subordinate clauses - Are “where” subclauses acceptable?

In mathematical texts, but not only there, one often first names an object and then explains what it is in a “where” clause, for example:

“Let x and y be any two numbers from a set M of natural numbers,
where M is given by the following property ...”

Is such a construction acceptable? Are there better choices instead of “where” for the same overall structure?

Clearly, overly long sentences are to be avoided. Whenever possible, one should try to decouple the two definitions. Here an alternative would be to first introduce M and then talk about x and y, but if we are mainly interested in x and y, first digressing to M feels like disrupting the flow of the text.

Simply splitting the above into two sentences creates, in my humble opinion, the impression that M should be known already, causing the reader to look back for its definition.

Answer

Yes. Subordinate clauses that start with "where" are more than acceptable; they're completely proper, both grammatically and semantically. By placing the "where" and its clause between commas, one is creating an non-restrictive clause, which is to say that one is providing parenthetical information that is not necessary to the operation of the sentence, information that one could scoop out of the sentence and still have the sentence make sense and mean the same thing. Essentially, it's proverbially putting one's hand by one's mouth and saying, "By the way..."

Are there better choices instead of this available? That's a judgment call and entirely depends on the situation. Clearly, in mathematics, as you've demonstrated, the overall opinion is that this particular construction is the best way to convey such information, a way that is clear and concise and so has become prolific in writing mathematical problems.