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?

  • $\begingroup$ Interesting idea, so +1. But would it really serve any purpose, except perhaps in providing some interesting statistics? $\endgroup$ – tomasz Aug 25 '12 at 19:18
  • $\begingroup$ @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. $\endgroup$ – alancalvitti Aug 25 '12 at 19:20
  • $\begingroup$ 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. $\endgroup$ – tomasz Aug 25 '12 at 19:23
  • $\begingroup$ 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. $\endgroup$ – alancalvitti Aug 25 '12 at 19:25
  • 2
    $\begingroup$ 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... $\endgroup$ – tomasz Aug 25 '12 at 19:29
  • A partial function is more general than a function. Equivalently, a function is more specialized than a partial function (and called total function if one wants to emphasize this).
  • The semidirect product of groups is a generalization of the direct product.
  • A preorder is a generalization of a (partial or linear) order.
  • A strictly increasing function is a special case of a (weakly) increasing function.
  • A (join- or meet-)semilattice is more general than a lattice.
  • Weak convergence in a Hilbert space is more general than (strong) convergence.
  • Non-deterministic automata are more general than deterministic automata.
  • The weak derivative is a generalization of the (strong) derivative.
  • An ultrafilter is a special case of a filter.
  • Uniform convergence is a special case of pointwise convergence.

The attributes "semi-", "pre-", "weak" tend to generalize concepts whereas "ultra-" or "strong" add further contraints. The prefix "non-", however, sometimes means "not" (as in non-archimedean) and sometimes "not necessarily" (as in non-deterministic or non-commutative). Furthermore, it seems that whenever a concept is weakened, using attributes like "pre-", "semi-" or "weak", the original concept may be called "strict" or "strong" or the like if one wants to emphasize explicitly that something falls under the more specialized original concept (e.g. "strong derivative", "total function", "strong convergence"). Similarly, if one wants to express that something does not fall under the specialization of a concept (like "strictly increasing", "absolutely convergent") this may also be described by additional attributes (e.g. "weakly increasing", "conditional convergence").

  • $\begingroup$ For finite automata, non-determinism is not a generalization. But I suppose they were intended to be when they were defined. $\endgroup$ – JeremyKun Nov 2 '12 at 4:40

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