MODEL THEORY OF PREGEOMETRIES
ERIN CAULFIELD
1. Introduction
1.1. Pregeometries. A pregeometry consists of a set M together with an operation cl : P(M ) → P(M ) satisfying the following properties:
(1)
(2)
(3)
(4)
(5)
For all A ⊆ M , A ⊆ cl(A).
If A ⊆ B ⊆ M , then cl(A) ⊆ cl(B).
For all A ⊆ M , cl(cl(A)) = cl(A).
For A ⊆ M , a, b ∈ M , and
S a ∈ cl(A ∪ {b}) \ cl(A), we have b ∈ cl(A ∪ {a}).
For all A ⊆ M , cl(A) = {cl(F ) : F ⊆ A finite}.
A subset A ⊆ M of a pregeometry (M, cl) is said to be closed if cl(A) = A.
A pregeometry (M, cl) is said to be modular if for all A ⊆ M closed and nonempty
and all b ∈ M , all x ∈ cl(A ∪ {b}), there exists a ∈ A such that x ∈ cl(a, b).
A subset A ⊆ M of a pregeometry (M, cl) is called independent if for all a ∈ A,
a∈
/ cl(A\{a}). For A, B ⊆ M , B is said to be a basis for A if B ⊆ A is independent
and A ⊆ cl(B). It can be shown that every A ⊆ M has a basis. Furthermore, for
any bases B1 , B2 of A ⊆ M , we have |B1 | = |B2 |. (See Theorem 10.3 in [1].) Thus,
we define the rank (or dimension) of A to be |B| for any basis B of A.
1.2. Model-theoretic preliminaries. Let L = {D0 , D1 , D2 , . . .}, where Dn is a
relation symbol of arity n + 1. Throughout this report, L will refer to this language.
Given a pregeometry (M, cl), we can make (M, cl) into a first-order structure as
follows. We define an L-structure M = (M ; D0 , D1 , D2 , . . .) by requiring that for
all n and a, a1 , . . . , an ∈ M , DnM (a, a1 , . . . , an ) if and only if a ∈ cl{a1 , . . . , an }.
There exists a set of universal L-sentences PG such that the models of PG are
exactly the L-structures obtained from pregeometries (M, cl). In particular, if M is
an L-structure and M |= PG, then we can define a function clM : P(M ) → P(M )
by taking
clM (A) = {c ∈ M : there exist n ∈ N, a1 , . . . , an ∈ A such that M |=
Dn (c, a1 , . . . , an )}.
for A ⊆ X. It can be shown that (M, clM ) is a pregeometry and for all n ∈ N and
all a, b1 , . . . , bn ∈ M , M |= Dn (a, b1 , . . . , bn ) if and only if a ∈ clM (b1 , . . . , bn ).
Note that since PG is a collection of universal sentences, for an L-structure
M |= T and A ⊆ M, (A, clA ) is also a pregeometry.
For an L-structure M |= PG, a line is a subset of M of the form clM (a) for
some a ∈ M with a ∈
/ clM (∅).
Consider the following collection of L-sentences, which we will denote by T2 :
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ERIN CAULFIELD
(1) ∃xD0 (x)
(2) ∀x∀y(D0 (x) ∧ D0 (y) → x = y)
(3) The set of sentences {∀a∀x1 . . . xn (¬D0 (a) → ((D1 (x1 , a)∧. . .∧D1 (xn , a)) →
∃y(y 6= x1 ∧ . . . ∧ y 6= xn )) : n ≥ 1}
(4) The set of sentences {∀x1 . . . xn ∃y(¬D1 (y, x1 ) ∧ . . . ∧ ¬D1 (y, xn )) : n ≥ 1}
(5) ∀x∀y∀z(D2 (x, y, z) ∨ D2 (y, x, z) ∨ D2 (z, x, y))
(6) The collection of sentences PG
Note that if M |= T2 , then (M, clM ) has rank 2.
For n ≥ 1, let σn be the L-sentence
σn = ∀a1 . . . an ∀b∀x(Dn+1 (x, a1 , . . . , an , b) → ∃a(Dn (a, a1 , . . . , an ) ∧ D2 (x, a, b)).
2. Results
The following are the results proved in this REGS project. The proofs of two
results are given.
Proposition. T2 has quantifier elimination.
Corollary. Let M |= T2 . The definable sets X ⊆ M are Boolean combinations of
singletons and lines.
Corollary. T2 is model complete. That is, for any L-structures M, N |= T2 with
M ⊆ N , we have M N .
Proof. This follows directly from the fact that T2 has quantifier elimination; see
Theorem 3.1.14 in [2].
Proposition. T2 is ℵ0 -categorical.
Corollary. T2 is complete.
Proof. Let V be a vector space of rank 2 over an infinite field K. For A ⊆ V , let
cl(A) be the K-linear span of A in V . Then (V, cl) is a pregeometry. Therefore,
we have an L-structure V = (V ; D0 , D1 , . . .) associated to (V, cl) as described in
Section 1.2. We have V |= T2 . Moreover, any model of T2 is infinite. Since |L| = ℵ0
and T2 is ℵ0 -categorical, T2 is complete by Vaught’s test.
Proposition. Let (M, cl) be a pregeometry and M = (M ; D0 , D1 , . . .) be the Lstructure associated to (M, cl) as described in Section 1.2. Then (M, cl) is a modular
pregeometry if and only if M |= σn for all n ≥ 1.
Proposition. Let M |= PG have rank d, where d ∈ N and d ≥ 2. Let A be a
proper substructure of M such that clM (∅) ⊆ A. Let N |= PG have dimension d,
and let f : A → N be an embedding. Suppose that for all X ⊆ A with |X| < d,
clM (X) = clA (X). Let x ∈ M \ A and let B ⊆ M have universe B = A ∪ {x}.
Then f can be extended to an embedding g : B → N if and only if there exists
y ∈ N \ f (A) such that for all independent A0 ⊆ A with |A0 | < d, f (A0 ) ∪ {y} is
independent.
3. Acknowledgements
The author acknowledges support from National Science Foundation grant DMS
08–38434 "EMSW21–MCTP: Research Experience for Graduate Students". The
author would also like to thank Professor Lou van den Dries for his help and advice
during this project.
MODEL THEORY OF PREGEOMETRIES
References
[1] C.W. Henson, Model Theory: Class Notes for Mathematics 571, 2013.
[2] D. Marker, Model Theory: An Introduction, Springer, 2002.
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