Elements of Finite Support form Submagma of Direct Product

Theorem

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

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

Let $T$ be the subset of elements of $S$ whose support is finite:

$T = \set {s \in S: \map \supp s \text{ is finite} }$

Then $T$ is a submagma of $S$.

Proof

From Finite Subsets form Ideal, the set of finite subsets of $I$ form an ideal of $I$.

From Elements with Support in Ideal form Submagma of Direct Product, $T$ is a submagma of $S$.