Mikhail R. Starchak
ABSTRACT
We consider positively existentially definable sets in the structure {Ζ; 1, +,⊥}. It is well known that the elementary theory of this structure is undecidable while the existential theory is decidable. We show that after the extension of the signature with the unary '-' functional symbol, binary symbols for dis-equality ≠ and GCD (.,.)=d for every fixed positive integer d, every positive existential formula in this extended language is equivalent in Ζ to some positive quantifier-free formula.
Then we get some corollaries from the main result. The binary order ≤ and dis-coprimeness /⊥ relations are not positively existentially definable in the structure {Ζ;1,+,⊥}. Every positively existentially definable set in the structure {N;S,⊥} is quantifier-free definable in {N;S,≠,0 ⊥}. We also get a decidable fragment of the undecidable ∀∃-Theory of the structure {Ζ;1,+,ł≤,|}, where | is a binary predicate symbol for the integer divisibility relation.
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Index Terms
- Positive Existential Definability with Unit, Addition and Coprimeness
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