}$ defines the formula ${\displaystyle (A\lor B)\land (\lnot A\lor C)}$.

In computer science, an operator is a ternary operator if it takes three arguments (or operands).