1
MAXIMAL NONFINITELY GENERATED SUBALGEBRAS
Harvey M. Friedman*
[email protected]
http://www.math.ohiostate.edu/~friedman/
Department of Mathematics
Ohio State University
September 24, 2000
October 19, 2001
Abstract. We show that “every countable algebra with a
nonfinitely generated subalgebra has a maximal nonfinitely
generated subalgebra” is provably equivalent to
1
1
CA
0
over
RCA
0
.
1. INTRODUCTION
For our purposes, a countable algebra
M
is a system whose
domain is a subset of
w
(
possibly empty), with at most
countably many constant and function symbols of various
arities.
A subalgebra of a countable algebra
M
is a subset of its
domain that contains the constants and is closed under the
functions. The empty subalgebra is allowed.
(
We could have
defined subalgebras to be algebras, but it is more convenient
for us to identify subalgebras with their domains).
A finitely generated subalgebra is a subalgebra
A
such that
for some finite
K
Õ
A
,
A
consists of all values of terms
involving constants and functions from
M
and arguments from
K
.
K
is called a set of generators for
A
. A subalgebra is
said to be nonfinitely generated if and only if it is not
finitely generated. Note that the empty set is automatically
finitely generated.
A maximal nonfinitely generated subalgebra is a nonfinitely
generated subalgebra
A
such that every nonfinitely generated
subalgebra is a subset of
A
.
THEOREM 1.1. Every countable algebra with a nonfinitely
generated subalgebra has a maximal nonfinitely generated
subalgebra.
Proof: Let
A
be a countable algebra with a nonfinitely
generated subalgebra
B
. We define a transfinite sequence
B
a
,
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a
<
w
1
, of nonfinitely generated subalgebras of
A
as follows.
B
0
=
A
. Suppose
B
a
has been defined. Define
B
a+1
to be a
nonfinitely generated subalgebra of
A
properly extending
B
a
if such exists;
B
a
otherwise. Suppose
B
b
,
b
<
l
, has been
defined. For limit ordinals
l
, set
B
l
to be the union of the
B
b
,
b
<
l
. By cardinality considerations, let
a
<
w
1
be such
that
B
a
=
B
a+1
. Then
B
a
is as required. QED
We wish to analyze Theorem 1.1 from the viewpoint of reverse
mathematics. The formalization of Theorem 1.1 within the
language of the base theory RCA
0
is straightforward.
Note that above proof is very set theoretic, and even uses
the axiom of choice both in the choice of the
B
a+1
and in the
use of the regularity of
w
1
.
We now give a second, more concrete proof.
Proof: We define a sequence
B
n
, n
<
w
, of nonfinitely
generated subalgebras of
A
as follows. Let
B
0
be any
nonfinitely generated subalgebra of
A
. Suppose
B
n
has been
defined. Define
B
n
+1
to be a nonfinitely generated subalgebra
of
A
properly containing
B
n
{
n
}
if such exists;
B
n
otherwise. Let
B
be the union of the
B
n
. Obviously
B
is a
nonfinitely generated subalgebra of
A
. To see that
B
is
maximal, let
B¢
be a nonfinitely generated subalgebra of
A
containing
B
, and let n be the least element of
B¢
\
B
. Then by
construction, n was thrown in at stage n
+1
, and so n
B
,
which is a contradiction. QED
LEMMA 1.2. Theorem 1.1 is provable in
1
1
CA
0
.
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 Fall '08
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 Math, Algebra, Empty set, Order theory, Finite set, tail, nonfinitely generated subalgebra

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