Elements with Support in Ideal form Submagma of Direct Product

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

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

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

Let $T = \set {s \in S: \map {\operatorname {supp} } s \in J}$ where $\operatorname{supp}$ denotes support.

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

Also see

 * Elements of Finite Support form Submagma of Direct Product