The definitions tag has no usage guidance.

**71**

votes

**25**answers

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### What are the most misleading alternate definitions in taught mathematics?

I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...

**64**

votes

**18**answers

8k views

### Can a mathematical definition be wrong?

This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...

**32**

votes

**2**answers

2k views

### Category Theoretic Interpretation of Matroids?

Hello everyone! First time poster, long time lurker here. I have a really basic question that has been bugging me for sometime. Specifically, I'm not exactly sure what the 'correct' category ...

**27**

votes

**1**answer

2k views

### What is Serre's condition (S_n) for sheaves?

The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...

**26**

votes

**11**answers

5k views

### What does the adjective “natural” actually mean?

Terms like "in the natural way" or "the natural X" are used frequently in mathematical writing. While it is certainly clear most of the time what is meant, on occasion, I have been confounded. The ...

**21**

votes

**3**answers

2k views

### When is a classification problem “wild”?

I hope someone can point me to a quick definition of the following terminology.
I keep coming across wild and tame in the context of classification problems, often adorned with quotes, leading me to ...

**17**

votes

**1**answer

948 views

### Surreal exponentiation — are the varying definitions equivalent? If not, is there agreement on which ones are better?

The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to ...

**16**

votes

**5**answers

1k views

### Definition of area

I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...

**14**

votes

**2**answers

614 views

### Is this a vertex algebroid?… What is vertex algebroid?

A couple of day ago, I was lamenting to a friend about the fact that I have no idea what vertex algebroids are.
During our discussion, I came up with a guess of what a vertex algebroid might be.
I'm ...

**12**

votes

**3**answers

865 views

### Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)?

Long ago, manifolds were embedded subsets of euclidean space defined by polynomials. Later, using the gluing of open sets, people realized they could define manifolds intrinsically. And in certain ...

**10**

votes

**1**answer

769 views

### What is the mathematical structure called if we replace commutative group by commutative monoid in the definition of linear space?

Could anyone tell me what the mathematical structure is called if we replace commutative group by commutative monoid in the definition of linear space?
Also, are there any names for "commutative ...

**10**

votes

**7**answers

483 views

### Does the notion of graphs with vertex multiplicity exist?

I need to use graphs where each vertex gets a natural number, $b(v)$, its multiplicity. These numbers indicate how many 'replications' of the vertex we have.
It is actually a way to write in a ...

**9**

votes

**2**answers

2k views

### What are examples of theorems which were once “valid”, then became “invalid” as standard definitions shifted?

That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in ...

**9**

votes

**3**answers

689 views

### What makes a distance?

In the answers to my previous question, I learned that there are different concepts of distance, that is of distance-like functions with the usual metric being only the most popular and important one.
...

**9**

votes

**0**answers

627 views

### What are the relationship between various definitions for quasi coherent sheaves?

It seems that there are many definitions of quasi coherent sheaves(modules). There is a nice page on nLab quasi coherent sheaves
My questions are:
Are there any other definitions of quasi coherent ...

**8**

votes

**4**answers

1k views

### Grothendieck Topologies versus Pretopologies

The wikipedia article(s) as well as the nlab article(s) about Grothendieck topologies and Grothendieck pretopologies are careful to differentiate the two very emphatically and to point out that ...

**8**

votes

**3**answers

997 views

### What are Penrose Tilings, and how do they relate to Quasicrystals?

The question is in the title, but let me elaborate a little.
Background
Penrose Tilings are really pretty and satisfy some remarkable properties. For instance, I believe the following is true: even ...

**7**

votes

**4**answers

2k views

### Definition of forgetful functor

I was wondering if it is possible to make a formal definition of what it means for a functor to be forgetful. That is, using only the terminology of categories. I have seen so many examples of ...

**7**

votes

**2**answers

883 views

### Who first introduced the functional definition of symmetry?

Who first introduced the definition of symmetry using functions explicitly? (That is, for instance, a symmetry of a subset $X$ of the plane is a function $F$ from the plane to the plane that preserves ...

**7**

votes

**1**answer

329 views

### BRST cohomology definition

Is there written anywhere a full definition of BRST cohomology? All I have found so far is BRST cohomology in _______.
As far as I can see, BRST cohomology is the ...

**6**

votes

**4**answers

1k views

### On similar concepts in mathematics whose similarity is a non-trivial fact.

Recently, while undertaking a study of commutative algebra, I learned three concepts: (i) a local ring, (ii) a regular local ring and (iii) a regular ring.
At the end, I found myself asking this ...

**6**

votes

**4**answers

1k views

### Locally constant functions with compact support = smooth ?

Hello,
I have a trivial question, but I hope that you don't mind helping. I often get confused with basic definitions.
Let F be a p-adic field. Then (from what I understand) $C_c^{\infty}(F)$ is the ...

**6**

votes

**1**answer

2k views

### Geometric picture of invariant differential of an elliptic curve

What is the geometric meaning of $\omega=dx/(2y+a_1x+a_3)$ for an elliptic curve?
This question is an adjunct to MO Q1 on formal laws and L-series, which motivated Q2. Q1 (Silverman) and Darmon (pg. ...

**5**

votes

**2**answers

674 views

### Unitary groups over number fields

When defining unitary groups over number fields, one usually takes $F$ to be a totally real number field, $E$ a CM quadratic extension of $F$, and $V$ a hermitian space attached to $E/F$. Then $U(V)$ ...

**5**

votes

**1**answer

715 views

### Why do mathematicians prefer one definition over the other when they both define the same concept?

Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of ...

**5**

votes

**2**answers

165 views

### Axiomatic approach to means

Recently I have been contemplating on a talk for high school children. One of my favorite topics in high school was the inequality of means. I had a great high school teacher who wrote some very nice ...

**5**

votes

**1**answer

226 views

### Meaning of $g_d^r$ in algebraic geometry

As an editor I often encounter the symbol $g_d^r$ as a noun. I tried googling but I only get papers where the symbol is used without a definition. Can someone supply a reference to a definition? ...

**5**

votes

**1**answer

283 views

### Canonical differential on Tate curve

I am starting studying the theory of (algebraic) modular forms, and I have some trouble in understanding completely the construction of the Tate curve. My problem is the following: as far as I know ...

**5**

votes

**0**answers

269 views

### Why is Pic^0(C) of a curve C a variety?

Let $C$ be an abstract non-singular curve.
I'm having a hard time finding a reference for why $\text{Pic}^0(C)$ is a variety.
Any pointers towards a reference would be appreciated.

**4**

votes

**3**answers

897 views

### Lattices: why require bilinear form to be integral?

This is a quite localized question, but I hope it won't be closed as unfit to MO. Well, a lattice $\Lambda$ in $\mathbb{R}^n$ is a discrete subgroup generated by a basis. Such a lattice gets a ...

**4**

votes

**1**answer

792 views

### Is there a mathematical object called “ivy”?

As the title says, is there a mathematical object referred to as "ivy" or "ivy type" or similar?
I have a type of graph where this name fits perfectly, but I don't want it to clash with something ...

**4**

votes

**5**answers

296 views

### procedure-based (as opposed to definition-based) concepts

Euler's work on divergent series was guided by computational procedures, rather than any definition of the "value" of such a series. E.g., he was happy to have half a dozen procedures that indicated ...

**4**

votes

**1**answer

1k views

### What is a proper stack?

I have seen the use of the word "proper Deligne-Mumford stack". Now, it is clear to me what it means for a morphism f of stacks to be proper: as usual it should be representable, and every morphism ...

**4**

votes

**2**answers

202 views

### On the definition of ‘smooth vectors’ in Rieffel's “Deformation Quantization for Actions of $ \mathbb{R}^{d} $”.

On the first page of Chapter 1 of Rieffel's Deformation Quantization for Actions of $ \mathbb{R}^{d} $, Rieffel defines a family of seminorms on the space $ A^{\infty} $ of smooth vectors of a Fréchet ...

**4**

votes

**2**answers

613 views

### A scheme simple over Spec(A)?

What does it mean to say that a scheme $X$ is simple over $Spec(A)$ ?
I stumbled on this terminology in a paper of S. Lubkin entitled "On a conjecture of Andre Weil".

**4**

votes

**0**answers

287 views

### Q-construction and Gabriel-Zisman Localization

It might be a stupid question.
When I took a look at the definition of Q-construction. It makes for an exact category $P$, one defines a new category $QP$ whose objects are the same as $P$ but ...

**3**

votes

**5**answers

359 views

### Strings and “co-subsequences”

Let $S$ be a string over some alphabet $\Sigma$. It is well known that a substring of $S$ is commonly defined as a sequence of contiguous elements from $S$, while a subsequence of $S$ is a sequence ...

**3**

votes

**1**answer

1k views

### What is “augmented algebra”?

Really sorry for this question, but googling for some time did not help me. I was trying to understand the meaning of the following phrase:
Let B be an augmented algebra over a semi-simple algebra T.
...

**3**

votes

**2**answers

2k views

### Set theory definition of addition, negative numbers, and subtraction? [closed]

Using the definition of natural numbers $0 = \emptyset$ and $S(n) = n \cup \lbrace n \rbrace$ where S is the successor function, what is the definition of addition on natural numbers?
Concerning the ...

**3**

votes

**1**answer

293 views

### How to define a generating subset for algebra in a category?

As is well known, the definition of an monoid can be generalised to the notion of a monoid $A$ in a monoidal category $C$ (see the n-lab entry here). What I would like to know is if the notion of ...

**3**

votes

**2**answers

308 views

### Definition modifications without choice

What definitions or equivalencies between definitions for standard set theory objects (such as large cardinals) do not hold or do not carry through in the expected manner to the world without choice? ...

**3**

votes

**0**answers

104 views

### Definition of a normed ring

A normed ring "should" be a monoid object in the monoidal category of normed abelian groups. There are (at least) two choices of morphisms of normed groups, namely bounded or short homomorphisms, ...

**3**

votes

**0**answers

154 views

### What does “control of a deformation problem” mean?

Is the expression "control of a deformation problem' ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ...

**3**

votes

**0**answers

112 views

### How to define Product of Conditional Measures?

I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below.
If $(X,\Sigma)$ is a measurable space, then the function $\mu : ...

**3**

votes

**0**answers

95 views

### On one class of Euclidean lattices

Let $\Lambda\subset \mathbb Z^3$ be 3D lattice with a basis
$$a_1=\left(\begin{smallmatrix} a_{11} \\ a_{21}\\
a_{31}
\end{smallmatrix}\right),a_2=\left(\begin{smallmatrix} a_{12} \\ a_{22}\\
a_{32}
...

**2**

votes

**2**answers

458 views

### Euclidean Function at 0

In a few places where I have looked the Euclidean Function of a Euclidean Domain is only being defined for non-zero elements. I am teaching an undergraduate course and I am trying to make things ...

**2**

votes

**3**answers

504 views

### Defining the integral of a function using the product measure

Imagine that we're trying to define the expression
$$\int_U f(x)dx$$
in a rigorous way.
Assume that $f:X \rightarrow \mathbb{R}^{\geq 0}$ where $(X,\mu)$ is a measure space, and suppose that $U$ is a ...

**2**

votes

**2**answers

686 views

### Non-split groups

I am looking for a reference with definitions on what it means for an algebraic group to be split, quasi-split, and non-split. I would like to see some examples of the different "types".
Thanks,
Tom

**2**

votes

**2**answers

697 views

### On figurate numbers

Do you know a text where I can find a definition of polygonal number that is both geometrically and operationally sound?
I've basically seen two ways in which this topic is approached in the ...

**2**

votes

**1**answer

175 views

### Notation: $Sigma$ and $Pi$ of intersections

In Jech - Set Theory, the proof of Theorem 31.7, I came along some notations I wish to understand correctly.
For a countable elementary substructure $M \prec H_\lambda$ and $A \in M$ and a generic ...