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One only needs to search MMA.SE, math journals, wikipedia, or god-forbid, n-cat lab, for keywords listed in the title, which can be extended with: uniform-, regular-, complete-, local-, partial-, non- (see below) &c&c, to be convinced that modified concepts are replete across maths, proliferating, and their diversity is likely accelerating.

Shafarevich: "it is the destiny of mathematics to expand in all directions."

This trend, coupled with the lack of standardized terminology, can make it difficult to compare results or in same cases even definitions.

It seems clear that in general a modifier term doesn't categorically reveal whether the modified concept is a specialization or generalization of the underlying concept (eg, subset versus superset, or subcategory versus supercategory). In some cases the modified concept might not bear a sub/super relation to the underlying, for exmaple, co- and op- in category theory and universal algebra (what's the relationship of universal co-algebra to algebra or co-induction to induction?).

So it appears we must be content with enumerating cases to discern the relation and then compare to see if a big picture emerges. Basic examples:

  • Semigroups are generalizations of groups but inverse semigroups are specializations of semigroups. (Quasicrystals are crystals - this got the Nobel - but their symmetries don't satisfy the crystal restriction theorem, eg, translation invariance, so are not groups, but might be modeled by inverse semigroups [ML]).

  • Quasimetrics are generalizations of metrics, but ultrametrics are specializations of the latter[VS] .

  • Noncommutative geometry, Connes stresses, includes commutative geometry so it is a generalization.

In the absence of an online OEIS-like database, would it be possible to crowd-source many more examples of mathematical concepts or categories noting sub/super (or other) relation to the underlying?

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    Interesting idea, so +1. But would it really serve any purpose, except perhaps in providing some interesting statistics?2012-08-25
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    @tomasz yes, this is research for a book on complexity in STEMs (science, technology, engineering, mathematics). One motivation is that the world is complex (biomedicine, healthcare, econometrics, finance, climate) but that in addition to the diversity of patterns and signals, there is also a "co"-diversity of methods to analyze those patterns, and results of, say, predictive analytics is complicated by both. Lack of standards in terminology complicates interpretation of methods.2012-08-25
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    I thought so, but I meant something more like actual practical use in studying, teaching, or developing mathematics, rather than writing *about* mathematics. :) Also, while an interesting idea, you should try to pose a concrete *question* someone can actually answer, I can't see any in there.2012-08-25
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    I was editing while you replied, but yes, I would say understanding what concepts are specializations of others as opposed to generalizations is something to be taught in high school or grade school.2012-08-25
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    I did the same thing, I suppose. :) Again: try to add some specific question to your question. As it is, I don't think it can be *answered*, which may get it closed.2012-08-25
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    It is a crowd-sourcing exercise. Hopefully people will either comment with specific examples from material they are familiar with, or perhaps give an extended list as an answer. That is my hope because there is no database and certainly AMS doesn't seem interested in XML markup and ontologies, ie, standardization.2012-08-25
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    I don't expect a single answer to nail the question, but rather the question may be completed by aggregating multiple comments or great examples.2012-08-25
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    I think you should make the question community wiki and post a partial answer of your own, so that we have better idea of what you want. Also, I think in most cases, what is a generalization and what is a specialization is self-evident, though I might be wrong...2012-08-25
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    I'm not familiar with the community wiki concept - but the point is, the modifier doesn't indicate which, so how is it self-evident?2012-08-25
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    My point is exactly that the modifier *does* indicate which. When you have "ultra"-something, it is obvious it is *more* than the thing. When you have "quasi"-something, it is obvious that it is not quite that. And if you have some adjective added, it is usually the case that it *adds* something to the concept, by further describing the object in question, so while it may sometimes be misleading, it is, I believe, self-evident in most cases.2012-08-25
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    I think your big picture is going to emerge. Although, it will be a bit like a Dali painting. I certainly think this is a great question either way.2012-08-25
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    I disagree with your opening remarks. Most authors either use standard terminology, or give full definitions and indicate where non-standard terminology has been used.2012-08-26
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    @ZhenLin, I'll give you just one counterexample: When I asked Prof. Bergman at Berkeley about a result of Herrlich written in his "Axiom of Choice" that in ZF+AD, there are no *free* ultrafilters on the naturals, he replied that the "standard" term is *nonprincipal*, after having asked others in the Math dept there. So it seems to vary by research group, or at least by continent.2012-08-26
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    They're both standard terms. I prefer to use "non-principal ultrafilter".2012-08-26
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    So standard that Bergman, an experienced algebra researcher didn't know about it? The lack of uniqueness is part of the issue. Multiply by ~31k technical journals now in existence and and an endless stream of emerging concepts (see n-cat lab). When you say "standard", I don't think you mean it in the engineering sense with a body like ISO or NIST or ANSI and revision control &c.2012-08-26

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