What are the most overloaded words in mathematics?

Question

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."

Answer 1

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.

Answer 2

Normal

• Normal distribution
• Normal vector
• Normal space
• Normal extension
• Normal subgroup
• Normal operator
• Normal convergence

Answer 3

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?

Answer 4

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.

Answer 5

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

Answer 6

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.)

Answer 7

"Cauchy theorem"

Answer 8

Obvious

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

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

Answer 9

Hedgehog.

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

Answer 10

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

Answer 11

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)

Answer 12

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.

Answer 13

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

Answer 14

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.

Answer 15

Elliptic.

• Elliptic Integral
• Elliptic Equation
• Elliptic Operator
• Elliptic Curve
• Elliptic(al?) Point
• Elliptic Function
• Elliptic (Moebius) Transformation

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

Answer 16

Deep

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

Answer 17

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.

Answer 18

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.

Answer 19

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

Answer 20

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.

• uniform polytope
• uniform convergence in machine learning (related but not the same)
• uniform distribution
• uniform convergence
• uniform continuity
• uniform integrability
• uniform boundedness
• uniform equicontinuity
• uniform space (from uniform continuity)
• The Riemann uniformization theorem
• uniform circuit family (complexity theory)
• Gowers uniformity norms
• uniform modules
• uniform matroid

Answer 21

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

Answer 22

Spectrum.

From http://en.wikipedia.org/wiki/Spectrum#Mathematics:

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

Answer 23

Closed:

Closed set Closed surface Closed geodesic Closed function This question is closed :-)

Answer 24

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)

Answer 25

Base/Basis

• Group Base
• Topological Base/Basis
• Algebra Basis
• Vector Space Basis
• Logarithm Base

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.

Answer 26

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

It is an adjective.

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

Answer 27

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

Answer 28

Index, Order, and Rank certainly qualify

Answer 29

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.

Answer 30

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

Answer 31

Elementary

Sense 1 = basic, simple, concerning the elements ("first steps") of a subject.

• elementary fact
• elementary introduction
• elementary problem
• elementary proof

Sense 2 = treating mathematical objects as elements of a collection.

• elementary number theory

Answer 32

-ary

-ary, as in $k$-ary numeral $s$, refers to the number $k$ of values in the domain $K = \lbrace 0, 1, \ldots, k-1 \rbrace$ that affords the basis of numeration.

-ary, as in $k$-ary relation $L$, refers to the number of domains $X_1, \ldots, X_k$ for which $L \subseteq X_1 \times \ldots \times X_k$.

-ary, as in $k$-ary operation $f$, refers to the number of domains in the domain of the function $f : X_1 \times \ldots \times X_k \to Y$, the rubric being, "a $k$-ary operation is a $(k+1)$-ary relation".

Some writers use Greek roots and the Greek suffix "-adic" for the number of domains in a relation, hence medadic, monadic, dyadic, triadic for relations of 0, 1, 2, 3 places, respectively. This usage actually has a degree of historical precedence and it can serve to sidestep conflicts with the domainance of binary numerals in our modern world, but of course the wrinkle but moves to other domains where writers are adicted to other habits.

NB. All puns are intended.

Answer 33

Space

Affine space

Banach space

Cauchy space

Euclidean space

Function space

Hardy space

Hilbert space

Inner product space

Kolmogorov space

Krein space

Klein space

Pontrjagin space

Lp space

Measure space

Metric space

Minkowski space

Normed vector space

Locally convex space

Linear topological space

F-space

Frechet space

Nuclear space

Operator space

Affine space

Projective space

Polish space

Quotient space

Sobolev space

Topological space

Uniform space

Vector space

Harmonic spaces

Conformal space

Complex analytic space

Affinely connected space

Algebraic space

Symplectic space

Measurable space

Measure space

Probability space

Riemann space

Lorentzian space

And so on.