## Works by Denis Richard

5 found
Order:
1. Alexis Bés & Denis Richard (1998). Undecidable Extensions of Skolem Arithmetic. Journal of Symbolic Logic 63 (2):379-401.
Let $be the restriction of usual order relation to integers which are primes or squares of primes, and let ⊥ denote the coprimeness predicate. The elementary theory of$\langle\mathbb{N};\bot, , is undecidable. Now denote by $the restriction of order to primary numbers. All arithmetical relations restricted to primary numbers are definable in the structure$\langle\mathbb{N};\bot, . Furthermore, the structures $\langle\mathbb{N};\mid, and$\langle\mathbb{N};=,+,x\rangle$are interdefinable. Direct download (7 more) Export citation My bibliography 1 citation 2. Patrick Cegielski, Yuri Matiyasevich & Denis Richard (1996). Definability and Decidability Issues in Extensions of the Integers with the Divisibility Predicate. Journal of Symbolic Logic 61 (2):515-540. Let M be a first-order structure; we denote by DEF(M) the set of all first-order definable relations and functions within M. Let π be any one-to-one function from N into the set of prime integers. Let ∣ and$\bullet$be respectively the divisibility relation and multiplication as function. We show that the sets DEF(N,π,∣) and$\mathrm{DEF}(\mathbb{N},\pi,\bullet)$are equal. However there exists function π such that the set DEF(N,π,∣), or, equivalently,$\mathrm{DEF}(\mathbb{N},\pi,\bullet)$is not equal to$\mathrm{DEF}(\mathbb{N},+,\bullet)$. Nevertheless, in all cases (...) Direct download (7 more) Export citation My bibliography 3. Denis Richard (1985). Answer to a Problem Raised by J. Robinson: The Arithmetic of Positive or Negative Integers is Definable From Successor and Divisibility. Journal of Symbolic Logic 50 (4):927-935. In this paper we give a positive answer to Julia Robinson's question whether the definability of + and · from S and ∣ that she proved in the case of positive integers is extendible to arbitrary integers (cf. [JR, p. 102]). Direct download (7 more) Export citation My bibliography 4. Denis Richard (1989). Definability in Terms of the Successor Function and the Coprimeness Predicate in the Set of Arbitrary Integers. Journal of Symbolic Logic 54 (4):1253-1287. Using coding devices based on a theorem due to Zsigmondy, Birkhoff and Vandiver, we first define in terms of successor S and coprimeness predicate$\perp$a full arithmetic over the set of powers of some fixed prime, then we define in the same terms a restriction of the exponentiation. Hence we prove the main result insuring that all arithmetical relations and functions over prime powers and their opposite are$\{S, \perp\}\$ -definable over Z. Applications to definability over Z and N (...)