Definition:Support of Element of Direct Product

Definition

Let $\family {\struct {S_i, \circ_i} }_{i \mathop \in I}$ be a family of algebraic structures with identity.

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

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

Let $m = \family {m_i}_{i \mathop \in I} \in S$.

The support of $m$ is defined as:

$\supp \set {i \in I: m_i \ne e_i}$

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Finite Support

The element is said to have finite support if and only if its support is a finite set.

Also see

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