Algebraically Closed subsets and strong amalgamation

Question

I came accross the following

Theorem: If $A$ is an $\aleph_0$- categorical structure, then the algebraically closed substructures of $A$ satisfy the strong amalgamation principle. (for definitions look at the end).

My questions are:

(1) If $T$ is an $\aleph_0$- categorical theory and $B$ is a model of $T$, not necessarily countable, do the algebraically closed substructures of $B$ satisfy strong amalgamation?

If $B_0,B_1,B_2$ are substructures of $B$ such that $B_0\subset B_1,B_2$, I am interested in particular in the case that $B_0,B_1$ (not $B_2$) are finitely generated.

(2) The terms disjoint and strong amalgamation, do they refer to the same property?

(3) Does anyone know a reference to the above theorem?

Definitions: If $A$ is a structure and $A_0\subset A$, $A_0$ is $algebraically\;\; closed$ if every finite set $B$ that is definable with parameters from $A_0$ is a subset of $A_0$.

If $A_0\subset A_1,A_2$, the triple $(A_0,A_1,A_2)$ have the $strong\; amalgamation \; property$, if there is a structure $A_3$ and embeddings $f:A_1\rightarrow A_3$ and $g:A_2\rightarrow A_3$ such that $f[x]=g[x]$, for all $x\in A_0$ and $f[A_1]\cap g[A_2]=f[A_0]$.

Answer 1

If your $f$ and $g$ are partial elementary maps, i.e. they preserve formulas, and if their images in $A_3$ are independent over the image of $A_0$ then your "stong amalgamation property" looks like a property more widely known as "2-existence". It holds in stable theories as follows from stationarity of types over algebraically closed sets (at this point one must either require elimination of imaginaries or mean the algebraic closure to be the algebraic closure in $M^{eq}$). 3-existence over models is a defining property of the so-called simple theories. It is hard to point to a particular reference as these properties are pervasive (and the notion of amalgamation was introduced long time ago by Shelah, if I am not mistaken).

I do not know much about $\aleph_0$-categorical theories, but the book by Cherlin and Hrushovski "Finite structures with few types" proves type amalgamation for a similar kind of structures (Chapter 5).

Answer 2

This is only a partial answer (now that you said the substructures are finite):

(2) As you said, they are the same. See for example ,

D.Macpherson, A Survey of Homogeneous Structures. (you can just google to get it.)

(3) A reference for the proof of the theorem you mentioned is Lemma 2.8 of the following:

David Evans, Examples of $\aleph_0$-categorical strucutres, in the book Automorphisms of first order structures (1994, Oxford Univ. Press) Edited by R. Kaye and D.Macpherson

Essentially what you need is a so-called "Separation Lemma" from permutation group theory. In this article, he also proved a converse. The article mentioned in (2) also provides a reference for it. (In Lemma 2.1.4, but I didn't check it.)

I also think (1) is probably false in general.

Answer 3

Here are a couple of references for "Strong Amalgamation". Most of them are from older papers/books and I thought that the terminology might have changed.

1) Paper: Decidability and $\aleph_0$-Categoricity of Theories of Partially Ordered Sets, James H. Schmerl, The Journal of Symbolic Logic, Vol. 45, No. 3 (Sep., 1980), pp. 585-611

Lemma 2.6 reads: Suppose $A$ is $\aleph_0$-categorical and admits quantifier elimination. Let K be the class of (isomorphs of) the finite algebraically closed substructures of $A$. Then K has the strong amalgamation property (SAP).

Schmerl defines SAP as: "K has the strong amalgamation property (SAP) if whenever $A_0,A_1\in K$ are such that $A_0\cap A_1\in K$, then there is $A\in K$ such that $A_0,A_1\subset A$.

Comments: (a) If $A$ is $\aleph_0$-categorical, then it has only an atomic model and quantifier elimination gives that all substructures are elementary. (b) This definition is equivalent to the one posted above, at least for finite algebraically closed substructures. (c) Schmerl does not prove the theorem, but he mentions a couple of references that do prove the theorem. So, the theorem is older than 1980. One of the references that I looked is the following book:

2) J. CROSSLEY and A. NERODE, Combinatorial functors, Ergebnisse Mathematische Grenzgebiete, Band 81, Springer, New York, 1974.

Lemma 5.5 (Duplication Lemma) states: Let $M$ be an atomic model. Let $A,B$ be finite algebraically closed subsets of $M$ and $p:A\rightarrow B$ an elementary isomorphism. Let $N$ be a finite subset of $M$ such that $N\cap B=\emptyset$. If $A'$ is a finite algebraically closed set containing A, then there an algebraically closed $B'\supset B$ such that $B'\cap N=\emptyset$ and an elementary isomorphism $p'$ extensing $p$ such that $p':A'\cong B'$. Moreover, there are infinitely many such $B'$ such that the intersection of any two distinct ones is $B$.

If $p:A\rightarrow B$ is a 1-1 and onto function, Crossley and Nerode define $p$ to be an elementary isomorphism if moreover, for all $a_0,\ldots,a_{n-1}\in A$ and for all formulas $\phi$, $$A\models\phi(a_0,\ldots,a_{n-1})\Leftrightarrow B\models\phi(p(a_0),\ldots,p(a_{n-1})).$$ I guess we would just say isomorphism today, rather than elementary isomorphism.