Problem in Math Writing

Question

I have a problem in math writing.

Consider $a_i$ with $1\leq i \leq n$ are some elements over a field.

My Question: Which of the following statements is correct?

A) Let $a_i$'s satisfy the following condition.

B) Let $a_i$'s satisfies the following condition.

Thanks for your suggestions.

Answer 1

I might say something like

Suppose the $a_i$ satisfy ...

or

Suppose $n$ field elements $\{a_i\}$ ( $1 \le i \le n$ ) satisfy ...

In any case the apostrophy s on $a_i$'s feels wrong to me. You do want the plural "satisfy" since you are talking about a sequence of several elements (plural).

Answer 2

In general, it is inappropriate to use an apostrophe-s to pluralize English nouns. However, the AMS style guide (which seems to be an appropriate style guide, given the context) instructs us to use the apostrophe-s to make variable names plural (see page 94). Hence that part of the construction is appropriate.

That being said, the heart of the question seems to be about the correct conjugation of the verb "to satisfy." The verb should be conjugated in order to agree with the subject of the sentence. In this case, the subject of the sentence is "the $a_i$'s", which demands a third person plural conjugation, i.e.

The $a_i$'s satisfy...

That being said, I agree with Ethan Bolker's assessment that the phrasing is a little awkward, in part because of the apostrophe-s. An equally readable alternative might be

The $a_i$ satisfy...

Personally, I might prefer a phrasing which emphasizes the common properties of the field elements in the set. Hence I would probably write a sentence more like

Let $k$ be a field and let $\{a_i\}_{i=1}^{N} \subseteq k$ be a collection which satisfies the condition...