Elements with Support in Ideal form Submagma of Direct Product

Theorem
Let $(S_i,\circ_i)_{i\in I}$ be a family of algebraic structures with unity.

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

Let $J\subset I$ be an ideal.

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

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

Also See

 * Elements of Finite Support form Substructure of Direct Product