Definition:Composite (Abstract Algebra)

Definition

Let $\left({S, \circ}\right)$ be an algebraic structure.

For each ordered $n$-tuple $\left({a_1, a_2, \ldots, a_n}\right) \in S^n$, the composite of $\left({a_1, a_2, \ldots, a_n}\right)$ for $\circ$ is the value at $\left({a_1, a_2, \ldots, a_n}\right)$ of the $n$-ary operation defined by $\circ$.

This composite is normally denoted $\circ_n \left({a_1, a_2, \ldots, a_n}\right)$.

If the tuple is empty, then the composite is assigned the value of the identity of the operation (if this is a structure with an identity, that is):

$\circ_0 \left({\varnothing}\right) = e_S$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 18$