Eva Leenknegt (Purdue)
In search of p-adic minimality: An exploration of weak p-adic structures
Abstract: Consider a structure (F,L), where F is a field and L is a language that is 'related' to the language of rings, in the sense that the Lring-definable subsets of F coincide with the L-definable subsets of F (that is, we require (F,L) to be 'Lring-minimal'). When F is a real closed field, a structure satisfying this property will be o-minimal, so all tools of o-minimality, such as the cell decomposition theorem, are at our disposal when studying such structures. The situation is less clear when F is a p-adic(ally closed) field. If L is an expansion of the ring languages, then (F,L) will be P-minimal, but very little is known in general for weaker structures (reducts of the ring language) When will a reduct L of the ring language give rise to an Lring-minimal structure? In the o-minimal case, the answer is easy: the only requirement is that the order should be definable in L. Our first challenge will be to find a p-adic equivalent of this 'minimal language' (<). Once such a language has been found, one can start constructing examples of weak p-adic Lring- minimal structures. While a general theory is still far away, individual examples show that there are some fundamental differences when comparing p-adic and o-minimal reducts of the ring language. I will give some examples of this. One of the questions that comes up is the existence of cell decomposition: could it possibly be true that every p-adic Lring-minimal language has cell decomposition? I will discuss some (partial) answers to this question, and show how we can use this to study examples of weak structures. |
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