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What does the $\times_i$ mean? For reference, $M_i$ is a set.

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mitmath514
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    Direct product. – anomaly Dec 20 '19 at 03:15
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    To elaborate on anomaly's comment: this means the product over all $i$ of the $M_i.$ Explicitly, if $I$ is some set and for each $i\in I$ you have a set $M_i,$ then $\times_i M_i$ is the product of all the $M_i.$ For example, if $I = {0,1,2,3}$ and you have sets $M_0, M_1, M_2, M_3,$ then $$\times_i M_i := M_0\times M_1\times M_2\times M_3.$$ However, it could be helpful for you to provide more context as to where you're seeing this notation. – Stahl Dec 20 '19 at 03:24

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Direct product. Think $M_1 \times M_2 \times \dots \times M_n$. The indexing set does not necessarily need to be finite. The more common notation for this is a capital pi: $$ \prod_{i \in I} M_i. $$

If the indexing set is understood from the context, e.g. $I = \{1,\dots,n\}$ is a common indexing set, then the "$\in I$" part of the notation may be dropped. We might also write

$$ \prod_{i = 1}^n M_i. $$

If you haven't seen direct products before, they're just tuples. For instance $\mathbb{R} \times \mathbb{R} \times \mathbb{C}$ is the set of all tuples $(x,y,z)$ where $x$ and $y$ are real and $z$ is complex.

For example, a function $f : \mathbb{R} \times \mathbb{Z} \times \mathbb{Z} \to \mathbb{R}$ would be declared in a programming language like

double f(double x, int y, int z);
Trevor Gunn
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