Expansions of Presburger arithmetic with the exchange property. Issue 4 (14th October 2021)

• Record Type: Journal Article
• Authors: Mariaule, Nathanaël
• Is Part Of: Mathematical logic quarterly. Volume 67:Issue 4(2021)
• Page Start: 409
• Page End: 419
• Publication Date: 2021-10-14
• Subjects: Mathematics -- Periodicals; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1002/malq.202000023
• Languages: English
• ISSNs: 0942-5616

Abstract

Abstract: Let G be a model of Presburger arithmetic. Let L be an expansion of the language of Presburger L Pres . In this paper, we prove that the L ‐theory of G is L Pres ‐minimal iff it has the exchange property and is definably complete (i.e., any bounded definable set has a maximum). If the L ‐theory of G has the exchange property but is not definably complete, there is a proper definable convex subgroup H . Assuming that the induced theories on H and G / H are definable complete and o ‐minimal respectively, we prove that any definable set of G is L Pres ∪ { H } ‐definable.