Suggestions for good notation

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 Σ1≤n<x [n prime] is the number of primes less than x, and the unmemorable and confusing Kronecker delta function δn 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 BA 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.)

• this question is broadly useful, so perhaps better as community wiki? – Suvrit Oct 20 '10 at 20:09
• In set theory we write ${}^B A$ for the set of functions from $B$ to $A$. – Andrés E. Caicedo Oct 20 '10 at 20:54
• I've always assumed that the notation $A^B$ is because of the "exponential law" $(A^B)^C = A^{B\times C}$ ... – Kevin H. Lin Oct 20 '10 at 23:18
• Arabic numerals ? Ah yes, they were transmitted to Europe by the Arabs. – Chandan Singh Dalawat Oct 21 '10 at 3:29
• Isn't $x \mapsto f(x)$ commonplace? As for homomorphisms, they are not simply maps, and $\mathrm{Hom}(A, B)$ denotes the whole class, while $A \to B$ denotes a single mapping. – Alexei Averchenko Oct 21 '10 at 3:38

I like the notation $A\mathrel{\in\in}\mathcal C$ for objects (rather than morphisms) in a category (not my own invention).

• It looks like a typo. – David Handelman Jan 28 '18 at 0:37

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 $$\lim_{x\to 0^+}f(x)$$.

• For $\lim_{x\to0^+}$ or $\lim_{x\searrow0}$ one could almost imagine $\lim_{0\leftarrow x}$... but then $\lim_{x \to0}$ would become ambiguous. – Zach Teitler May 13 '19 at 19:14
• I have seen $\lim\limits_{x\downarrow 0}$ and $\lim\limits_{x\uparrow 0}$ – Mr Pie May 13 '19 at 21:32
• @ZachTeitler, obviously the solution is to write $\lim_{x \rightarrow 0 \leftarrow x}$ for the two-sided limit. :-) – LSpice Jul 7 '19 at 12:40
• Obviously, not $\lim_{0 \leftarrow x \rightarrow 0}$, that would be ridiculous... :-) – Zach Teitler Jul 8 '19 at 23:13

I recently saw the following notation in the context of divisors on algebraic varieties, and I liked it very much.

Suppose that $D$ and $E$ are reduced divisors on a normal algebraic variety $X$. One can use $D \vee E$ to denote the reduced divisor with support equal to $D + E$ and $D \wedge E$ to denote $(D + E) - (D \vee E)$. I could imagine variants on this if $D$ and $E$ are non-reduced (involving taking max's, respecitvely mins, of the divisors component-wise).

EDIT: I'm slightly curious as to why this was downvoted. I guess it's too common to be interesting?

• It is not me , I do not know algebraic variety, but as a guess may be there is some order there and so you have an inf and sup or even a lattice that would fully justify these notation and be rather common at that. – Jérôme JEAN-CHARLES Nov 23 '10 at 23:54

Has anyone come across any more similar examples of good notation that should be better known?

Some interesting glyphs:

1. Combinatorial Principles in Set Theory:

2. Bisimulation:

• Given two states p and q in S, p is bisimilar to q, written p ~ q, if there is a bisimulation R such that (p,q) is in R.
3. Boxplus operator in Coding Theory

• I always considered 1. to be a prototypical example of bad notation. – Emil Jeřábek supports Monica Dec 19 '11 at 13:27
• @EmilJeřábek, I think that (2) has to be a still more prototypical example, in the sense that, without additional context, no two people will agree which equivalence relation is denoted by $\sim$ (although I guess we all agree it is an equivalence relation?). – LSpice Jul 7 '19 at 12:31

In algebra, it is very useful to write $J\cap A$ for the inverse image of an ideal $J$ in a ring $B$ under a homomorphism $f:A\to B$, rather than $f^{-1}(J)$. I normally also omit the morphism and write $IB$ for the ideal generated by the image in $B$ of an ideal $I$ in $A$, rather than $f(I)B$.

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:

• I learned from Indranil Biswas to use $X \times^G Y$ for $(X \times Y)/G$. I think this notation is becoming standard. – Ben McKay Jul 15 '17 at 10:26
• I also think the notation $X\times^G Y$ is the standard one. – Qfwfq Jan 14 '18 at 0:49
• Unfortunately not standard in differential geometry yet---everywhere I look people are still using the ambiguous fibre-product notation. – ಠ_ಠ Jan 14 '18 at 13:09

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.)

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.

Many years ago, I invented the notation $\mathcal{Lex}_{T}(X)$ as the function equal to $1$ if and only if the consideration of $X$ in the theory $T$ entails no contradiction and $0$ otherwise. Lex is both "law" in Latin and a shortcut for "Logical existence", which in some sense is the only law of mathematics.

• Why the down vote? Could someone familiar with the subject please explain. – Ben McKay Jul 15 '17 at 10:27
• I'm afraid nobody works on this. I got interested in the subject more than 15 years ago, after a friend of mine starting to study maths told me something like "the integers of the empty set are even" and "the integers of the empty set are odd" were both equally true. I felt shocked and tried to figure out a way to forbid the consideration of "integers of the empty set" and other impossible concepts. – Sylvain JULIEN Jul 15 '17 at 10:37
• Isn't this just $\operatorname{Con}(T \cup \{X\})$ or am I misinterpreting? – user76284 Oct 27 '19 at 0:12
• It seems $\operatorname{Con}$ is a unary predicate, while $\mathcal{Lex}$ is a map with values in $\{0,1\}$. But otherwise, the idea is the same. – Sylvain JULIEN Oct 27 '19 at 9:32

$x\gtrless 0$ to denote that $x\in\mathbb{R}\setminus\{0\}$. Or, if $x\notin\mathbb{R}$ but rather $x\in\mathbb{Z}\setminus\{0\}$ for example, then we write $x\in\mathbb{Z}_{\gtrless 0}$. I prefer the $\gtrless$ notation because it is simpler and more understandable when first met with the eye, and I hope it is set in stone sooner or later.

I also came across the following notation: $$\sum_p'f(x).$$ The little dash $'$ on top of the sigma $\Sigma$ denotes that $p$ is prime. I like this, but I don't feel this is necessary because we can write $$\sum_{p \ \text{prime}}f(x)$$ but when looking at both conventions, the former looks a lot neater.

Also, we have notation $>$, $\gg$ and $\ggg$ with $<$, $\ll$, and $\lll$ but I was thinking, is there a notation to denote that a value $a$ is not much less than a value $b$? I came up with $$a\overline{<} b\tag*{a is not much less than b}$$ $$a\overline{>}b\tag*{a is not much greater than b}$$

I have also seen that the symbol for concatenation is sometimes $||$, but if I was to see $$A \ || \ B$$ then my first thought would be $A$ is parallel to $B$, but that is just me. With some research, I discovered that there is however some kind of official notation, such that $A^\frown B$ but I don't like it.

I also went extreme and invented this: $$\mathop{\LARGE\Omega}_{\substack{x=k \\ \\ R}}^{x_0} (x, y)$$ to define a set of coordinates; a relation. Here, $x\to x_0$ and $R$ simply denotes the rule, $y = \cdots$. This way, we can write stuff like $$\mathop{\LARGE\Omega}_{\substack{x=36 \\ \\ y = 2x+1}}^{144}(x, y)$$ And then if we don't have a rule, but something like $y\geq0$ for example, then we can write a double index to refer to what value $y$ tends towards, namely $y_0$.

Oh and I almost forgot, I thought that maybe we can symbolise contradiction? For instance, I want to prove that $\sqrt{2}$ is irrational. I would first suppose it is rational, then come to a conclusion that contradicts this statement. Since most of us use $\Box$ in notation of completing a proof, I decided that I could use $\Diamond$ in notation of a contradiction. I thought about it because it is like a titled box $-$ a bit like approaching the proof $\Box$ on a different angle $\Diamond$, if you get what I am saying.

I am not taking this too seriously, for I was just being creative, but I believe there's nothing wrong in trying new things out. Any thoughts?

• Re: your second-to-last paragraph, there are multiple symbols used for proof-by-contradiction - my personal favorite being "$\perp$," but "$\Rightarrow\Leftarrow$" is also common, as is some form of "lightning bolt." I would strongly object to "$\Diamond$," however, given the meaning of $\Diamond$ and $\Box$ in modal logic. – Noah Schweber May 13 '19 at 18:08
• Is your $\Omega$ thing any different than $\{(x,y) \colon 36 \leq x \leq 144, y=2x+1\}$? – Zach Teitler May 13 '19 at 19:18
• @ZachTeitler nah, it's the same thing. The $\Omega$ thing is easier to write, but the "normal way of writing it" is easier to read ;) – Mr Pie May 13 '19 at 21:30

I like the notation $f:A\cong\subseteq B$ for "$f$ is an embedding of $A$ into $B$." The idea is that the relation of embeddability is obtained by composing the relations "isomorphic to" and "substructure of."

• Hmm, others seem to disagree. What do you think of $\hookrightarrow$? – David Roberts Nov 24 '11 at 23:11
• @David: I tend to use $\hookrightarrow$ for maps that are literally inclusions. – Andreas Blass Nov 25 '11 at 4:00
• I've mostly seen people use $\subset$ or $\subseteq$ for literal inclusions and $\hookrightarrow$ for embeddings (or whatever kind of injection is suitable). Reserving $\hookrightarrow$ for literal inclusions seems kind of pointless when $\subset$ exists. – Ketil Tveiten Nov 25 '11 at 8:34
• ... and then you can decorate the arrow with o or | to incorporate meaning like "open immersion" or "closed immersion". – Konrad Voelkel Dec 19 '11 at 10:43

$\sin^{-1}(x)$ as opposed to $\text{arcsin}(x)$. This encapsulates the fact that it is an inverse function.

• And, combined with $sin^2(x)$ for $(sin(x))^2$, it make one ponder the meaning of $sin^n(x)$ when $n=-1$. – Harald Hanche-Olsen Oct 20 '10 at 20:10
• I have to go with this notation being anti-useful, since it is in fact not an inverse function except on a restricted domain (and that domain is different than the one for $\cos^{-1}$). Not that I have much call for it, but I decided once that I would never use $\sin^{-1}$. It takes greater fortitude to abandon $\sin^2$. – Ryan Reich Oct 20 '10 at 20:17
• I have also permanently abandoned $\sin^{-1}$ after having numerous students mysteriously convert an $\arcsin$ into a $\csc$ without realizing it. (Of course, the notations $\sec$, $\csc$ and $\cot$ are almost completely worthless themselves.) – JBL Oct 20 '10 at 21:34
• On this subject, $\arcsin(x)$ is actually great notation because it reminds you what the restricted domain is. That is, it gives outputs which are lengths of arcs (measured, as always, from the positive x-axis). Of course, this is never mentioned. – Ryan Reich Oct 20 '10 at 21:56
• Gauss raged against the deplorable sin$^2x$ notation more than 150 years ago (sorry, I don't have the reference.) I guess we're stuck with it, though, since it's concise and ubiquitous in school maths. Also, how often do we really need the multiple composition of trig functions? – John Bentin Oct 21 '10 at 11:37