# What are the most overloaded words in mathematics? [closed]

This is community wiki. In each answer, please list one word at the top and below that list as many different meanings of that word in mathematics as you can think of, preferably with links or definitions. ("Adjective" and "adjective noun" count as the same adjective.) People should edit previous answers as appropriate.

(This is mostly just for fun, but I'm also curious if there have been successful attempts to rename concepts that involve overused words.)

Edit: I may have been slightly unclear about the intent of this question.

• When I say "overused" I don't mean "used too often," I mean "used in too many different ways." So I'll change the title of the question to reflect this.
• Different concepts named after the same mathematician, while potentially confusing, are understandable.
• I mostly had in mind adjectives that get recycled in different disciplines of mathematics. Different uses of the same noun tend to be less confusing, e.g. the example of "space" below. I think it's good to be intentionally vague about what we consider a "space."
• I'm -1'ing this because I think it's a rather uninteresting question. Dec 1, 2009 at 12:47
• I'm +1'ing it because the answers are useful. Dec 1, 2009 at 18:54
• Here's a harder and possibly even more useful project: grouping the multiple answers into related meanings. Is that worth it? Dec 3, 2009 at 6:13
• I don't like the question because it invites too many bad answers. I (-1)ed a bunch of answers for the following reason. Just because an adjective is used with many different nouns or a word is used frequently doesn't mean that it's overloaded; the use may be perfectly consistent. I realize the the question was edited, so this is a slightly unfair criticism. Dec 14, 2009 at 16:55
• ProofWiki's list of definition disambiguation pages has some nice examples Dec 30, 2017 at 12:43

Regular. To start off:

The regular representation of a group $G$ over a field $k$ is the action on $k[G]$ given by group multiplication.

A topology is regular if a closed set and a point not in that set can be separated by disjoint open sets.

A point $\zeta_0$ on the boundary of a domain in $\mathbb C$ is called regular if there exists a subharmonic barrier function $b(z)$ defined within $D$ near $\zeta$. This may not be the standard definition but Gamelin's complex Analysis defines it as a subharmonic function $\omega(z)$ on $\{|z-\zeta_0|<\delta\}\cap D$ which is negative everywhere, tends to 0 at $\zeta_0$, but $\limsup(\omega(z))<0$ as $z$ tends to any other boundary point of $D$ within distance $\delta$ of $\zeta_0$.

I've borrowed/paraphrased the following from the Wikipedia disambiguation page but removed a couple that either are not too relevant to pure math or qualify the "regularity" more. Feel free to put them in too.

Regular cardinal, a cardinal number that is equal to its cofinality

Regular category

Regular element, and regular sequence and regular immersion.

Regular code, an algebraic code with a uniform distribution of distances between codewords

Regular graph, a graph such that all the degrees of the vertices are equal

The regularity lemma, which has nothing to do with regular graphs

Regular polygon, and regular polyhedron

Regular prime: a prime $p$ that does not divide the class number of the $p$th cyclotomic field $\mathbb Q[\zeta_p]$.

Regular surface in algebraic geometry

Regularity of an elliptic operator

JS Milne's comment: A regular map is a morphism of algebraic varieties.

Regular value of a differentiable map

Regular ring (Note: this definition can be made noncommutative. A right noetherian ring R is said to be right regular if every finitely generated right R-module has finite global dimension. See Lam's Lectures in Modules and Rings, Section 5G.)

(von Neumann) Regular ring

Regular language, a language that can be accepted by a finite state machine.

Absolutely regular is a synonym for $\beta$-mixing in stochastic processes.

Regular matroid, a matroid which is representable over every field. In this sense, all graphs are regular (their cycle matroids are regular), which has nothing to do with regular graphs.

• "Strong" and "weak" are also used quite a bit. I would say that these are not overloaded a la programming languages in the same "strong" sense that "regular" is. Dec 1, 2009 at 8:06
• I suggest the word "denominal" for "regular element" in a ring, because it makes for a nice denominator :) And then the related terms "denominal sequence" instead of "regular sequence", "denominal immersion" instead of "regular immersion"... Dec 1, 2009 at 8:09
• regular map: morphism of algebraic varieties Dec 1, 2009 at 8:12
• All right, I've tried to add those and am now done editing. Please do add more and organize or paraphrase as you see fit! I am not familiar with all of the terms; that's why it's community wiki. Dec 1, 2009 at 8:21
• I've added two notions of "regular ring." I would add a third notion of Artin-Schelter regular ring, if I was familiar enough with the concept. Dec 1, 2009 at 23:26
• en.wikipedia.org/wiki/Normal_(mathematics) has the whole list! Dec 1, 2009 at 8:57
• What makes this worse is that it often means something different that "regular." If somebody can teach me a way to remember the difference between a regular space and a normal sapce, I'd be extremely grateful. Dec 1, 2009 at 18:56
• @John: Sometimes a picture is all it takes: brownsharpie.courtneygibbons.org/?p=395 Dec 1, 2009 at 21:37
• @Andrew: Now I'm curious, but I can't get that to load (internal server error). Do you know what's wrong? Dec 3, 2009 at 6:22

The most overloaded word in mathematics is the empty word. The one that comes between $a$ and $b$ in $ab$, meaning multiplication. Or does it mean the binary operator in a more general monoid or group? Or one of the two binary operators in a ring? Or the action of a monoid or group on a set, or the action of the base ring on a module? (And if so, is it a left or right action?) Or the application of a function (or functor) on its argument? Or even three or four of these in one expression, or, even worse, two at the same time in the very same place, exploiting associativity to ensure the ambiguity is mostly harmless? Or one of countless other things?

Perfect

• A perfect integer is the sum of its proper divisors.
• A perfect complex is locally quasi-isomorphic to a bounded complex of finitely generated projective modules.
• A perfect field is a field whose algebraic extensions are all separable.
• A perfect square is a natural number of the form $n^2$ for some $n \in \mathbb{N}$.
• A perfect group is equal to its own commutator subgroup.
• A perfect set is a closed set with no isolated points.
• A perfect graph is a graph such that each induced subgraph's chromatic number is equal to its clique number.
• I never understood why people call it a perfect square, rather than a square. (frustrating)
Apr 19, 2010 at 22:21
• There are also perfect codes, which are those that attain the Hamming bound; see en.wikipedia.org/wiki/Hamming_bound#Perfect_codes Apr 20, 2010 at 3:55
• Perfect post.......LOL Apr 20, 2010 at 4:24
• There is also a perfect equilibrium in game theory. Nov 22, 2017 at 20:22
• @muad because then every non-negative real number would a square (or if you count complex numbers, every complex numbers). Nov 22, 2017 at 22:16

Simple

• Simple field extension
• Simple group
• Simple ring
• Simple module
• Simple algebra
• Simple graph
• Simple polygon
• Simple curve
• Simple zero
• Simple function
• Simple connectivity
• Simple root
• I disagree with this one as being "overuse" because all of the ones that I know about do have some connection to the ordinary meaning of the word "simple". Dec 2, 2009 at 8:34
• The word "overused" was changed to "overloaded" in the question, but I can see your point. Dec 2, 2009 at 9:58
• @Surb, do we need "simple root" on the list, when we already had "simple zero"? Nov 22, 2017 at 21:10
• @GerryMyerson Actually I was thinking more to a "simple eigenvalue" but simple root came out.
– Surb
Nov 22, 2017 at 21:22

If you happen to work on del Pezzo surfaces, don't make the mistake of standing in an airport security line talking about "blowing up a plane at eight points".

(Yes, this really happened, and ended happily, or at least not in Guantanamo.)

• This answers belongs to the mathematical jokes question not here. (A good joke though.) Apr 30, 2010 at 7:16

"Cauchy theorem"

Obvious

"'Obvious' is the most dangerous word in mathematics." -- E. T. Bell

Example: all examples You are using to answer this post are obvious.

• I think it's synonym "clearly" is even more dangerous. (At least, I rarely write "obviously", but I do write "clearly" all the time to justify statements that should be true, but I can't explain why very clearly). Dec 19, 2009 at 20:41
• I love when a famous mathematician says it's obvious and it takes me 5 pages to justify it. That's him just being too lazy to write it out and he makes you feel like an idiot for not using the Force to see it like he does. Apr 20, 2010 at 4:07

Hedgehog.

Just one use of this word in mathematics is "overuse".

• I got 143 hits on MathSciNet (that was in 'anywhere') and 127 on the arXiv. My favourite is "Hedgehogs of Hausdorff dimension one". Dec 1, 2009 at 15:19
• Really? I appreciate the use of funny words in mathematics. Dec 1, 2009 at 16:16
• Really? It should be used more often. "Wait, by 'hedgehog," do you mean a topological space, a Lie superalgebra, or a cohomology theory?" Dec 1, 2009 at 18:53
• I was thinking something similar about the word "coconut." No one wants to study a coconut or it's dual, the cococonut. Apr 20, 2010 at 15:10
• It's too late for me to work this out for myself, but would the natural morphism go "nut -> coconut" or "coconut -> nut"? Or, in other words, is a "coconut" a "nut" or is a "nut" a "coconut"? Enquiring minds want to know ... Apr 20, 2010 at 21:42

Natural

Very often I read things like "Now, it is natural to ask...", ""X is a natural generalization of Y" or "A natural question is..." when the problems are by no means natural, and the feeling of "naturalness" is only achieved post factum.

• To complicate matters, "natural" has a technical meaning in category theory that may or may not coincide with the colloquial use of the word. Dec 1, 2009 at 16:54

Separable

• Separation axioms ($T_0$,$T_1$, etc.)
• Separable space (countable dense subset)
• Separable differential equation
• Separable scheme (although analogous at least in spirit to the Hausdorff axiom)
• Separable field extensions / polynomials
• Separable subgroup (ie a subgroup that's closed in the profinite topology)
• Separable quantum state (it means mixed unentangled)
• Although these are all used in completely different ways, because they do all (at least, those that I know what they mean) actually link to the common notion of separation then I don't find them at all confusing. Dec 1, 2009 at 8:46
• I've never understood what having a countable dense subset has to do with the common notion of separation. Dec 2, 2009 at 2:48
• Well, for the real line (say) it means that every pair of distinct real numbers is separated by a rational number. Dec 2, 2009 at 3:01
• Also, there's separated subsets of a topological space: A and B are separated if they are respectively contained in disjoint open sets. A space X is disconnected iff it is the union of two separated sets, which is sometimes called a "separation" of X. (Munkres, for instance, uses this terminology.) Apr 20, 2010 at 14:26
• @Qiaochu: that shows why one wants to say something like “the reals are separated by the rationals”. But a space could just as well be separated in that sense by some uncountable set — so this would justify a terminology like countably separable,κ-separable, etc., but not why separable is used just for the countable case. Dec 19, 2010 at 18:01

Admissible, a colorless synonym for "which lies in the class of objects we're studying".

• But, like "nice" mentioned above, a good term to use if you eventually want this class of objects to be named after you :) Do you have some more colorful alternatives that you prefer? Apr 20, 2010 at 14:31
• Teleman classes. :) Apr 20, 2010 at 22:32
• This is so true. Admissible is the least useful adjective ever. Apr 30, 2010 at 4:59

Deep

I'm not sure whether it has one meaning or zero. Either way, I think that it is deeply overused.

trivial

Besides being a synonym to 'obvious', like in 'the proof is trivial', it has the meaning of 'shallow' ('the question is trivial') and moreover denotes a bunch of mathematical notions:

trivial group

trivial representation

trivial topology

trivial solution (in ODE/PDE)

etc.

Sometimes it produces confusion as it is not quite clear which sort of triviality is meant.

• I'm with you. And worse, for trivial topology, I've heard people mean both "indiscrete topology" and "contractible" because contractible spaces, of course, are "topologically trivial" Dec 1, 2009 at 23:30
• I think this should be split. I agree on the casual use (first sentence) but disagree on the others because they do connect to the ordinary meaning of the word "trivial" and so I don't regard them as "overuse". Charles' point about "trivial topology" is more a question of ambiguous use than overuse. Dec 2, 2009 at 8:37

Elliptic.

Of course, these are not entirely independent, but there are several distinct meanings involved.

• I'll let someone else do "hyperbolic" :-) Dec 1, 2009 at 9:28
• Parabolics are more mysterious! ;-) E.g. consider parabolic subroup/subalgebra. Dec 2, 2009 at 2:05
• Go for it... . Dec 2, 2009 at 3:34

Primitive.

• Primitive polynomial (in the sense of finite field theory, namely minimal polynomial of field generator)
• Primitive polynomial (in the sense of ring theory, namely gcd of coefficients is 1)
• Primitive element (and primitive extension)
• Primitive function (antiderivative)
• Primitive permutation group (no non-trivial equivalence relation preserved)
• Primitive polytope (rarely used, I think).
• (left) Primitive ring
• Primitive recursion (in logic and complexity theory)
• Primitive (nonnegative) matrix

Nice

Mostly because it is such a local word - 'such and such is called 'nice' if...'

Segal once formally defined a 'nice simplicial space' - these days called simplicial spaces satisfying the Segal condition, a very sensible move.

• So, if you want to have something named after you ... ;-) Dec 22, 2009 at 20:11
• Good is used in the same way... Apr 21, 2010 at 10:54

The word "stable" is used in many different contexts. Also "elementary" has many usages. The word "lattice" has two entirely different meanings which are ar time confusing. So is the word "field".

I have two more: "deterministic" refers sometimes as "not random" and it is also a central concept in computational complexity where "non deterministic" has another meaning (very different from random). The word "classical" is used in various confusing ways.

• Can you comment about where "stable" is used confusingly? The only applications I know of are differential equations and homotopy theory, and I never thought these two notions could be confused somehow ... Dec 15, 2009 at 13:05
• Interestingly, both pairs of meanings you mention are in fact distinguished in German: Verband / Gitter and Körper / Feld. Since I assume the origin is German, I wonder why this got lost in translation. "Bond" could've been used for lattices in logic, and, obviously, "body" for fields in algebra. Too late now. Apr 20, 2010 at 18:09

Complete/Completion

complete metric spaces,

complete measure spaces,

completing a ring at an ideal,

complete graph

complete category

complete lattice

and many more uses (a lot in computation theory/logic) at

http://en.wikipedia.org/wiki/Completeness

• I disagree that these are "overuse" as they do all link to the idea of "nothing missing" (at least, the ones that I know about do). Dec 2, 2009 at 8:38
• @AndrewStacey, it seems that that logic would rule out most of the answers; for example, one could argue that (to pick one that's relevant to me) the word 'good' is not overused because, in all contexts, it refers to some desireable quality. Feb 13, 2014 at 14:54

Uniform. Most of these do have the intuitive sense of "being locally the same everywhere," but by no means all of them, and their sheer number gets pretty confusing.

"Let" (which does not meet the 15 character minimum)

• I agree with Deane. It's too easy to write prose of the form "Let n be a natural number. Let m = n^2 + 1." The two uses of "let" mean quite different things. While there's no danger of misinterpretation, it has a slightly jarring effect, and it's one of those things that makes the text just a little bit less transparent, just a little bit less easy to read. It would be better to change the second "let" to something like "put" or "write". Dec 1, 2009 at 15:02
• Fix, of course! Dec 15, 2009 at 3:50

Spectrum.

Spectrum (homotopy theory)

Spectrum of a matrix, in linear algebra

Spectrum of an operator, in functional analysis (a generalisation of the spectrum of a matrix)

Spectrum of a ring, in commutative algebra

Spectrum of a C*-algebra

Spectrum of a theory, in mathematical logic

Stone space of Boolean algebra

• I disagree! Most of these are closely related, and it's right that they should be denoted by the same word. It's only the homotopy usage that's really different. E.g. let T be a linear operator on a fin-dim vector space V. Then T has a spectrum, its set of eigenvalues. But T also gives rise to a ring, the subring of End(V) generated by T. The set of prime ideals of this ring - its spectrum - is in one-to-one correspondence with the set of eigenvalues. (The prime ideals are (T-lambda), where lambda is an eigenvalue.) So, the two uses of "spectrum" are really the same. Dec 2, 2009 at 15:54
• How "spectral sequences" fit it? Dec 22, 2009 at 19:58

Closed:

• I'm not convinced that "closed" is overused. It is certainly used a lot, but is it really too much? Dec 1, 2009 at 14:56
• "Closure" certainly has many meanings.
– lhf
Dec 1, 2009 at 15:46
• "Closed" is certainly overused in Mathoverflow ;-) Apr 21, 2010 at 7:32

Abelian.

1. Abelian Group (also other commutative algebraic structures, and related structures like Abelian extensions)
2. Abelian theorem
3. Abelian Variety (as well as surface)
4. Abelian function
5. Abelian integral
6. Abelian Category
7. Abelian equation (used in web geometry, also appears as Abelian relation)
• They are all connected to the mathematician Abel! I think if you were going to say something like this the word Riemann or Euler should come to mind first. Dec 1, 2009 at 15:01
• I'm willing to let things named after mathematicians slide. Dec 1, 2009 at 16:13
• To be fair, abelian groups, abelian varieties, and abelian categories are all pretty closely related. Dec 2, 2009 at 0:53
• Well... I guess, although I don't know exactly how "abelian variety" is related to "abelian group", other than it's a group (not necessarily abelian!) If I knew what "abelian group" was and had to guess what "abelian variety" means, I would surely have failed miserably. Dec 2, 2009 at 1:57
• According the the wikipedia page, en.wikipedia.org/wiki/Abelian_variety , the group law of an abelian variety is always commutative. Feb 11, 2010 at 21:55

Base/Basis

Edit: It's been clarified that we're really more interested in adjectives but I think the use of base in these examples are quite substantially different.

Reflexive (relation, locally convex (Banach) space, operator algebra, module, a.s.o.)

Proposition Every infinite dimensional von Neumann algebra is reflexive, and also it is not reflexive.

• P.S. The above (intentionally confusing) Proposition is both true and non-trivial.
Dec 3, 2009 at 19:40

Canonical... would be a canonical example. I guess.

• No, that would be a bad example. Dec 15, 2009 at 3:49

Index, Order, and Rank certainly qualify

Generically

A word used a lot when you don't want to precisely specify under which conditions something is true, but its true in most cases. An example would be that generically all square matrices are invertible.

The precise meaning of this - at least in algebraic geometry - is that whatever property we are talking about is true on a dense open subset. Another example would be given a function between two smooth manifolds then a generic point in the target manifold is not a critical value of the function.

Well-defined. Overused not because it has too many definitions but because it has too few.

• Dec 1, 2009 at 17:49
• I think Gowers has effectively ended this long-suffering confusion with this blog post-glad you found it,Qiachu. Apr 20, 2010 at 4:04