Nonequivalent definitions in Mathematics

I would like to ask if anyone could share any specific experiences of discovering nonequivalent definitions in their field of mathematical research.

By that I mean discovering that in different places in the literature, the same name is used for two different mathematical objects. This can happen when the mathematical literature grows quickly and becomes chaotic and of course this could be a source of serious errors. I have heard of such complaints by colleagues, mostly with respect to definitions of various spaces and operators, but I do not recall the specific (and very specialised) examples.

My reason for asking is that I'm currently experimenting with verification using proof assistants and I'd like to test some cases that might be a source of future errors.

EDIT: Obviously, I would be interested to see as many instances of this issue as possible so I would like to ask for more answers. Also, it would be even more helpful if you could provide references.

• A classic example is "compact", which can mean either "Hausdorff and every open cover has a finite subcover", or just "every open cover has a finite subcover". Nov 22, 2017 at 19:12
• @ArturoMagidin This leads to the classic dialogue "The space $X$ is quasicompact, so..." "What does quasicompact mean?" "It means compact." Nov 22, 2017 at 19:35
• In my experience at least, the only people who think "compact" includes Hausdorff are algebraic geometers. I've never heard a topologist use "compact" to mean anything other than "every open cover has a finite subcover". Nov 23, 2017 at 3:29
• @JohnPardon: there is also a cultural thing here. In France, it is standard to include "Hausdorff" in the definition of a compact space. Nov 23, 2017 at 14:49
• @Taladris, maybe in france, everyone's an algebraic geometer Nov 29, 2017 at 0:54

A partition of a set.

A partition of an integer.

A partition of an interval (in defining Riemann and Riemann–Stieltjes integrals.

A partition of unity.

A partition of a sum of squares in multiple regression, multi-way analysis of variance, experiments with both random effects and fixed effects, the partition of residual sums of squares into pure-error and lack-of-fit sums of squares, etc.

A partition of a matrix.

Quotition versus partition.

• $$6\div2=3$$ because $$3$$ is how many $$2$$s must be added to get $$6$$ (quotition).
• $$6\div2=3$$ because when $$6$$ is a sum of $$2$$ equal terms, then each term is $$3$$ (partition).

When is a function concave? When is it convex? Do you determine this by looking at the graph "from above", or "from below"?

• I can never remember which way this goes without looking it up, but are there really varying conventions in upper-level mathematics (as opposed to calculus textbooks)? I thought one spoke exclusively about convex functions in, say, measure theory. Nov 29, 2017 at 18:59
• @LSpice Instead of looking it up, you could just look up (i.e., look at the graph from below) and you'd have the standard convention. Dec 1, 2017 at 23:55
• @AndreasBlass, hah, very nice! Indeed, once I know the answer I can justify it ex post facto, but unfortunately that doesn't help me personally remember, as @‍RodrigoA.Pérez puts it, whether to look "from above" or "from below". Maybe thinking of it as looking up will help. Dec 2, 2017 at 1:39
• The definition of "convex function" is universally agreed and standard, luckily Jun 12, 2019 at 12:11

The wide range of choices in the definition of an automorphic form is particularly annoying. Depending on the purposes, it could be a meromorphic function fully invariant by a certain discrete group of transformation, a holomorphic function almost-invariant, a differential form, a subrepresentation of an $$L^2$$ space, a classical or an adelic object, a solution to a partial differential equation, etc.

These are sometimes related, sometimes definitely different, and the lack of vocabulary consistency in such an active field requires some care when dealing with object (leading authors to usually restate a precise definition of the objects they are dealing with, or at least quoting the literature associated with the paradigm they have chosen.

Let $$X_1,X_2,X_3,\ldots$$ be independent random variables.

Sometimes it is said that a stopping time for this random process is a random variable $$T$$ for which the truth value of $$T=n$$ (for $$n=1,2,3,\ldots$$) is determined by the values of $$X_1,\ldots,X_n.$$

And sometimes it is said that a stopping time for this random process is a random variable $$T$$ for which the each of the events $$\big[T=n\big]$$ for $$n=1,2,3,\ldots$$ is independent of the sequence $$X_{n+1}, X_{n+2}, X_{n+3},\ldots\,.$$

The second definition is more inclusive than the first.