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My question, right up front, is: what is the term for a modifier that behaves this way? But "this way" takes some explanation, and that is the rest of the question.

I am a mathematician, and my question makes the most sense in a context where words are formally defined anyway, but you can freely substitute "foo" and "bar" for any technical jargon.

A ring is defined to be a set with two operations that satisfy certain properties. Among these properties, there is not universal agreement: should, or should not, the ring be required to admit a multiplicative identity ('unit')? For a mathematician who does not require this, it is easy to indicate when they wish temporarily to impose the hypothesis: they can just refer to a "unital ring".

A mathematician who will almost always consider only unital rings might decide to make that part of the definition of a ring, so that they can say "ring" where a more permissive mathematician would say "unital ring". (This lightweight controversy is discussed in the Wikipedia article.) However, this latter kind of mathematician, on making the rare encounter of a ring that does not have a unit, must either make up an entirely new term for it, or call it a "non-unital ring".

This usage is almost universally understood, but somewhat puzzling: while it makes sense to understand, e.g., a "commutative ring" as being a structure that it is a ring, and also satisfies the requirement that its multiplication be commutative, there is no way that one can so interpret "non-unital ring" if a ring, by definition, has a unit.

So, what's happening here is that "non-unital" is not refining "ring" by adding conditions, but changing the meaning of "ring" by dropping existing conditions. That is, you know something about a commutative ring even without cracking open the definition of "ring" (namely, that it has a commutative operation); but, to know something about a non-unital ring, you must not only crack open the definition of "ring", but recognize which among the properties one is expected to remove.

I'm looking for a word describing the behavior or function of modifiers that behave like "non-unital", in the sense outlined above.

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  • Which do you want, a modifier that changes the meaning by widening its scope, or one that specifically refers to objects that are inside the wide scope but outside the narrow scope? In contexts where "ring" is used to mean "unital ring", the term "rng" is used to mean ring but with the unit condition waived. By contrast, "non-unital ring" would indicate specifically a ring with no unit.
    – Rosie F
    Jul 12, 2023 at 7:14
  • Like how a "fake diamond" isn't a diamond? If you put something like that instead of going on about rings it'd help us all.
    – Stuart F
    Jul 12, 2023 at 9:09
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    Would further examples of what you are looking for be non-alchoholic beer, non-profit business, childless parents, non-residential house, irreligious church, unskilled tradesman, inedible snacks etc.? And if not, how are the two concepts different?
    – DW256
    Jul 12, 2023 at 9:55
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    The answer to your question is found at Name for when an adjective modifying a noun changes the class of objects the noun describes Jul 12, 2023 at 10:05
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    @RosieF, re, you'd think, but, in my circle (of non-nLab habitués), "rng" tends to be less used and "non-unital ring" more so, even when, the logical issues described in this question aside, it really means "not necessarily unital ring". (Similarly, almost everyone calls Lie algebras "non-associative algebra", even though there is a (boring) associative Lie algebra in each dimension.)
    – LSpice
    Jul 12, 2023 at 14:11

1 Answer 1

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The concept described is covered by privative adjectives.

From Wikipedia:

In linguistics, a privative adjective is an adjective which seems to exclude members of the extension of the noun which it modifies.

...

Partee (1997) argued that privative adjectives ... coerce a broader interpretation of the nouns they modify.

So a non-unital ring brings to mind something that would not usually fit into the category of things designated by ring (lacking the designated critical attribute), but rather some new interpretation of the category of things designated by the privative adjective non-unital and noun ring.

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  • Thanks! I didn't get a chance to accept before the question was closed, but I hope you still get the reputation from my doing so now.
    – LSpice
    Jul 12, 2023 at 14:18

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