Minimal (unstable) expansions of (Z,+,0)

par Christian d'Elbée (ICJ)

jeudi 2 févr. 2017, 16:15 → 17:15 Europe/Paris

Salle 125 (ICJ, bât. Braconnier, UCBL - La Doua)

The two structures (Z, +, 0, <) and (Z, +, 0, |p) (where x|py; vp(x); vp(y)) are strict expansions of (Z, +, 0). Take (Z, +, 0, . . .) a reduct of (Z, +, 0, <) which is a strict expansion of (Z, +, 0) then (Z, +, . . .) defines <, possibly with parameters. The same holds for (Z, +, 0, |p). In that sense, they are minimal expansions of (Z, +, 0). G. Conant proved in [1]: (Z, +, 0, <) is a minimal expansion of (Z, +, 0). We propose another proof of the result of Conant, as well as a proof that (Z, +, 0, |p) is a minimal expansion of (Z, +, 0). In [2] is proven that (Z, +, 0) has no stable dpminimal strict expansions. As both (Z, +, 0, <) and (Z, +, 0, |p) are dp-minimal, it suffice to consider an unstable reduct (Z, +, 0, . . .). Besides, with the cost of getting in a saturated model, we will only need to study 1-dimensional definable sets. While the result of Conant will follows pretty quickly with this approach, the case of (Z, +, 0, |p) will need a good understanding of 1-dimensional definable sets and their arithmetic. This is joint work with E. Alouf. [1] G. Conant. There are no intermediate structures between the group of integers and Presburger arithmetic. May 2016. Available at https://arxiv.org/pdf/1603.00454.pdf. [2] G. Conant, A. Pillay. Stable groups and expansions of (Z, +, 0). January 2016. Available at https://arxiv.org/ pdf/1601.05692.pdf