Fraïssé limit explained
In mathematical logic, specifically in the discipline of model theory, the Fraïssé limit (also called the Fraïssé construction or Fraïssé amalgamation) is a method used to construct (infinite) mathematical structures from their (finite) substructures. It is a special example of the more general concept of a direct limit in a category.[1] The technique was developed in the 1950s by its namesake, French logician Roland Fraïssé.[2]
of finite
relational structures, if
satisfies certain properties (described below), then there exists a unique
countable structure
, called the Fraïssé limit of
, which contains all the elements of
as
substructures.
The general study of Fraïssé limits and related notions is sometimes called Fraïssé theory. This field has seen wide applications to other parts of mathematics, including topological dynamics, functional analysis, and Ramsey theory.[3]
Finitely generated substructures and age
. By an
-structure, we mean a
logical structure having
signature
.
Given an
-structure
with domain
, and a subset
, we use
to denote the least
substructure of
whose domain contains
(i.e. the closure of
under all the function and constant symbols in
).
A substructure
of
is then said to be
finitely generated if
for some
finite subset
.
[4] The
age of
, denoted
, is the class of all finitely generated substructures of
.One can prove that any class
that is the age of some structure satisfies the following two conditions:
Hereditary property (HP)
If
and
is a finitely generated substructure of
, then
is isomorphic to some structure in
.
Joint embedding property (JEP)
If
, then there exists
such that both
and
are embeddable in
.
Fraïssé's theorem
As above, we noted that for any
-structure
,
satisfies the HP and JEP. Fraïssé proved a sort-of-converse result: when
is any non-empty, countable set of finitely generated
-structures that has the above two properties, then it is the age of some countable structure.
Furthermore, suppose that
happens to satisfy the following additional properties.
Amalgamation property (AP)
For any structures
, such that there exist embeddings
,
, there exists a structure
and embeddings
,
such that
(i.e. they coincide on the image of A in both structures).
Essential countability (EC)
Up to isomorphism, there are countably many structures in
.
In that case, we say that K is a Fraïssé class, and there is a unique (up to isomorphism), countable, homogeneous structure
whose age is exactly
.
[5] This structure is called the
Fraïssé limit of
.
Here, homogeneous means that any isomorphism
between two finitely generated substructures
can be extended to an automorphism of the whole structure.Examples
The archetypal example is the class
of all finite linear orderings, for which the Fraïssé limit is a
dense linear order without endpoints (i.e. no
smallest nor largest element). By
Cantor's isomorphism theorem, up to isomorphism, this is always equivalent to the structure
, i.e. the
rational numbers with the usual ordering.
As a non-example, note that neither
nor
are the Fraïssé limit of
. This is because, although both of them are countable and have
as their age, neither one is homogeneous. To see this, consider the substructures
and
, and the isomorphism
between them. This cannot be extended to an automorphism of
or
, since there is no element to which we could map
, while still preserving the order.
Another example is the class
of all finite
graphs, whose Fraïssé limit is the
Rado graph.
.
The Fraïssé limit of the class of finite abelian p-groups is
(the direct sum of countably many copies of the
Prüfer group). The Fraïssé limit of the class of all finite abelian groups is
opluspprimeZ(pinfty)(\omega)\simeq(Q/Z)(\omega)
.
The Fraïssé limit of the class of all finite groups is Hall's universal group.
The Fraïssé limit of the class of nontrivial finite Boolean algebras is the unique countable atomless Boolean algebra.
ω-categoricity and quantifier elimination
The class
under consideration is called
uniformly locally finite if for every
, there is a uniform bound on the size of
-generated (substructures of) structures in
. The Fraïssé limit of
is
ω-categorical if and only if
is uniformly locally finite.
[6] If
is uniformly locally finite, then the Fraïssé limit of
has
quantifier elimination.
If the language of
is finite, and consists only of relations and constants, then
is uniformly locally finite automatically.
For example, the class of finite dimensional vector spaces over a fixed field is always a Fraïssé class, but it is uniformly locally finite only if the field is finite.The class of finite Boolean algebras is uniformly locally finite, whereas the classes of finite fields of a given characteristic, or finite groups or abelian groups, are not, as 1-generated structures in these classes may have arbitrarily large finite size.
See also
Notes and References
- Web site: The n-Category Café. golem.ph.utexas.edu. en. 2020-01-08.
- Book: Hodges, Wilfrid.. A shorter model theory. 1997. Cambridge University Press. 0-521-58713-1. 468298248.
- Lupini. Martino. November 2018. Fraïssé limits in functional analysis. Advances in Mathematics. 338. 93–174. 10.1016/j.aim.2018.08.012. free. 0001-8708.
- Web site: An introduction to model theory (lecture notes), Defn 2.2.1. Schlicht. Philipp. January 7, 2018. Mathematical Institute of the University of Bonn.
- Book: Notes on infinite permutation groups. 1998. Springer. Bhattacharjee, M. (Meenaxi), 1965–. 3-540-64965-4. Berlin. 39700621.
- Book: Hodges, Wilfrid. Model Theory. 1993-03-11. Cambridge University Press. 978-0-521-30442-9. 350. 10.1017/cbo9780511551574 .
