Elements with Support in Ideal form Submagma of Direct Product

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

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

Let $J \subset I$ be an ideal of $I$.

Let $T = \left\{ {s \in S: \operatorname{supp} \left({s}\right) \in J}\right\}$ where $\operatorname{supp}$ denotes support.

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

Also see

 * Elements of Finite Support form Submagma of Direct Product