Explanation of the form of the summation notation used in set covering and other similar problems

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Does the colon appearing at the bottom of the summation notation reads as "such that?"

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Is there an alternate way of writing the notation in more familiar forms?

Thank you.

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With $S$ in $T$ and $e$ in $S$ or with $S$ as an element of $T$ and $e$ as an element of $S$ or something along those lines depending on what kind of objects $S$ $T$ and $e$ are.