In the book Roads to Infinity: The Mathematics of Truth and Proof, the author stated:
For each n, one has the n-string braid group $B_n$, consisting of all the n-string braids under the braid product operation.
I don't understand what this sentence means, because I don't know it means for a braid to be "under" the braid product operation. Google didn't turn up anything, and the context doesn't seem to help. So, what does it mean for an object to be under an operation?
In this context, the term "under the operation" means that $B_n$ is a group when we consider it with the given operation.
More generally, if $X$ is a set and we say that we have the group (or some other structure) $X$ under the operation $*$ (or some collection of operations), then we mean that we are putting the operation $*$ (or the collection of operations) on $X$ and this gives a group (or some other structure).
The word "under" has the meaning of "provided with". For example
The set of integers $\Bbb{Z}$ is a group under addition, while it is not a group under multiplication.
You may find the exposition An Introduction to Braid Theory by Maurice Chiodo to be of use. He talks about the Braid Group on page 20, and the definition of a braid product is given on the same page as Definition 3.1. For those not interested in opening the file:
