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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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.

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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.

how widespreadit 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 Sleziak Dec 23 '20 at 8:33meaningexactly 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 Sleziak Dec 23 '20 at 14:24