I've seen expressions like $x\lt y\in S$ used on this site to mean "$x\lt y$ and $x\in S$ and $y\in S$." This has me worried because I sometimes use $x\lt y\in S$ to abbreviate "$x\lt y$ and $y\in S$". How widespread is the former usage? Should I avoid using abbreviations such as $x\lt y\in S$ and $x\ne y\in S$ etc. because I'm likely to be misunderstood?
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6$\begingroup$ It's ambiguous. It's terrible notation. $S$ could be a set of ordered pairs and $\lt$ could be a relation thereon. You should avoid it at the cost of a few more keystrokes. $\endgroup$– David Roberts ModDec 23, 2020 at 2:44

1$\begingroup$ Since you asked how widespread it is, here is search for $x\lt y\in S$ on SearchOnMath with the domain limited to MO. (You can also change to other domains  just to have some comparison.) Already on the first page, it shows many different expressions. (I am not sure how exactly this search engine decides which two expressions are similar.) Here is the same search on ApproachZero. (A0 indexes Mathematics and AoPS, but not MO.) $\endgroup$– Martin SleziakDec 23, 2020 at 8:33

$\begingroup$ @bof Of course, in both cases  in SearchOnMath and in Approach Zero you can use this as a search query, too. (I am not sure to which extent this could be helpful. But at least among the search results you can see some occurrences  still, you'd have to check them individually to see what meaning exactly the author of the post has in mind. As you said in the last comment, for this you can't really rely on computer.) $\endgroup$– Martin SleziakDec 23, 2020 at 14:24

$\begingroup$ @bof all three of your examples are unambiguous because of how those relation work. The problem with $x\lt y \in S$ is, as you pointed out, is that people can quite reasonably assume it means one of two mutually exclusive things. $\endgroup$– David Roberts ModDec 23, 2020 at 22:42

2$\begingroup$ I’m voting to close this question because it should be posted on main and not on meta. $\endgroup$– Federico PoloniJan 3, 2021 at 11:18

$\begingroup$ @bof Your question does not apply only to writing mathematics on the main site, but to writing mathematics in general. In its current version, you didn't even write "on MO" in it (and even if you would that would be a boatprogramming specification). Anyhow, we have other users and moderators to take this decision; this is just my opinion and I do not claim I know more. :) $\endgroup$– Federico PoloniJan 3, 2021 at 13:23

$\begingroup$ @FedericoPoloni, re, what is "a boatprogramming specification"? $\endgroup$– LSpiceJun 30, 2023 at 1:35

$\begingroup$ @LSpice This page should explain it: meta.stackexchange.com/questions/14470/… $\endgroup$– Federico PoloniJun 30, 2023 at 6:18
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2 Answers
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Yes, ambiguous notation should be avoided in places with a general audience, like journals and here on MathOverflow. It's useful to have shorthand in specific places where all readers can agree on the meaning but it is evidently incorrect to assume that everyone everywhere agrees on that same meaning.

$\begingroup$ Nonetheless, there are surely universally accepted ambiguities. Not only does (almost?) noone refer to "the group $(G, \cdot)$", and almost everyone to "the group $G$" (in general; of course one might write something like "the monoid $(\mathbb Z, \cdot)$ versus the monoid $(\mathbb Z, +)$"), doing so would surely be regarded as obscuring, not clarifying. So I think one cannot respond: always write precisely!; but rather: only use almost universally understood ambiguities! And from this point of view the question can be understood as: is this amiguity universally understood, and, if so, how? $\endgroup$– LSpiceJun 30, 2023 at 1:40

$\begingroup$ @LSpice Ambiguous usually means something that has two or more plausible readings. My personal favorite is "$1=0!$" I don't think "the group $G$" is ambiguous in that sense. Nobody would say the phrase "this is a pen" is ambiguous just because you don't know the color. $\endgroup$ Jun 30, 2023 at 2:56

$\begingroup$ Re, I think that "the monoid $\mathbb Z$" has at least two plausible readings, and that it is quite common to omit the operation if the context makes it clear (though also quite common to specify it). (For that matter, it is possible, though not likely, to put different group operations on $\mathbb Z$, so even "the group $\mathbb Z$" is technically ambiguous.) But I was not claiming that ambiguity is good, or even acceptable, in this particular case, only that I think that the dictum "never write ambiguously" is not commonly practiced. $\endgroup$– LSpiceJun 30, 2023 at 3:48

$\begingroup$ Re, also, I think that that analogy may not be perfect; a pen is a pen regardless of its colour, but a set is simply not a group, even if it has a canonical group structure (sometimes a unique one up to choice of identity, but never, except the onepoint set, genuinely a unique one). $\endgroup$– LSpiceJun 30, 2023 at 3:49

1$\begingroup$ Ironically, I think we're arguing about the meaning of ambiguity, not about mathematical practice. What you describe is what I would call imprecise or inexact. These are cases where the meaning can be clarified with additional context. When you say "the group $G$" it is (unambiguously) clear that you mean that $G$ is endowed with a group structure that satisfies the group axioms. It is not immediately clear what this structure is but that just missing context. I think this is part of language and it has nothing to do with mathematical practice per se. $\endgroup$ Jun 30, 2023 at 5:15

$\begingroup$ By "ambiguous" I mean something that has two or more plausible meanings even with full context. For example, every paper or book on partial combinatory algebras says somewhere that $abc$ means $(ab)c$. That's because the application operation is not associative and thus $abc$ is ambiguous. Interestingly, this is done even though everybody in the field uses this convention. $\endgroup$ Jun 30, 2023 at 5:21

$\begingroup$ The only "universally accepted ambiguity" I can think of is precedence of operations. Everywhere, even outside mathematics, $ab+c$ means $(ab)+c$ and not $a(b+c)$. $\endgroup$ Jun 30, 2023 at 5:24

1$\begingroup$ Actually, there's more like the annoying $\sin^2(x)$ convention. $\endgroup$ Jun 30, 2023 at 5:33

1$\begingroup$ Re, I have waged a quixotic battle against that all my years of teaching. I use $\sin(x)^2$, which students insist on reading as $\sin(x^2)$ because parentheses seem to mean nothing to them, so I am stuck with the abomination $\bigl(\sin(x)\bigr)^2$, which makes noone happy. $\endgroup$– LSpiceJun 30, 2023 at 13:43
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The short versions of your two suggested meanings are: $$x,y\in S,\, x<y$$ or $$y\in S,\, x<y$$ Neither require much more typing or space than the original ambiguous expresion.

$\begingroup$ I can't help pointing out that even the first can potentially be ambiguous, since commas are implicitly standing for conjunction in contextdepend ways—i.e., $x, y \in S$ means $x \in S \land y \in S$, but $x, y \in S, x < y$ means $(x \in S \land y \in S) \land (x < y)$. For an extremely contrived example of ambiguity, one could imagine $S$ being something like a set of strings in a certain language, with $<$ being the superstring relation, in which case we could have $1 = 1, 1 \in S, 1 = 1 < 1$. $\endgroup$– LSpiceJun 30, 2023 at 1:38