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## 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 AB for the set of functions from B to A is really confusing, and I find it much easier to write this set as B→A.

• 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.)

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this question is broadly useful, so perhaps better as community wiki? – suVRit Oct 20 2010 at 20:09
In set theory we write ${}^B A$ for the set of functions from $B$ to $A$. – Andres Caicedo Oct 20 2010 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 Lin Oct 20 2010 at 23:18
Arabic numerals ? Ah yes, they were transmitted to Europe by the Arabs. – Chandan Singh Dalawat Oct 21 2010 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. – Kallikanzarid Oct 21 2010 at 3:38

I like to interpret $f(x)$ as meaning $f\circ x$, otherwise known as the pullback $x^*f$. For instance $x$ could be the standard real valued coordinate on a line. This makes rigorous sense of the concept of a "variable" and hence also dependent and independent variables ($y=f(x)$). In the example of functions on a line, $f'=dy/dx$ is simply a ratio of 1-forms.

Such an interpretation also answers the common complaint that $f=f(x)$ confuses a function with its values. Instead it represents the very common shorthand of omitting pullbacks!

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This point of view is espoused in Munroe's 1958 AMM article "Bringing calculus up to date" (jstor.org/stable/2308879). He gives a very little bit of history, mostly without references. – L Spice Jun 28 at 22:01

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.

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D) Perhaps {$(a)^2, b, c$} for your multiset {{a,a,b,c}} – Qfwfq Sep 24 2011 at 21:04
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I found the notation $K_\bullet$ for a complex (in with objects an abelian category or as an objects of the derived category) is very helpful. Otherwise people have to write something like $\cdots \to K_{n}\to \cdots \to K_{2}\rightarrow K_{1} \to K_{0}$ which just contains exactly the same amount of information.

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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.

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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.

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I use $\rightharpoonup$ for partial functions. – Andreas Blass Oct 20 2010 at 22:23
Use a broken arrow instead: f : A - - - > B. I have no idea how three dots is suggestive of the domain being something smaller than A. – KConrad Oct 21 2010 at 18:15
I just meant that it suggests that $f$ is something like a function from A to B, without being intrusive. This notation is really useful in situations where you have numerous partial functions of different arities running around. – Joel David Hamkins Oct 21 2010 at 18:22
Never saw the three-dot notation. It looks like a smudge or tiny dead gnat to me. I use the notation mentioned by Andreas. – Todd Trimble Oct 30 2010 at 18:28

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.

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In Ravi Vakil's thread on teaching schemes, there was some discussion of using the notation $\mathcal{F}(x)$, since given a section $f$ of the sheaf, its values as a function on $X$ are written $f(x)$ and lie in the fibers. – Ryan Reich Oct 21 2010 at 6:38
It's this answer, and the many, many comments below it: mathoverflow.net/questions/28496/…. Note that what I wrote is given by BCnrd about halfway down. – Ryan Reich Oct 21 2010 at 6:48
@Ryan: I appreciate the $\mathcal{F}(x)$ notation, but there's a (minor?) clutching with the widely used notation $\mathcal{O}_{X}(x)$ to denote the line bundle on the algebraic curve $X$ twisted by the divisor given by the point $x\in X$. – Qfwfq Oct 21 2010 at 11:44
It's true, this notation would lead to the unfortunate equality $\mathcal{O}_X(x) = \mathcal{O}_X/\mathcal{O}_X(-x)$. – Ryan Reich Oct 21 2010 at 16:22
• 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)

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Majorization theory involves also the useful notation $x_\uparrow$ and $x_\downarrow$ for vectors with real entries. – Denis Serre May 18 2011 at 15:14
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(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.

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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.

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$$a^{\cdot\, n}=a\cdot a\cdots a$$ $$a^{\wedge\, n}=a\wedge a\wedge\dots\wedge a$$ $$a^{,\,n}=a,a,\dots,a$$ For example one could write $$\langle(x+10y-z)^{,\,2}\rangle= \langle(x+10y-z),(x+10y-z)\rangle.$$ or $$\sin^{\circ(-1)}x=\arcsin x$$ or $$\sin^{\cdot(-1)}x=\frac1{\sin x}$$

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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.

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One partial compromise is to subscript a function $f: X \to Y$ as $f_{Y \leftarrow X}$. For instance, the composition of two inclusion maps $\iota_{Z \leftarrow Y} \circ \iota_{Y \leftarrow X}$ becomes $\iota_{Z \leftarrow X}$. – Terry Tao Oct 21 2010 at 17:11
Hehe, objectively right'. – Greg Muller Oct 21 2010 at 18:42
@Terry Tao: I've seen that one from time to time, and it is pretty nice when you are dealing with different spaces (especially when you have a category of them and, as you write, a functorial assignment of maps). For self-maps of X, it leaves...something to be desired. – Ryan Reich Oct 21 2010 at 20:36
Abstractly an arrow from $X$ to $Y$ has and an orientation but no direction. It could be drawn either up/down/right/left/slant 30°/slant-45°/... . But we are heavy victims of the typographical habits, and this in spite of modern computer possibilities (not means :Tex being a terrible tool!). The best thing to do is to think of an arrow as slanted +200°or in three dimensions, then the typographical induction tends to disappear completely (at least for me and I guess for Hebrew/Arabic writers too). So the motto (not necessarily a categorist's one) is "let's do multidimensional algebra!". – Jérôme JEAN-CHARLES Nov 23 2010 at 23:50

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.

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You could use $\perp$ for that as well... – Harry Altman Oct 22 2010 at 13:28
You can use $A \parallel B$ for $A\cap B=\varnothing$, since parallel lines don't meet, and so do disjoint sets. – Asaf Karagila Oct 13 2011 at 17:31

Multi-factorials are handy. Sometimes results can be expressed compactly by introducing a double factorial or possibly higher factorial. For example

$$\int_0^{\pi/2} \sin^{2n+1} \theta \:\: d\theta = \frac{(2n)!! }{ (2n+1)!!}$$

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I wish there were a notation that didn't scream "iterated factorial", though (not that one sees this very much). I forget: does $n?$ mean anything? The question mark is handy because it suggests having to make a choice, as in "even or odd?" – Ryan Reich Nov 16 2010 at 10:48
The question mark has a meaning in C and programming languages derived from C. The notation a ? b : c; means to do b if a is true, otherwise do c. The question mark also means "optional" in regular expressions. For example, the regular expression ab?c matches abc or ac. I don't know whether either of these notations would make sense imported into math. On a related note, sometimes I would like to import C's % operator into math notation. – John D. Cook Nov 16 2010 at 23:30
True, but C also doesn't have a factorial operator, and ! means something entirely different again. There's not much reason to make mathematical notation agree with programming design choices. As for %, we always have "mod". – Ryan Reich Nov 29 2010 at 9:08
The problem with "mod" is that it is usually an equivalence relation and not a function. That is, you see "a equiv b mod m" more than "a mod m". I'm not sure the latter is common notation or that people agree in detail what it means. – John D. Cook Nov 29 2010 at 15:22

I think that Inuit numerals are cool. (http://en.wikipedia.org/wiki/Inuit_numerals) They are useful for vigesimal type things.

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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".

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The lack of a nice obviously symmetric notation for $\binom{a+b}{b}$ has bothered me; Dijkstra suggested in EWD 782 the notation $P(a,b)$, generalizing it also to $P(a_1,\ldots,a_k)$ for $\binom{a_1+\ldots+a_k}{a_1,\ldots,a_k}$. (Though I certainly disagree with him about $\binom{n}{k}$ being useless - you certainly do want to think about it that way a lot of the time.) I haven't actually had any reason to use this since I saw it but I can certainly think of times I would have.

Also the double-parentheses multichoose notation $\left(\!\binom{n}{k}\!\right)$ is nice because it lets you say "...and this is n multichoose k (which is equal to this binomial coefficient)" instead of just jumping directly to a binomial coefficient whose relevance may not be immediately obvious. But I suppose that's not really on the level of giving you a better way to look at things.

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Isn't there a nice obviously symmetric notation for ${a+b} \choose b$, namely ${a+b} \choose {a,b}$? – Rasmus Oct 21 2010 at 18:34

In the notation of Time scale calculus, the ordinary calculus derivative df/dt and the forward difference operator $\Delta f$ are both written as $f^\Delta$. Indefinite sums and indefinite integrals are both written as $\int{f(t)\Delta t}$ and called indefinite integrals. The context would say $\mathbb{T}=\mathbb{Z}, \mathbb{T}=\mathbb{R}$ or other $\mathbb{T}\subset\mathbb{R}$.

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Cauchy-Binet as a generalized Pythagoras theorem.

Let $X$ be an $n \times k$ matrix with $n \ge k$. For any $k$-index $I=i_1...i_k, \; 1 \le i_1 < ... < i_k \le n$, there is some advantage to denote by $X_I$, the determinant of the $k \times k$ submatrix of $X$ with rows indexed by $I$. For any two such $X,Y$, we can state the Cauchy-Binet formula as a pairing $$\det (X^TY)= \sum_{I} X_I Y_I$$ where the sum is over all $n \choose k$ $k$-indices. This is a Pythagoras theorem for $X=Y$ since it says that the the volume-squared of the parallelepiped spanned by the $k$ columns of $X$ in $\mathbb{R}^n$ is the sum of squares of the volume of the projections on the $n \choose k$ $k$-dimensional coordinates.

For any $n \times m$ matrix $A$ with $m,n \ge k$ and $k$ indices $I,J$, we also denote by $A_{IJ}$ the determinant of the $k \times k$ submatrix of $A$ with rows indexed by $I$ and column indexed by $J$. Then for $X(m \times k)$ and $Y(n \times k)$, we have by Cauchy-Binet twice, $$\det(X^TAY)=\det(X^T(AY))=\sum_{I}X_I(AY)_I =\sum_I X_I \det(A^IY)=\sum_I X_I \sum_J A_{IJ} Y_J,$$ where $A^I$ is the $k \times n$ matrix given by the rows of $A$ indexed by $I$ and we note that $(AY)_I= \det(A^IY)$ and $(A^I)^T_J=A_{IJ}$. This notation thus allows us to view Cauchy-Binet (usually stated with $m=n,A=I$) as an extension of the usual $x^TAy=\sum_{ij}A_{ij}x_iy_j$ for $k=1$.

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This is not really about notation, but it's a very good point. – darij grinberg Nov 2 2010 at 0:09

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) $\mathsf{TOP}(n)=\mathrm{Aut}_{\mathsf{TOP}}(\mathbb{R}^n)$.

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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:

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 $A[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$.

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

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 $A[[X_{1..n},Y_{1..n}]]$.

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

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I use the notation

$V \oplus^{\perp}W$

to denote orthogonal direct sum [Edit: direct sum of, say, subspaces of a given inner-product space].

Or

$(M,g) \times^{\perp} (N,g')$, or simply $M \times^{\perp} N$, to denote (orthogonal) cartesian product of Riemannian manifolds.

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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...

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Using $(a, b, ... )$ is handy to denote a column vector, which is the transpose of the row vector $[a, b, ... ]$, especially in linear text. Correspondingly, all displayed matrices should be written with brackets, not parentheses. This notation agrees with the usual identification of coordinates with column vectors.

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I take it you also hate having to write the transpose when you're constrained to writing on one line? :) – J. M. Oct 21 2010 at 11:45
Yes, particularly when stacking vectors $a, b, ...$, in which case you have to write the artificial $[a^T, b^T, ... ]^T$ without this convention. – John Bentin Oct 21 2010 at 13:52
It is, of course, not $[a, b, \dots]^T$ but $[a\;b\;\dots]^T$. This is an awful convention because even once you remind yourself why the $T$ is there you (or I at least) am not convinced that it has any meaning other than to satisfy a badly chosen precedent. I don't know why anyone would vote this down. – Ryan Reich Oct 21 2010 at 16:27
Don't forget the invariant literature which uses $\left[v_1,...,v_n\right]$ not for the matrix formed by the columns $v_1$, ..., $v_n$, but for its determiannt... – darij grinberg Oct 21 2010 at 17:04

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?

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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 2010 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

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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.

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Instead of writing $$|x-y|\le \varepsilon,$$ I used to write $$x\lessgtr y\pm \varepsilon.$$ You may read it as $x$ is more-or-less $y$ plus-minus $\varepsilon$.

One may also write something like $$x\lessgtr e^{\pm\varepsilon}\cdot y$$ which is much better than $$|\ln(y/x)|\le\varepsilon$$

It is easier to read, especially if instead of $x$ and $y$ you have long expressions.

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What would you write for $|x-y|\geq \epsilon$? – Nate Ackerman Jan 24 at 23:24
I'm not happy about the "principle" in Anton's reply to Nate's comment. In the body of the question, the upper inequality (with $<$ and $+\varepsilon$) and the lower inequality (with $>$ and $-\varepsilon$) are to be understood as combined by "and", whereas in the comment, the upper and lower inequalities are intended to be combined by "or". Allowing both uses of the notation seems to be inviting confusion. – Andreas Blass Jan 26 at 16:22
Andreas, you are right, I made this reply without much thinking :) – Anton Petrunin Jan 30 at 19:10

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 $f(I)B$.

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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."

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@David: I tend to use $\hookrightarrow$ for maps that are literally inclusions. – Andreas Blass Nov 25 2011 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 2011 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 2011 at 10:43
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$\sin^{-1}(x)$ as opposed to $\text{arcsin}(x)$. This encapsulates the fact that it is an inverse function.

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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 2010 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 2010 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 2010 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 2010 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 2010 at 11:37