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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 $\operatorname{gcd}$.

Bill Dubuque
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Zlatan der Zechpreller
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    I'm far from a professional, but I find $(a,b)$ awful. – Git Gud Jul 01 '14 at 23:20
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    I also think that $(a,b)$ is a better notation! – evinda Jul 01 '14 at 23:21
  • by the way the phrasing of the last sentence shouldn't be taken too seriously, I hope this is ok here. I just noticed that some mathematicians tend to use the $(a,b)$ notation, whereas the man of the street always uses gcd – Zlatan der Zechpreller Jul 01 '14 at 23:23
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    The (bad) notation $(a,b)$ is often used for the greatest common divisor; somebody uses $[a,b]$ for the lowest common multiple. Why are they bad, in particular the former? Because $(a,b)$ is already used for several other purposes. – egreg Jul 01 '14 at 23:23
  • @egreg: I think bill dubuque uses $[a,b]$. I agree that they are bad in the sense that several things are denoted $(a,b)$, but confusion almost never arises imo. amusingly atiyah and macdonald use hcf for $(a,b)$, very weird notation indeed – Zlatan der Zechpreller Jul 01 '14 at 23:37
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    The best notation I've seen and use is $(a,b)$ for gcd and $[a,b]$ for lcm. I would never use ${a,b}$ for either; it's a two-element set. – anon Jul 01 '14 at 23:52
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    I find nothing unprofessional about using $\operatorname{lcm}(a,b)$ or $\gcd(a,b)$. – Bonnaduck Jul 01 '14 at 23:54
  • @blue By the same token $(a,b)$ is an ordered pair (and it's not unreasonable to thing about ordered pairs in this context). – Git Gud Jul 01 '14 at 23:57
  • @Bonnaduck: see my first comment. – Zlatan der Zechpreller Jul 01 '14 at 23:59
  • There is always a cost-benefit analysis attached to any notational choice between abusive convenience and unambiguous clunkiness. If almost any realistically-occurring potentially ambiguous usage is actual trivial to correctly interpret in context, then the choice is obvious. – anon Jul 02 '14 at 00:02
  • The ambiguities and confusions arising from this notation have already been mentioned by plenty others here, but I have seen no mention of how pretentious this particular notation is. How about defining a radically different meaning for the plus sign, while we're at it? I'd like to think I'm fairly literate, but realistically, even if I saw a note at the beginning explaining this notation, I would still get confused later on in the paper. I don't like $(a, b)$ either, but I guess that's one we just have to learn to live with. – Robert Soupe Jul 02 '14 at 03:11

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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.

Bill Dubuque
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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.

J. W. Perry
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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.

mr_e_man
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  • Using $a \sqcup b$ for arithmetic minimum might be intelligible, but using anything cup-like for the GCD is going to confuse a great many people. The relation $a\mid b$ (which you are unnecessarily writing as $\lt!!!!{\cdot};$) is a partial order. In this context, symbols like $\lor$ and $\cup$ are invariably used to mean the join operation, which is the LCM, and not the meet operation, which is the GCD. Compare this with the completely analogous relation of set containment. In this partial order, the analogue of max and GCD is set union ($\cup$), not anything like $\sqcap$. – MJD Dec 13 '23 at 19:11
  • Similarly in a Boolean lattice, the join operation (max) is logical or $\lor$, and the meet operation (min) is logical and $\land$. See for example the Wikipedia illustration for “Join and meet” – MJD Dec 13 '23 at 19:15
  • I know divisibility is usually denoted $a\mid b$. I don't like that because the relation is asymmetric but the symbol is symmetric, and it doesn't have a reversed form. And as I suggested with the link (but maybe my intention wasn't clear), I believe the standard meet and join symbols are upside-down. – mr_e_man Dec 13 '23 at 19:23
  • The logical symbols make sense to me; $\land$ increases information, and $\lor$ decreases information. But the corresponding set symbols are opposite. To have a larger amount of information about a number, is to have a smaller set containing that number. Consider that logical implication $p\to q$ is sometimes denoted $p\supset q$, even though the corresponding expression in terms of sets is $P\subset Q$ (which means $(x\in P)\to(x\in Q)$). The union symbol should point upward, as it increases the set, but $\cup$ points downward. Oh well, I can't change that. Flip my symbols if you want. – mr_e_man Dec 13 '23 at 20:04
  • I forgot to mention another problem with $a\mid b$: vertical lines have so many other meanings. There's the numeral $1$, the letters I and l, absolute value $|x|$, set-builder notation ${x\mid x^2<2}$, function restriction $f\big|_S$.... See the comments under https://math.stackexchange.com/questions/2642658/absolute-value-notation-is-ambiguous, also https://math.stackexchange.com/questions/128004/is-the-notation-a-mid-b-standard-notation-for-a-divides-b – mr_e_man Dec 18 '23 at 22:19
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"$(a,b)$" 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.

Eric Towers
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