Notation for the least common multiple and greatest common divisor

Question

Suppose I use the notation $\{a,b \}$ for the least common multiple of integers $a$ and $b$. Is this common notation in number theory (it is from Hardy and Wright), and would a mathematically literate human being know that it is notation for the least common multiple? I don't want to use the high school student's $\operatorname{lcm}$, since it doesn't fit to the professonial's $(a,b)$ notation for the high school student's $\operatorname{gcd}$.

Answer 1

Clarity is key here. If you define your notation beforehand then your subsequent text is safe, but that does not mean that you made the best choice (your peers and readers make that call).

Assuming that a notation used by Hardy and Wright is understood without definition by a mathematically literate human today would be a mistake. If you have to ask if something is commonly understood, then it is probably not commonly understood.

The notations $\gcd(a,b)$ and $\operatorname{lcm}(a,b)$ are very well known by most literates. However, when one wants to use an operator many times in a dense amount of text it is understandable that one would want a non-conflicting abbreviated notation. I personally find $(a,b)$ for $\gcd$ to be quite common in modern texts. Your curly brackets for $\operatorname{lcm}$ would not be so common. I have however seen hard brackets in modern number theory lit.

In the end, when in doubt, and you need to repeatedly use an operator to the extent that an abbreviation makes sense, and you are not absolutely convinced that your choice of notations is absolutely commonly understood (merely having to ask puts the notation in this category), just define your notation and be careful to use a socially agreeable notation that does not introduce latent ambiguities.

Answer 2

"$(a,b)$" is exactly the right thing to write. It is the ideal generated by $a$ and $b$. In the integers, this happens to be the ideal $(\gcd(a,b))$.

Belaboring my comment, below: "$\{a,b\}$" is the two element set containing $a$ and $b$. Anyone who says differently is selling something. Since you should naturally want to take GCDs of more than two integers at a time, you'll just apparently be writing larger subset of the integers.

Additionally, $[x,y]$ and $\{x,y\}$ drag all sorts of commutator and anticommutator baggage when I see them. Also, Poisson brackets. If instead of explicitly writing what operation you intend, you choose to abuse brackets, you will replace with precision with obscurity.

Answer 3

In my experience, in recent number theoretical literature that employs abbreviated notation for the gcd and lcm, the notation $\,(a,b) := \gcd(a,b)\,$ is almost universal, and $\,[a,b] := {\rm lcm}(a,b)\,$ is the most common lcm abbreviation, but far less universal than the gcd notation (so define it if you use it). The natural (multi)set $n$-ary extensions $\gcd S$ and ${\rm lcm}\, S$ are also commonly used. For ideals it is common to define the gcd as the sum $A+B,\,$ and the lcm as the intersection $A\cap B$. Some authors also employ lattice theoretic notation such as $\,a\vee b\,$ and $\,a\wedge b\,$.

One of the primary reasons for adopting the notation $\,(a,b)\,$ for the gcd is that it serves to strongly emphasize the analogy between gcds and ideals that holds in many familiar domains, e.g. in a PID we have $\,(a,b) = (c)\,$ iff $\,c\,$ is a gcd of $\,a,b.\,$ So we can view $\,(a,b)\,$ as either an ideal or a gcd (defined up to a unit factor), and this allows us to give proofs that work both for ideals and gcds, since both satisfy common laws, e.g. associative, commutative, distributive, and $\,(a,b) = (a)\,$ if $\,a\mid b.\,$ For example, see this answer and see this proof of the Freshman's Dream $\,(a,b)^n = (a^n,b^n),\,$ which works for both gcds and invertible ideals.

Remark $ $ The abbreviated notations chosen in much older textbooks were often constrained by the typesetting technology available at the time. Nowadays, no such constraints exist (e.g. using $\TeX).$ Indeed, now we can design new symbols for such purposes that avoid any possibility of confusion with existing notation. While some authors have done just that, none of these notations have yet percolated into the mainstream. That may occur someday if a good design is employed in an influential publication.

Answer 4

I have used the notation $a\sqcup b$ for the minimum of $a$ and $b$, and $a\sqcap b$ for the maximum. (No, my symbols aren't upside-down. Compare this with floor notation; $\lfloor a\rfloor\leq a$, and likewise $a\sqcup b\leq a$.)

Since the GCD is the "multiplicative minimum", we could denote it as $a\,{\sqcup}\!\!\!{\cdot}\;\,b$, and similarly the LCM as $a\,{\sqcap}\!\!\!{\cdot}\;\,b$.

In conjunction with these, the divisibility relation could be $a\lt\!\!\!\!{\cdot}\;\,b$. It means there exists $x$ such that $a\cdot x=b$, just like $a\leq b$ means there exists $x$ such that $a+x=b$. (This is assuming all variables are in $\mathbb N\ni0$.)

$$a\,{\sqcup}\!\!\!{\cdot}\;\,b\quad\lt\!\!\!\!{\cdot}\;\,\quad a\quad\lt\!\!\!\!{\cdot}\;\,\quad a\,{\sqcap}\!\!\!{\cdot}\;\,b$$

$$a\cdot(b\;{\sqcup}\!\!\!{\cdot}\;\;c)\quad=\quad a\cdot b\;{\sqcup}\!\!\!{\cdot}\;\;a\cdot c$$

It's certainly not a common notation; it must be defined where it's used.