MaxSubalg101901

# MaxSubalg101901 - 1 MAXIMAL NONFINITELY GENERATED...

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1 MAXIMAL NONFINITELY GENERATED SUBALGEBRAS Harvey M. Friedman* [email protected] http://www.math.ohio-state.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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2 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 be a nonfinitely generated subalgebra of A containing B , and let n be the least element of \ 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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## This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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MaxSubalg101901 - 1 MAXIMAL NONFINITELY GENERATED...

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