# 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". – Arturo Magidin Nov 22 '17 at 19:12
• @ArturoMagidin This leads to the classic dialogue "The space $X$ is quasicompact, so..." "What does quasicompact mean?" "It means compact." – Will Sawin Nov 22 '17 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". – John Pardon Nov 23 '17 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. – Taladris Nov 23 '17 at 14:49
• @Taladris, maybe in france, everyone's an algebraic geometer – Vivek Shende Nov 29 '17 at 0:54

Variety has a number of slightly different definitions. Apparently some authors use reduced of finite type over a field, whereas I would want separated to rule out the line with two origins. Most people use scheme-theoretic language to solve this problem.

Conway-Sloane jokes the discriminant of a quadratic form is a function of the form and the author. They are not wrong. That same subject has two distinct definitions of integral, which are basically the same (multiply by $2$) but which are a pain to keep track of. The Hasse-Witt invariant has two different definitions as well.

• Re: quadratic forms: What's odd to me is Gauss went "twos in" in Disquisitiones and "everyone else" went "twos out". This has led to pairs of theorems that address this both ways, e.g. the 15 and 290 theorems. – Eric Towers Nov 24 '17 at 6:45

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...]

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.

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.

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

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.

• What do you have in mind for the different definitions of 'regular'? I am used to seeing it used sometimes to mean 'regular' in the sense that the centraliser is of minimal dimension, and sometimes to mean 'regular' in that sense + semisimple; is that what you mean by the definitions being equivalent "in the semisimple setting"? – LSpice May 25 '19 at 17:09

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

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.

• Indeed; and likewise the Bohr compactification $\mathbf R\hookrightarrow b\mathbf R$ is not an embedding, hence not a compactification, in the sense of general topology : $\mathbf R$ is dense but doesn’t have the subspace topology. Misunderstandings ensue that I have experienced first hand. – Francois Ziegler Nov 27 '17 at 8:40

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.

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.

• Also, do edges have their own identities or are they just pairs of vertices? – darij grinberg Dec 20 '17 at 23:03
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.

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.

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.

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

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.

• Another example is the Geometric distribution. In some books it is the trial of first success and in others it is the number of failures before the first success. Not a big deal perhaps, but I do find the ambiguity annoying at times. Different supports, different means, different MGF, but the same name. – John Coleman Nov 24 '17 at 4:24
• @D.Thomine I have never stumbled upon the Mittag-Leffler distribution so I do not know about it. Since this is a community wiki, perhaps you can add it to the answer? – Therkel Nov 25 '17 at 20:37
• @Therkel : good idea. That's done. – D. Thomine Nov 26 '17 at 10:34

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.

Is a parallelogram also a trapezoid?

• Does the disagreement about this question correlate with the level of the mathematics? I don't recall ever seeing a negative answer to this question (or analogous ones, like square vs. rectangle, circle vs. ellipse, equilateral vs. isosceles) in research-level mathematics, but I can easily imagine a negative answer in high-school textbooks. – Andreas Blass Nov 23 '17 at 16:46
• Trapezoid or trapezium, depending on US or UK. – Gerald Edgar Nov 23 '17 at 17:16
• For square vs. rectangle and other your examples I have also never seen negative answer, but for parallelogram vs. trapezoid/trapezium it is usual at least in Russian tradition. It has the following sense for me: parallelogram does not behave like (other) isosceles trapezoids, which, for instance, are always inscribed in a circle. The only parallelogram which is a limit of isosceles trapezoids is rectangular. I would like rectangulars being called isosceles trapezoids, but not other parallelograms. – Fedor Petrov Nov 24 '17 at 8:19
• @FedorPetrov Your comment (which I saw only now) suggests to me that perhaps the problem lies not in the definition of "trapezoid" but in the definition of "isosceles trapezoid". That is, a parallelogram (other than a rectangle) might reasonably be called a trapezoid, but we might (for the reasons in your comment) decide not to call it isosceles. – Andreas Blass Jan 2 '19 at 15:18
• See matheducators.stackexchange.com/questions/13700/… for an interesting historical take. – kjetil b halvorsen Jan 15 at 1:25

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?

• For (1), if you ever talk to probabilists, you will have another question of whether the Laplacian includes a factor of $1/2$. This is somewhat related to Fedor Petrov's note about Hermite polynomials. – Nate Eldredge Nov 28 '17 at 0:38

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".)

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 $$n$$.

• A meta post about the ambiguous tag "ra.rings-and-algebras": meta.mathoverflow.net/questions/3625/… – YCor May 25 '19 at 10:06
• I think the choice of the misleading terminology "universal algebra" is one of the main reason it is much less known than it ought to be. – YCor Jun 12 '19 at 10:06

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".

• And also : injective object in a category with respect to a class of maps, injective set map in the sense of one-to-one set map, or the notion of injective object in homological algebra. – Philippe Gaucher Nov 23 '17 at 9:50

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

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.

• "A perfect ring is a ring, such that every left module admits a project cover." Who says that? Never heard this one. – darij grinberg Nov 24 '17 at 23:35
• @darij: For example T. Y. Lam in A First Course in Noncommutative Rings, Springer GTM 131 (2001). (Well, of course he says "left perfect".) – Fred Rohrer Nov 25 '17 at 11:57

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.

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.

• I seem to recall once seeing a generalization of Euclidean domains, where $d$ or $\phi$ or whatever took values not necessarily in $\mathbb{N}$ but in any well-ordered set. But I don't think they used a different name; I think they still just called them "Euclidean domains"! (I mean, to my mind, this should probably be the standard definition, but it isn't, so, clarity would be a good thing...) – Harry Altman Dec 1 '17 at 1:51

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



[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

• So the difference to the old definition is that $E\to Aut(G)$ does not need to be split? Am I understanding this correctly? – Johannes Hahn Dec 11 '17 at 17:31
• An nontrivial inner automorphism $e$ of $G$ occurs twice in a holomorph $E$ in the old sense: once as an element $e_1$ of $\mbox{Aut}(G)$ and once as an element $e_2$ of $G$. So $e_3 = e_1 e_2^{-1}$ is in $E \setminus G$ and also in the centralizer $C_E(G)$ of $G$. There is no such $e_3$ in a holomorph in the sense of R. Griess. – Martin Seysen Dec 11 '17 at 21:45

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.

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.

The notion of spectrum in operator theory. Some people assume that if $\lambda$ has a property that $T-\lambda I$ is injective with dense range which is not the whole space and with continuous inverse then $\lambda$ is not an element in spectrum since $(T-\lambda I)^{-1}$ can be extended continuously other authors consider such $\lambda$ as an element of spectrum.

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

• Have you had a chance to look it up? – Gerry Myerson May 24 '19 at 23:38
• @GerryMyerson, probably math.stackexchange.com/questions/2560348/… . (Of course I think almost no-one writes $Dn$, as opposed to $D_n$, so that is probably a typo.) Brian Conrad's (and DavidWheeler's) argument was that one can use $\operatorname D_{2n}$ for the symmetries of an $n$-gon exactly when one uses $\operatorname S_{n!}$ for the symmetric group on $n$ letters. – LSpice May 25 '19 at 17:17

In set theory a forcing notion can be a pre order with a largest element or lowest element, depending on the style of the author.

• This is a nonexample. Both versions are trivially equivalent. – Andrés E. Caicedo May 25 '19 at 23:37

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. – LSpice Nov 29 '17 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. – Andreas Blass Dec 1 '17 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. – LSpice Dec 2 '17 at 1:39
• The definition of "convex function" is universally agreed and standard, luckily – Pietro Majer Jun 12 '19 at 12:11