Triple System

In algebra, a triple system is a vector space $V$ over a field $K$ with a ternary operation which is a $K$-trilinear mapping $V \times V \times V \rightarrow V$. They are used in the theory of non-associative algebras and appear in the construction of Lie algebras. Examples include Freudenthal–Kantor triple systems, Jordan triple systems, Lie triple systems, Anti-Lie triple systems and Allison-Hein triple systems.

In combinatorics, a triple system is a class of block design with blocks of size $3$. Examples include Steiner triple system and Kirkman triple system.