Suggestions for good notation

Question

I occasionally come across a new piece of notation so good that it makes life easier by giving a better way to look at something. Some examples:

• Iverson introduced the notation [X] to mean 1 if X is true and 0 otherwise; so for example Σ_1≤n<x [n prime] is the number of primes less than x, and the unmemorable and confusing Kronecker delta function δ_n becomes [n=0]. (A similar convention is used in the C programming language.)

• The function taking x to x sin(x) can be denoted by x ↦ x sin(x). This has the same meaning as the lambda calculus notation λx.x sin(x) but seems easier to understand and use, and is less confusing than the usual convention of just writing x sin(x), which is ambiguous: it could also stand for a number.

• I find calculations with Homs and ⊗ easier to follow if I write Hom(A,B) as A→B. Similarly writing B^A for the set of functions from A to B is really confusing, and I find it much easier to write this set as A→B.

• Conway's notation for orbifolds almost trivializes the classification of wallpaper groups.

Has anyone come across any more similar examples of good notation that should be better known? (Excluding standard well known examples such as commutative diagrams, Hindu-Arabic numerals, etc.)

Answer 1

$f_*$ and $f^*$ for direct and inverse image. We really should use this right from the beginning, for functions $f\colon X\to Y$, where $f_*\colon P(X)\to P(Y)$ ($P(X)$ being the power set) and $f^*\colon P(Y) \to P(X)$ instead of the awful notations $f(A)$ and $f^{-1}(B)$ for subsets $A$ of $X$ and $B$ of $Y$.

Answer 2

Dirac's bra-ket notation

This notation is very useful when applying the Hilbert spaces in Quantum Theory. It exploits some properties of duality, eigenvalues/eigenvectors, projectors and self-adjoint operators. In mathematics, perhaps it is difficult to adopt, because mathematicians are using notations that are more general, and cannot exploit these particularities. But if you know Hilbert spaces you can learn this notation in one minute, and then it makes visible many of these nice properties.

Answer 3

Here is a notation in algebraic geometry that in my opinion is very useful and self-explanatory but not used widely.

For a birational morphism $f:X\to Y$ there exists an open dense set $U\subseteq Y$ for which $f$ induces an isomorphism $f^{-1}U\to U$. For a closed subset $Z\subseteq Y$ such that $Z\cap U\neq\emptyset$ the strict transform is defined as $$\overline{f^{-1}(Z\cap U)}\subseteq X,$$ i.e., the closure of the preimage of the part of $Z$ that lies on the part where the morphism is an isomorphism. This is a very important construction and there isn't a universally accepted notation for it.

János Kollár invented the following notation for this: $$f^{-1}_*Z:= \overline{f^{-1}(Z\cap U)}\subseteq X$$ The genius of the notation is that anyone familiar with basic notation in algebraic geometry should understand what it is:

1) As $f$ is birational, $f^{-1}: Y\dashrightarrow X$ exists as a rational map.

2) For any map $g$, it is common to use $g_*$ to denote push-forward of cycles.

The strict transform is really just the push-forward of cycles via the rational map $f^{-1}$.

Answer 4

The ever-controversial reverse Polish notation for functions: $f(x) = xf$. Thus in composition, the order makes sense: $(g \circ f)(x) = x f g$ (this point is moot for the fortunate Hebrew- and Arabic-speaking mathematicians). I hate this notation in practice but I can't deny that it is objectively right and "just makes sense" in more or less the same way that the original post discusses writing $B^A = A \to B$. Please no one vote this up.

Answer 5

Diagrammatic notation for tensors (Penrose diagrams, birdtracks, etc.). It makes many things like the invariance of tr(A B C) under cyclic permutation into empty statements.

Answer 6

I like $ A^{\text{H}} $ for the conjugate transpose of the matrix $ A $, ananlogously to how $ A^{\text{T}} $ and $ A^{\text{C}} $ means the transpose and the conjugate. You call it the Hermitian of the matrix for short. I learnt this notation from Rózsa Pál, but I can't tell who invented it.

Answer 7

For quite some time, Giovanni Sambin has been advocating the notation $A\between B$ for "overlapping" sets, that is, for inhabited intersections, where $\exists_{x} \ x \in A \cap B$. He makes his case in a constructivist frame, but I think that a lot of ink, chalk---and keyboard pushing...---would be spared if one wrote $A\between B$ instead of $A\cap B \neq \emptyset$, even in non-constructive mathematics.

Answer 8

The universal property of the univariate polynomial ring: For any commutative ring $A$, any commutative $A$-algebra $B$ and any $x\in B$, there exists one and only one $A$-algebra homomorphism from the polynomial ring $A\left[X\right]$ to $B$ which maps $X$ to $x$.

This is the so-called evaluation homomorphism at $x$. I denote this homomorphism by $\lim\limits_{X\to x}$. This has the advantage that we have $\lim\limits_{X\to 0}\dfrac{\left(X+1\right)^n-1}{X}=n$ and similar properties hold just as in classical analysis. The polynomial $\dfrac{\left(X+1\right)^n-1}{X}$ is well-defined (since $X$ is not a zero divisor in $A\left[X\right]$ and divides $\left(X+1\right)^n-1$), but if we would blindly replace $X$ by $0$ we would obtain a $\dfrac{0}{0}$ error.

Answer 9

In commutative algebra with many variables, repeating lists of variables in polynomial arguments and various rings gets very tedious. I suggest using $X_{1..n}$ instead of $X_1,\ldots,X_n$.

Here's an excerpt from Bourbaki's Commutative Algebra, page 222:

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For every formal power series $f\in A[[X_1,\ldots,X_n]]$, $$f(X_1,\ldots,X_n)-f(Y_1,\ldots,Y_n)=\sum_{i=1}^n (X_i-Y_i)h_i(X_1,\ldots,X_n,Y_1,\ldots,Y_n)$$ where the $h_i$ belong to $A[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$.

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And here's how it looks with my suggested notation:

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For every formal power series $f \in A[[X_{1..n}]]$, $$ f(X_{1..n})-f(Y_{1..n})=\sum_{i=1}^n (x_i-Y_i)h_i(X_{1..n},Y_{1..n})$$ where the $h_i$ belong to $A[[X_{1..n},Y_{1..n}]]$.

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I think this notation is maximally succinct, and helps a reader from getting lost in long lists of variables.

Answer 10

A) Two notations I love are the rising factorial $x^\overline n$ and its falling factorial twin $x^\underline n$. They are used and advocated in the great book see http://en.wikipedia.org/wiki/Concrete_Mathematics . In passing this book uses great notations.

B) A general trick with binomials to reuse them with sets instead of numbers, here are some typical examples.

1) $\binom S k $ to denote the set of all $k$-sets of the base set $S$ .

2) $S^\underline 2$ to denote the pairs $(x,y)$ of $S$ where $x$ and $y$ are different.

3) $S^\underline k $ to denote the $k$- uplets of $S$ (each uplet has $k$ different elements).

C) Another notation I find useful when listing some (big) families of examples in a combinatorial setting. Use as variables the very numerals $1$ $2$ .. themselves instead of $x_1$ , $x_2$ ... . For example ( very untelling because too small an example) : the intersection of $123$ and $34$ is $3$.

D) I also often use {{ a,a,b,c}} for multiset. Any other standard or suggestion (or a way to avoid speaking about multiset) is welcome.

Answer 11

I find $$\lim_{x\nearrow0} f(x)$$ $$\lim_{x\searrow0} f(x)$$ for the limit from below and from above much more intuitive that all other notation I've seen, including $\lim_{x\to 0^-}f(x)$ and $\lim_{x\to 0^+}f(x)$.

Answer 12

If one needs to denote the fiber (not the stalk which is standardly denoted $\mathcal{F}_{x}$) of a sheaf $\mathcal{F}$ at the closed point $x$ of the $\Bbbk$-scheme $X$, one can write

$\mathcal{F}\mid_{x}$

After all, the fiber $\mathcal{F}\otimes_{\Bbbk}\;\kappa (x)$ is the restriction (pullback) of $\mathcal{F}$ to the point $x:\rm{Spec}\;\Bbbk\rightarrow X$.

The problem is that, when you identify vector bundles with locally free sheaves, the above notation clatches with the usual notation $E_x$ for the fiber of vector bundles. On the other hand almost always the context would be sufficient to clarify which of the two notations is being used.

Answer 13

All of the notations created to simplify writing category theory. For instance, the idea of drawing a circular arrow inside of a diagram to indicate that that diagram is commutative. As well as the idea of putting an angle in the top left or bottom right of a square diagram to indicate that it is a pushout or pullback. And finally, the notation of augmenting any of these notations with $\simeq$ to indicate that the diagram is only "up to homotopy".

Answer 14

I would love for topologists to start differentiating between properties that hold “locally” and properties that hold “regionally”, that is: A local property holds on some neighbourhood basis of every point, a regional property holds on some neighbourhood of every point. For any space, having a property locally implies having it regionally.

This would resolve the permanent confusion of “which sort of local” is meant in a given context.

Answer 15

This is probably the exact opposite of what the thread starter intended, but here's an instance where it might have been useful to have overloaded notation! I recently found that in propositional logic, $p \to q$ obeys much the same rules as exponentiation $q^p$: for instance, we have $(r^q)^p = r^{q \times p}$, and similarly, $p \to (q \to r)$ iff $p \land q \to r$. This is apparently due to the universal property for exponential objects, as applied to a Boolean algebra viewed as a poset category. I suppose it's also an instance of the Curry—Howard correspondence.

More generally, it seems like it isn't such a bad idea to conflate exponentiation and arrows - it looks nicer, to me at least, to write that a function of the type $A \to (B \to C)$ is naturally isomorphic to a function of the type $A \times B \to C$, than to write about $A \to C^B$. Even, as some have suggested, $A \to {}^B C$ or $C^B \leftarrow A$ would look nicer. On the other hand we'd lose the association with cardinal arithmetic if we do this...

Answer 16

If $\mathcal{C}$ is a category and $X,Y\in\mathrm{obj}(\mathcal{C})$, I like the notation $\mathcal{C}(X,Y)$ to denote $\mathrm{Hom}_{\mathcal{C}}(X,Y)$.

So, $\mathcal{C}(X,X)=\mathrm{End}_{\mathcal{C}}(X)$.

What do you think of the notation $\mathcal{C}(X):=\mathrm{Aut}_{\mathcal{C}}(X)$ ?

This would be consistent with the notation (or similar notations) $\mathsf{DIFF}(S^1)$ (resp. $\mathsf{TOP}(S^1)$ ) for diffeomorphisms (resp. homeomorphisms) of the circle, i.e. the $\mathrm{Aut}$ in the category $\mathsf{DIFF}$ of smooth manifolds (resp. $\mathsf{TOP}$ of topological manifolds), sometimes used in topology (see e.g. here and here. And (see e.g. here) $\mathsf{TOP}(n)=\mathrm{Aut}_{\mathsf{TOP}}(\mathbb{R}^n)$.

Answer 17

Forgive me for bumping such an old big-list question, but I can't resist. In linear algebra texts you will sometimes find the notation $[v]_B$ for the vector of coordinates of a vector $v \in V$ with respect to a basis $B$ of $V$, but you will not (to my knowledge) find corresponding notation for the matrix of a linear transformation $T : V \to W$ with respect to two bases $B, C$ of $V$ and $W$, except perhaps in the special case $V = W, B = C$, where I have seen $[T]_B$. The notation I use and advocate for this matrix is

$$_C[T]_B$$

and I taught this in a few linear algebra classes at UC Berkeley as a TA although I don't think it caught on. It has the pleasant property that its definition can be written

$$_C [T]_B [v]_B = [T(v)]_C.$$

This lets us give a transparent proof of the equally pleasant "functoriality" property that if $S : U \to V$ is another linear transformation and $A$ is a basis of $U$ then

$$_C[TS]_A = (_C[T]_B)(_B[S]_A)$$

("the $B$s just cancel"). Try even stating this result without notation for it! The notation makes it hard to misstate the basic facts about how such matrices change upon change of basis etc., because the correct statements "typecheck" properly and you won't be tempted to multiply matrices that haven't been expressed in compatible bases. Although things would be slightly better if either 1) the vector notation were written with the basis on the left or 2) matrices acted on the right and we switched the order of the two bases.

Importantly, the change-of-basis matrix between two bases $B, B'$ can just be written $_B[I]_{B'}$ and all of its properties become obvious (everything follows from "canceling" bases) and easy to remember, including e.g. that $_{B'}[I]_B$ is the inverse matrix and that if $T : V \to V$ is an endomorphism then

$$_{B'}[T]_{B'} = (_{B'} I_B) (_B[T]_B)(_B I_{B'}).$$

$\LaTeX$ balks a bit at the repeated subscripts, though.

(Incidentally, it's annoying that we use "vector" to denote both an "abstract vector" and a "concrete vector" while we have different terms for "linear transformation" and "matrix." Perhaps we should start using something like "array" or "list" or "tuple" to mean a concrete vector.)

Answer 18

• I also like the notation $x \prec y$ to denote majorization of a vector $x$ by a vector $y$; once defined, this notation relieves quite lot of burden.

• On a related note, I also prefer the notation $A \succeq 0$ to signify that $A$ is a positive semidefinite matrix (some prefer to use the perhaps "more natural" $A \ge 0$, but since I frequently deal with nonnegative matrices, the $\ge$ is out)

Answer 19

The three-dot notation $f\mathrel{\scriptsize\vdots}A\to B$ to indicate that $f$ is a partial function from $A$ to $B$, meaning that $\text{dom}(f)\subseteq A$ rather than $\text{dom}(f)=A$. Partial functions are pervasive in logic, especially computability theory and set theory, and this notation is both compact and suggestive.

Answer 20

I really like $(-)^n$ instead of $(-1)^n$ for alternating signs in series etc. Its more aesthetic and slightly easier to write. I found it in a 1976 paper about multipolar expansions.

Answer 21

For a Lie group $G$ with Lie algebra $\mathfrak{g}$, and element $A\in\mathfrak{g}$, denote the left invariant vector field $\overrightarrow{A}(g)=L_{g*} A$, and the right invariant vector field $\overleftarrow{A}(g)=R_{g*}A$. On any matrix group, $\overrightarrow{A}(g)=gA$ and $\overleftarrow{A}(g)=Ag$, easy to remember. The notation uses fewer subscripts than the more common notation $X_A$, which is also not easy to adapt to right versus left invariant vector fields. When discussing group actions, we usually use left actions. The left action of the group on itself is generated by the right invariant vector fields. Some authors use $X_A$ for left invariant vector fields, but also for the generators of any action of the group, which can lead to sign mistakes when discussing the left action of the group on itself.

Answer 22

To say that $u$ and $v$ are orthogonal you can spell out "The scalar product of $u$ and $v$ is equal to zero", i.e.:

$\langle u,v \rangle=0$

but you can also use the binary symbol $\perp$ to write the sentence "$u$ orthogonal to $v$" more directly, i.e. $u\perp v$.

Analogously, to say that sets $A$ and $B$ have empty intersection, of course you can spell out "$A$ intersection $B$ equals the empty set", i.e.:

$A \cap B = \emptyset$

But it would be nice if there was a binary symbol (like a barred $\cap$ symbol, not to be confused with the $\pitchfork$ symbol for transversality) to say directly "$A$ does not intersect $B$ (nontrivially)".

I don't think this symbol already exists in LaTeX.

Answer 23

Has this already been mentioned? If a group $G$ acts on a commutative group $A$ by homomorphisms, $G \to Aut(A)$, then use $a^g$ to denote the action. Especially if the group multiplication on $A$ is written multiplicatively, where we can say things like $(ab)^g = a^g b^g$. This can come up especially in Galois theory; I remember Lang using this notation in his Algebra to prove Hilbert's Theorem 90, and I thought it was very neat, and enhanced the readability of notation as well.

Answer 24

Whenever $X$ is a singleton set, denote by $!X$ the unique element in $X$, or put else (assuming some standard material set theory), $$! \colon \mathrm{Singletons} → \mathrm{Set}$$ is the unique inverse for $\{\,\} \colon \mathrm{Set} → \mathrm{Singletons},~x ↦ \{x\}$. So $!\{x\} = x$ for all things $x$ and $X = \{!X\}$ for all singletons $X$.

This makes it possible to formulaically talk about “the unique element $x$ such that …” (which happens a lot) and it fits well with the notation “$∃!$” for saying “there exists some unique …”.

For example, one may define the minimal polynomial of an element $α$ in an algebraic field extension $E / F$ as $$\operatorname{minpol}_F α = ~!\{f ∈ F[X];~\text{$f$ monic, irreducible with $f(α) = 0$}\}.$$

Answer 25

Round brackets as Cartesian coordinates and square brackets as homogenous coordinates.

I picked this up idea from Needham's Visual Complex Analysis, and I'm not sure how commonly it's used elsewhere. In the book, the convention was to use round-bracketed matrices $$ \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} $$ for representing linear transformations, and square-bracketed matrices $$ \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ for representing Möbius transformations. More generally, we can let round brackets be for all "ordinary", "Cartesian", "vector"-ish things, and square brackets be for all "projective" or "homogeneous" things. It's useful to be able to tell them apart quickly, as the algebra looks the same but the meaning is very different.

Outside of complex analysis, there's an even plainer context where this convention would help: computer graphics (a.k.a. the actual "real-life" application of projective geometry). In computer graphics, there are two common ways to think of a point, say, in 2-dimensional space. One way is a pair $(x,y)$, the way people normally think of coordinates. The other way is to add an extra coordinate: $x$ and $y$ along with an extra $1$. This is a practical representation, since it works naturally with affine and projective transformations, which we represent in homogeneous coordinates too. The use of both systems leads to a problem. Without any notational convention, is $(3,-4,1)$ supposed to mean a 2D point or a 3D point? Is $$ \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} -2 \\ 3 \\ 1 \\ \end{bmatrix} $$ supposed to be a translation applied to a 2D point, or a shear mapping applied to a 3D point? You need context and you can't tell at a glance. If we used round/square brackets with our designated interpretations, though, we'd know right away that $(3,-4,1)$ means a 3D point and the square-bracketed matrix above means a 2D translation.

Unfortunately, they don't actually make any notational distinction at all in computer graphics. They're just going to stay confused ;) - but you don't have to. You can use the round and square to distinguish ordinary and projective coordinates. It deserves to be more standard.

(There's another convention out there to make the distinction with commas and colons, i.e. $(a,b,c)$ is Cartesian and $(a \mathbin{:} b \mathbin{:} c)$ is homogeneous. It works well for some purposes, but not if you need the notation to generalize to matrices.)

Answer 26

For the value at $x$ of the conditional probability density function of a random variable (capital) $X,$ given the event that the value of a random variable (capital) $Y$ is $y,$ I prefer $f_{X\,\mid\,Y\,=\,y}(x)$ to the more frequently seen $f_{X\,\mid\,Y}(x\mid y).$ It emphasizes that we are concerned with a function of $x.$

Answer 27

(This would be a comment on notation for partial functions, but I don't have the reputation points, as I just joined MO.) Though this is by no means standard, for personal use I've adopted the following system of arrow decorations that captures many standard types of binary relations. For a relation f from A to B, use $\rightharpoonup$ to indicate $\forall x\in A ~\exists y \in B~~xfy$, $\rightharpoondown$ to indicate $\forall x\in A~\exists^{\leq 1} y\in B~~xfy$, $\leftharpoondown$ to indicate $\forall y\in B~\exists x\in A~~xfy$, and $\leftharpoonup$ to indicate $\forall y\in B~\exists^{\leq 1}x\in A~~xfy$. So, $\rightarrow$ is for functions, $\leftrightarrow$ is for bijections, $\leftharpoonup\hspace{-1em}\to$ is for injections, $\leftharpoondown\hspace{-1em}\to$ is for surjections, $\rightharpoondown$ is for partial functions, $\rightharpoonup$ is for serial relations, and so on.

Answer 28

I find the standard notation $X \times_G Y := \frac{X \times Y}{G}$ for balanced products of $G$-spaces annoying, because it conflicts with the well-established notation for fibre products. But on the other hand writing out the full quotient $(X \times Y)/G$ is cumbersome, especially if you're dealing with associated bundles to principal bundles, or if you have spaces with compatible left and right actions of different groups, and you need to write something like $X \times_G Y \times_H Z$.

So I came up with that following notation which combines the product symbol "$\times$" and the quotient symbol "$\_$" into a single binary symbol:

The last one, without the group decorating it, is used in the same way as tensor products $\otimes$ when the ring involved is obvious (e.g. if you're working with associated bundles to a $G$-principal bundle $P$, and you're not planning on taking any reductions of the structure group).

It's just a rotated semidirect product, so it's easy to implement in LaTeX, and looks similar enough to the usual notation that it's easy to introduce in a talk. Here's my LaTeX code:

Answer 29

In type theory, the notation $(x : A) \to B (x)$ instead of $\Pi_{x : A} B (x)$ for the dependent product and $(x : A) \times B (x)$ instead of $\Sigma _{x : A} B (x)$ for the dependent sum. For non-type-theorists, a dependent product is a set-indexed product of a collection of sets, and a dependent sum is a set-indexed disjoint union of a collection of sets[1]. The former notation is used in the Agda, the latter is less widely spread but both are used for example in cubicaltt. Some advantages of this notation are:

• It is consistent with notation for non-dependent sums and products $A \times B$ and $A \to B$ (the latter is in fact an instance of the third example of good notation given in the original question, and is already widespread among type-theorists as a notation for the set of functions from $A$ to $B$)

• The fact that $A$ does not appear as a subscript certain complex types easier to follow, as in the equation

$$ (x : A) \times ((y : B (x)) \times C (x, y)) \cong (u : (x : A) \times B (x)) \times C (\pi_1 u, \pi_2 u) $$

• The fact that $A$ does not appear as a subscript makes it easier to represent this notation in plaintext, as is frequently done when programming with dependent types.

[1] Here and in the rest of my answer, I am ignoring the distinction between a set and a type.

Answer 30

In computations in differential geometry, when we need to omit one entry of a list, I like to write $x^{\hat\imath}$ instead of $(x_1,\dots,\hat{x}_i,\dots,x_n)$, and similarly write $x_{\hat\imath}$ instead of $(x^1,\dots,\hat{x}^i,\dots,x^n)$. This makes the Einstein convention work, and simplifies the computations of relations between exterior and Lie derivatives of differential forms, for example. Similarly, $x^{\widehat{\imath\jmath}}$ to omit two entries. Careful that $x^{\hat\imath\hat\jmath}\ne x^{\widehat{\imath\jmath}}$