An ubiquitous pattern of questions

Question

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 · ... · p_k for some k > 0, where the p_i are distinct primes and no p_j-1 is divisible by any p_i (cyclic numbers, see Sloane's A003277)

Further examples from MO:

• Which graphs are Cayley graphs?

• Can we recognize when a category is equivalent to the category of models of a first order theory?

• 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?

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

Answer 1

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

• The Craig Interpolation Theorem
• The Robinson Joint Consistency Theorem
• The Beth Definability Theorem

[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.