What is the difference between a maximal element and largest element?

[wild cliff]

These seem to be the same

[drifting linden]

<@&987246746478460948> please have a look, thanks.

[drifting linden]

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[acoustic rose]

maximal and largest are basically synonymous yeah, they mean the same thing

the only nuance is if you're talking about numbers which may be negative, then largest will always mean literally the largest, whereas maximal might mean maximal in absolute value

so like the largest out of {-3, 2} will always mean 2, but if someone says maximal they might mean -3 because |-3| = 3 > 2 = |2|

as usual common sense and context determine what the writer means

also "largest" almost always refers to the normal ordering on real numbers, or on sizes of sets, whereas "maximal" gets used for almost any partial ordering, such as containment of subsets and things like this as well as for sizes of numbers, so the contexts where you will see largest vs. maximal differ slightly

that covers basically all the linguistic nuance