# Naming a Lemma In Math Thesis

In my pure math thesis I have an important lemma and I want to name it. The statement is quite technical so I will omit quite a lot of details.

I want to call this lemma “the adding lemma”. My supervisor thinks it should be “the addition lemma”.

I feel like it should be “adding” because there are other results like the pumping lemma (formal languages) or the lifting lemma (polynomial algebra). Both pumping and lifting have -ing suffix and they somehow describe the “action” the lemma performs (pumping lemma: a string can be pumped—that is, have a middle section of the string repeated an arbitrary number of times) (lifting lemma: ...root can be lifted to a unique root modulo...).

The lemma in my thesis basically allows us to add objects to a structure. I will disregard plenty of details. If I have a formula which satisfies something, the lemma asserts there is exactly one witness which satisfies something. Thus we can add the object expressed by the formula to our structure.

I feel like “the adding lemma” follows the same logic as the names of the pumping and the lifting lemma.

• Personal opinion: the "extension lemma" or the "existence lemma" sound to me like a better fit than either of these options. Both the "adding lemma" and the "addition lemma" sound a bit strange to me, although it's hard to pinpoint exactly why. Mar 14, 2023 at 19:53
• Lemma 3 very pretty, and the converse pretty too. But the proof of the poor lemma is impossible to do. Mar 14, 2023 at 20:29

Generally, when a theorem is about a particular action or operation, it's named with the noun form of that operation.

The verbs "pump" and "lift" don't have special noun forms, at least not that I can think of, and so English defaults to using the verbal noun (aka gerundial noun) to describe the corresponding actions. These are usually formed with "-ing" like the present participle, so we have pumping lemma and lifting lemma.

The verb "add" does have such a form, namely "addition". (Apparently this is called a deverbal noun). So when talking about this action as a noun, we would normally prefer that form.

As other examples, we would say existence lemma instead of existing lemma, extension theorem instead of extending theorem, and so on.

(As another example, if you rewrite this answer replacing "action" by "acting", and "operation" by "operating", it sounds very strange.)

Addition would seem to imply a numerical action. Adding can mean that or other things; adding carrots to the soup. But there are other possibilities with other words.

How about "insertion lemma" since that seems more descriptive, though there is already something with that name: https://arxiv.org/abs/2203.13543

But you might think a bit more about the "best" name, not the one that came to mind first.

And, fighting with your supervisor is seldom the best career move. It really only matters if this lemma is significant enough that the idea can be generally applied and is likely to be adopted by others.