There is an ubiquitous pattern of questions concerning assumedly any kind of mathematical object or structure: groups, graphs, numbers, categories, and so on. It goes like this (informally):



> 
> Can a class of objects or structures of a given kind *X* that is characterized by some "external condition" *Y* be defined by a condition *Z* in their respective "internal"
>  language, and if so: how?
> 
> 
> 


Well-known examples ("external condition" **=** "internal condition"):


1. groups $G$ isomorphic to a subgroup of the symmetric group on $G$ **=** all groups (*Cayley's theorem*)
2. graphs embeddable in the plane **=** graphs not containing a subgraph that is a subdivision of $K\_5$ or $K\_{3,3}$ (*Kuratowski's theorem*)
3. numbers *n* of trees on *k* labeled vertices **=** numbers *n = k**k*-2 for some *k* > 1 (*Cayley's theorem on trees*)
4. numbers *n* with only one group of order *n* **=** numbers *n = p1 · p2 · ... · pk* for some *k* > 0, where the *pi* are distinct primes and no *pj-1* is divisible by any *pi* (*cyclic numbers*, see [Sloane's A003277](http://www.research.att.com/~njas/sequences/A003277))


Further examples from MO:


* [Which graphs are Cayley graphs?](https://mathoverflow.net/questions/14830/which-graphs-are-cayley-graphs)
* [Can we recognize when a category is
equivalent to the category of models
of a first order theory?](https://mathoverflow.net/questions/13155/can-we-recognize-when-a-category-is-equivalent-to-the-category-of-models-of-a-fir)
* [Can you determine whether a graph is
the 1-skeleton of a polytope?](https://mathoverflow.net/questions/9255/can-you-determine-whether-a-graph-is-the-1-skeleton-of-a-polytope)



> 
> **Question #1:** What's the proper way to
>  characterize this pattern of
>  questions? What's the common
>  context / rationale?
> 
> 
> 




---



> 
> **Question #2:** How is the introductory question to be
>  posed properly?
> 
> 
> 



An ubiquitous pattern of questions
There are three sibling theorems of logic which guarantee that such characterizations are bound to happen.


* [The Craig Interpolation Theorem](http://en.wikipedia.org/wiki/Craig_interpolation)
* [The Robinson Joint Consistency Theorem](http://en.wikipedia.org/wiki/Robinson%27s_joint_consistency_theorem)
* [The Beth Definability Theorem](http://en.wikipedia.org/wiki/Beth_definability)


[The wikipedia entries need some work. I suggest you look up these theorems in a good logic book, for example Hodges's *Model Theory* or his more accessible *Shorter Model Theory*.]


I will focus on the Beth Definability Theorem, though the other two siblings lead to similar conclusions in slightly different contexts.


Suppose you have a first-order language L0 and a larger language L. Let T be a theory in L and let φ(x) be a formula of the larger language L with the following property. Whenever A1 and A2 are two models of T which have the same universe A and the same interpretation for all parts of the small language L0, then A1 ⊧ φ(a) iff A2 ⊧ φ(a) for all a ∈ A. Beth's Definability Theorem says that there must be a formula φ0(x) of the smaller language L0 such that T ⊦ ∀x(φ(x) ↔ φ0(x)).


The connection with your question is as follows. The base language L0 is the 'internal' language of the structures you really care about, while the larger language L has some additional 'external' data. The theory T characterizes the structures with external data that you care about, and φ(x) is a property of such structures that you are interested in. If φ(x) is sufficiently independent of the external data, then φ(x) must be equivalent to an internal formula φ0(x).


Not all of the examples you give are easily cast into this formalism, but the basic flavor is the same. Unfortunately, the Beth Definability Theorem (and its proof) does not say much on how to find the internal formula φ0(x) but, at least, it says that the search will not be in vain.


