Definition:Support of Element of Direct Product

Definition
Let $\left({S_i, \circ_i}\right)_{i \mathop \in I}$ be a family of algebraic structures with identity.

Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be their direct product.

Let $e_i$ be an identity of $S_i$ for all $i \in I$.

Let $m = \left({m_i}\right)_{i \mathop \in I} \in S$.

The support of $m$ is defined as:
 * $\operatorname {supp} \left\{ {i \in I: m_i \ne e_i}\right\}$

Finite Support
The element is said to have finite support its support is a finite set.

Also see

 * Elements with Support in Ideal form Submagma of Direct Product
 * Identity is Unique