Nonequivalent definitions in Mathematics

Question

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.

Answer 1

The limit of a function may be a deleted limit or a non-deleted limit.

A number $L$ is the deleted limit of $f$ as $x$ approaches $p$ if, for every $\epsilon>0$, there exists a $\delta>0$ such that $0<|x−p|<\delta$ and $x$ is in the domain of $f$ implies $|f(x)−L|<\epsilon$.

The non-deleted limit is exactly the same except that instead of $0<|x-p|<\delta$ you have $|x−p|<\delta$.

For example, if

$$f(x)=\begin{cases} 1,&x=0,\\0,&x\ne 0,\end{cases}$$

then $\lim_{x\to 0}f(x)=0$ if deleted limits are used, but $\lim_{x\to 0}f(x)$ does not exist if non-deleted limits are used.

Answer 2

I believe the number of proper definitions of the term "quantum group" is some element of $\mathbb R \setminus \{1\}$. Even the simplest example, the quantized enveloping algebra of $\mathfrak{sl}_2(\mathbb C)$, has at least two definitions (more if you count the coproduct as part of the definition).

If you are careful with the details, as for example in Jantzen's book "Introduction to Quantum Groups", you either get bogged down in the details of left-right compatibilites and twisted coproducts or, more in the spirit of the subject, you change the definition midway.

Answer 3

Are Hermite polynomials $H_n(x)$ orthogonal w.r.t. the weight $e^{-x^2}$ or $e^{-x^2/2}$?

Answer 4

What some call the adjugate matrix is often called the adjoint (or sometimes classical adjoint) matrix, while adjoint matrix commonly also means the transpose or Hermitian transpose. I doubt this double use ever causes confusion to anyone but students, but it is an example of the sort the questioner seeks.

A more serious example is that regular elements of (algebraic) Lie groups are defined differently by different authors. Correspondingly one encounters various definitions of what is a Cartan subalgebra. Here the issue tends to be the scope of definition; usually two different definitions are equivalent in the semisimple setting, or over characteristic zero, but not in more general settings.

Answer 5

I can't believe this hasn't mentioned already.

What is a graph? Everybody agrees that a graph has vertices and edges (which are usually not oriented). Depending on whom you ask, it might not be allowed to have multiple parallel edges between two vertices. Often a graph is not allowed to have loops at a vertex. Also, is a graph necessarily finite?

And I am not even talking about all the non combinatorial definitions of a graph, e.g. $1$-dimensional CW-complex.

Answer 6

One of the most classical examples is the word «algebra», which denotes not only a branch of mathematics, but also the following mathematical objects:

In linear algebra an algebra is a vector space, equipped with a bilinear operator (called product).

In set theory an algebra is a collection of sets closed under finite union, finite intersection and complement.

In universal algebra an algebra is a set, equipped with collection of finitary operations.

Note, that any algebra in linear algebraic sense is also an algebra in universal algebraic set, but not vice versa.

A quite similar thing happens with varieties:

In algebraic geometry a variety is the set of solutions of a system of algebraic equations.

In universal algebra a variety is a class of all algebras (in universal algebraic sense) with a given signature, satisfying a given set of identities.

Moreover, both those «varieties» are translated to Russian as «многообразия» - the same word, that is used for manifolds (topological spaces, such that each point of them has a neighbourhood, that is homeomorphic to $\mathbb{R}^n$ for some fixed $n$)

And if I have reached the theme of ambiguous translations, I think, that I should mention that «perfect groups» (groups equal to their derived subgroup), «complete groups» (centerless groups isomorphic to their automorphism group) and «immaculate groups» (finite groups, whose order is equal to the sum of orders of their proper normal subgroups) are all translated to Russian as «совершенные группы».

Also, the following examples deserve to be mentioned:

Two abstract groups are called commensurate (or commensurable), if they have isomorphic subgroups of finite index. Two subgroups of a group are called commensurate (or commensurable) if their intersection has finite index in both of them. Note, that two subgroups of a group may be commensurate as abstract groups, but not commensurate as subgroups.

Artinian groups are groups, that satisfy the minimum condition on subgroups. Artin groups are groups, that have a presentation of specific form. Both of them are «Артиновы группы» in Russian.

A right (left) ideal of a ring is a subring, that is closed under right (left) multiplication on arbitrary element of the ring. A right (left) ideal of a semigroup is a subsemigroup, that is closed under right (left) multiplication on arbitrary element of the semigroup. Note, that an ideal of the multiplicative semigroup of an associative ring is not always an ideal of that ring (because it does not need to be closed under addition).

Cubic graphs are usually defined as finite simple 3-regular graphs. However, the Hamming graph $H(3, 2)$ is also referred by some authors as «The Cubic Graph». Well, it is indeed finite, simple and 3-regular, but not the only one with this property.

The definition of simple graph I am used to is "graph without loops and multiple edges", however I know, that some people define simple graph as "graph without multiple edges" (loops are allowed).

In different sources $D_{2n}$ means either $C_{2n} \rtimes C_2$ or $C_n \rtimes C_2$.

The word "automaton" is sometimes used as a synonym for "acceptor" and sometimes for "transducer"

Sometimes $C_n^k$ means binomial coefficient $\frac{n}{(n - k)!k!}$ and sometimes direct product of $k$ isomorphic copies of a cyclic group of order $n$.

Answer 7

An $n$-category used to mean a strict $n$-category (a category enriched in the cartesian closed category $(n-1)$-$\mathsf{Cat}$), and now is often used to mean a weak $n$-category (which itself is defined in multiple ways).

I'd like to mention also that many of the examples in the answers and comments are examples of what the nLab calls red herrings, which need be neither red nor herrings, or sometimes all herrings are red herrings. For example, a multi-valued function is not actually a function, but all functions are multi-valued functions. A manifold with boundary is not actually a manifold, but all manifolds fit the definition of manifold with boundary, hence the locution "manifold without boundary". The category of non-associative algebras includes associative algebras.

Answer 8

A distribution can either be related to probability or to generalized functions

Answer 9

In functional analysis, an embedding of one topological vector space $X$ into another $Y$ is simply a continuous linear injection $T$ from $X$ to $Y$. In particular it typically is not an embedding in the usual sense of general topology, since $T$ need not be a homeomorphism onto its image.

Answer 10

Some people take division ring, division algebra, and skew field to be the same thing; namely a (possibly noncommutative) ring with $1\neq 0$ such that all nonzero elements are units. Some authors however add one or two extra qualifiers to some of these definitions. [Such as forcing noncommutativity, or being finite dimensional over the center, or...]

Answer 11

The distinction between compact and quasi-compact spaces is another one. If a topological space has the property that any open cover contains a finite subcover, then to some people this space is compact, to others it is quasi-compact. In the latter case, a compact space would then be a quasi-compact space that is also Hausdorff.

Answer 12

The notion of a category:

It sometimes happens that some people define a category as a locally small category without stating it, or probably without knowing that the general definition is where a category consists of a class of morphisms, instead of that morphisms between any two of its objects has to be a set. Thus, if you see the term "category" in the literature/ in lecture notes, the definition of a category might denote a category which is locally small.

The notion of a topological category is another example from this area https://en.wikipedia.org/wiki/Topological_category.

Furthermore, it often happens that there are definitions of something which are equivalent for cartain classes of objects, but not in general:

In $C^*$-algebras: There are notions like "stably finite" (and other notions like this), which can be non-equivalent, see here https://math.stackexchange.com/questions/2073741/stably-finite-c-ast-algebras for instance, which are defined in many ways and which are equivalent for a certain classes of $C^*$-algebras (simple $C^*$-algebras in this case).

The notion 'quasidiagonality' of $C^*$-algebras is another example.

Answer 13

Several distributions from probability theory share names:

The log-gamma distribution. Similarly to the log-normal distribution where we say $X$ is log-normal if $Y = \log X$ is normally (Gaussian) distributed, we can say that $A$ is log-gamma if $B = \log A$ is gamma distributed.

But I have also seen $A$ being called log-gamma if $B = \exp A$ is gamma distributed.

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The Weibull distribution may refer to the heavy-tailed distribution function or one of the three max-stable distributions.

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Another example is given by Mittag-Leffler distributions, which are distributions on $\mathbb{R}_+$. For $\alpha \in (0,1]$, let $E_\alpha (z) := \sum_{n \geq 0} \Gamma (1+n\alpha)^{-1} z^n$ be the Mittag-Leffler function of index $\alpha$. Then a (normalized) Mittag-Leffler distribution may refer to:

• a distribution whose CDF is defined with the Mittag-Leffler function:

$$\mathbb{P} (X_\alpha \geq t) = 1-E_\alpha (-t^ \alpha) \quad \forall t \geq 0;$$

these are heavy-tailed distributions, with Laplace transform $(1+\lambda^ \alpha)$, and used mostly in statistics, economics, etc.

• a distribution whose MGF is defined with the Mittag-Leffler function:

$$\mathbb{E} (e^{zX_\alpha}) = E_\alpha (z) \quad \forall z \in \mathbb{C};$$

their densities decay quickly at infinity, and they appear as the limit of local times of Markov processes.

Answer 14

Non-cooperative game theory has the odd property that essentially all authors have their own notions of what a game in extensive form is and thus prove results about principally different mathematical objects.The used notions tend to be equivalent under some reasonable isomorphism, but (almost) nobody ever bothers to make these isomorphisms explicit.

Answer 15

For a commutative ring with unity $R$, a primitive polynomial in $R[x]$ is a polynomial whose coefficients generate the ideal $(1)$. If $R$ is a UFD, a primitive polynomial in $R[x]$ is a polynomial whose coefficients have the greatest common divisor $1$. These defintions coincide iff $R$ is a PID, hence for higher-dimensional UFD's (like $k[x,y]$) we get two different definitions of the same notion.

A perfect ring is a ring, such that every left module admits a project cover. In positive characteristics, a perfect ring is a ring on which the $p$-th power map is an isomorphism.

Also note the slightly different definitions of a projective morphism in EGA and Hartshorne. To be fair Hartshorne adresses the difference after his definition.

Another (not perfectly fitting) example would be a noetherian scheme and a scheme whose underlying topological space is noetherian.

Answer 16

1. The category of models of a (finitary or infinitary) first-order theory
2. A model category which is an abstract setting for doing homotopy theory

The first notion is expounded for example in the Adamek and Rosicky book "Locally Presentable and Accessible categories", chapter 5. As for the second notion, a possible starting point is https://ncatlab.org/nlab/show/model+category.

Answer 17

Is a parallelogram also a trapezoid?

Answer 18

Differential geometry seems to have many cases of conflicting conventions:

1) Is the Laplacian a non-negative operator? Or a non-positive one?

2) Is the mean curvature the... mean... of the principal curvatures? Or their sum?

3) Is the directional derivative linear in the magnitude of the direction? Or invariant to it?

Answer 19

The notion holomorph in finite group theory

Robert L. Griess writes in [1], Definition 2.21:

A holomorph of a group $G$ is a group $E$ containing $G$ as a normal subgroup such that $C_E(G) = C(G)$ and the action of $E$ by conjugation on $G$ induces $\mbox{Aut}(G)$.

Later in [1] he explains:

Our term holomorph replaces an older and rare usage which means the semidirect product of a group with its automorphism group.

The term holomorph in the new sense is also used in [2]. The term holomorph in the old sense is used in

https://groupprops.subwiki.org/wiki/Holomorph_of_a_group ,

https://en.wikipedia.org/wiki/Holomorph_(mathematics) .

$ $

[1] Robert L. Griess, Jr., Twelve Sporadic groups, Springer Verlag 1998

[2] A. A. Ivanov, The Monster Group and Majorana Involutions, Cambridge University Press 2009

Answer 20

Injective: Semigroups can be completely (right/left) injective, while a Banach algebra is injective if the multiplication induces a continuous linear map of the injective tensor product $X\check{\otimes}X$ into $X$ (Varopoulos). I discovered the former when a reviewer insisted on a hyphen in the title of a paper I had submitted: "Injective seimgroup algebras".

Answer 21

I'm surprised nobody has mentioned "range" which can by synonymous with "co-domain" or "image" depending on personal preference.

Answer 22

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).

Answer 23

An affine stratification can be a stratification into copies of affine space ($\mathbb{A}^{n_i}$) or into affine schemes ($\operatorname{Spec} R_i$).

Answer 24

In graph theory, there are at least two different meanings of the word "hereditary".

Some definitions first. Let G be a graph. If graph H is obtained by deleting 0 or more vertices from G, then H is an induced subgraph of G. If graph K is obtained by deleting 0 or more vertices, and deleting 0 or more edges from G, then H is a subgraph of G. So every induced subgraph is a subgraph, but not vice versa.

Let X be a set of graphs. If, for any G in X, any subgraph H of G is also in X, then X is hereditary. If, for any G in X, any induced subgraph H of G is also in X, then X is induced hereditary. So every hereditary set is also induced hereditary.

Now, some authors use the term "hereditary" to refer to induced hereditary sets.

Answer 25

Everybody agrees that an isometry is a distance-preserving map. In the context of functional analysis and in particular operator algebras, this is indeed the definition. But in geometry, an isometry is usually required to be bijective, leading e.g. to the isometry group of a metric space.

If there is a danger of confusion, one can still say isometric embedding or isometric isomorphism for disambiguation, whenever no specific term such as unitary is available. (Incidentally, unitary is also an overloaded term meaning either "unitary operator" or "having a unit element".)

Answer 26

I believe that the notion of Euclidean domain provides another example of this situation.

Some authors define Euclidean domain as an integral domain $D$ endowed with a function $d \colon D\setminus \{0\} \to \mathbb{Z}^{+}$ such that

(i) If $a,b \in D \setminus \{0\}$ and $a \mid b$, then $d(a) \leq d(b)$.

(ii) If $a, b \in D\setminus \{0\}$, then there exist $q, r\in D$ such that $a=bq+r$ where $r=0$ or $d(r)<d(b)$.

(cf. Herstein's Topics in Algebra, Stewart's Algebraic number theory and Fermat's Last Theorem, etc.). Since this is the definition I was more accustomed to, I must confess I felt a wee bit uneasy when I learnt that, in their book on problems in algebraic number theory, R. Murty and J. Esmonde had chosen to define this concept as follows:

If $R$ is [an integral] domain with a map $\phi \colon R \to \mathbb{N}$, and given $a, b\in R$, there exist $q, r \in R$ such that $a=bq+r$ with $r=0$ or $\phi(r)<\phi(b)$, we call $R$ an Euclidean domain.

Later on, I would come to the conclusion that they'd decided to disregard the first condition in the definition à la Herstein because in their examples, etc., they were to analize "euclidianness" with respect to the norm in the extension which determines an application that is even completely multiplicative.

Answer 27

Singular support of a sheaf seems to be a subset of the cotangent bundle, whereas the singular support of a distribution is a subset of the base space. The former is more like the wavefront set, as far as I can intuit.

People try to avoid the confusion by denoting the former by S.S., and the latter by sing supp, but in my mind they read the same.

Answer 28

I think it's fair to mention that in mathematical physics many terms used in pure mathematics with some definition are applied to describe cognate objects with another definition, in a potentially irking way. :)

1. A Lie algebra may be called "Lie group".
2. The renormalization group is usually only a semigroup.
3. Distributions may be called "functions", like "the Dirac delta function"
4. If you go into gauge theory you have many terms with different meanings from mathematics. You can check the dictionary in the wiki page
5. Talking of "gauge", there is also the gauge of the "gauge or Henstock-Kurweil integral", a generalization of the Riemann integral.
6. A vector field on Minkowski spacetime will be called a "scalar field", if it transforms under Lorentz transformations in the trivial representation (on $\mathbb R^4$, but it can be generalized), instead of the defining representation. We can define these terms on space or galilean spacetime with respect to orthogonal or galilean transformations.
7. One example i came across is in stochastic PDEs: "stochastic quantization" may refer to adding a fictitious time dimension à la Parisi-Wu and averaging over a stochastic process to get a quantization of a classical PDE (say of the equations of motion of euclidian 3D $\phi^4$-theory), but "stochastic quantization" may be just à la Nelson, without additional dimension, with the equilibrium measure of a Markov process corresponding to the ground state of the associated Schrödinger equation. It may be confusing, especially when considering yet other dynamics related to the above by Wick-rotation.

Generally when there are different mathematical models for a single physical theory we may find a given mathematical term "interpreted" differently, applied to objects of a different mathematical nature. Examples of more physical than mathematical terms come to mind easily: "state", "position", "momentum", "energy"; in classical physics those are real numbers or functions on state space which is a manifold (finite dimensional for point physics or infinite-dimensional for continuum physics), while in quantum physics those are self-adjoint operators on state space -though actually they are more general than that, as a single self-adjoint operator only corresponds to position/momentum along a single dimension, and if we want the full position we must consider several such operators, and this is usually not described in textbooks. It is more difficult to find purely mathematical terms with different interpretations. Mathematics tends to use physically-inspired terms: for instance a "sheaf" or a "bundle", "separated", "compact", "rigid",... "Rigid" is actually an example of a term with different definitions: in topology "rigid" refers to classes of spaces which are topologically determined by their homotopy type, whenever in that class 2 spaces have the same homotopy type they are homeomorphic (or diffeomorphic, or even isomorphic with respect to generally finer structures), like in Mostow's rigidity theorem. But "rigid" is also used in "rigid analytic space". But i'm leaving mathematical physics with these examples, and this term has probably been given in another answer.

Coming back to different physical models of the same phenomenon: in general relativity you have black holes, which are solutions with a light-trapping bounded region of space (or the bounded region itself). But there are also several theories of quantum gravity, with black holes as they describe the same natural objects. In string theory we have a Hilbert space of states and a black hole would be certain states or "ensembles" of states (a set with a locally finite measure). We may even add dualities to this picture, and then black holes may correspond to free states on a related space, or thermal (equilibrium) states. My description is vague but it is only meant to give a sense of how words can become overloaded: the point is that practitioners would find it more practical to use the same word and implicitly sort out possible ambiguities, than to be super-rigorous. This tradeoff happens in mathematics, but mathematics users tend to nitpick more on possible ambiguities.

Answer 29

For some people, the dihedral group $D_n$ has order $n$ and for others, its order is $2n$. See the discussion on MathStackExchange. I'll look it up when I get a chance.

Answer 30

In the field of dynamical systems, you would see various definitions for a stable system. There are even different definitions for specific names, e.g. Lyapunov stability in the literature.

I have seen that some call the limit cycle a stable manifold, while others consider it an unstable one.