## Alternative techniques in the decision problem for Abelian groups |

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### Contents

Szmielews Product Construction | 10 |

Sentences Determining Elementary | 23 |

Eliminating Quantification from | 26 |

2 other sections not shown

### Common terms and phrases

arbitrary Abelian group Assume basic classes basic sentences cardinal number CHAPTER class of Abelian classes are elementary closed formula construct couples p,k cyclic group decidable theory DECISION PROBLEM decompose F direct products Ehrenfeucht criterion elementarily equivalent elementary classes elementary theory elements x0 eliminating quantification Ershov Feferman-Vaught finite number finite sets following theorem free variables function F game Gn game of length Godel number H 3(k+J Hence implies independent mod pk isomorphism kind iff linearly independent mod linearly independent modulo M-j_ M2 are indiscernible maximal set natural numbers number of elements nx=C obtain order pk pk and strongly pk+1 player II wins positive integer Proof q 0 otherwise recursive axiom system recursively axiomatizable theory sequence of elements set Nom(T set of elements specified by sentence strongly linearly independent stsf stsfM2 Szmielew theory of Abelian torsion-free TT TT winning strategy xk+-j_