# Triple system

From Encyclopedia of Mathematics

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.

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Triple system.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Triple_system&oldid=42986