 
Summary: Archive for Mathematical Logic manuscript No.
(will be inserted by the editor)
Vincent Astier
Elementary equivalence of some rings of
definable functions
Received: date / Revised: date
Abstract We characterize elementary equivalences and inclusions between von
Neumann regular real closed rings in terms of their boolean algebras of idempo
tents, and prove that their theories are always decidable. We then show that, under
some hypotheses, the map sending an Lstructure R to the Lstructure of definable
functions from Rn to R preserves elementary inclusions and equivalences and gives
a structure with a decidable theory whenever R is decidable. We briefly consider
structures of definable functions satisfying an extra condition such as continuity.
Keywords von Neumann regular ring · elementary equivalence · ring of definable
functions
Mathematics Subject Classification (2000) 03C60 · 06F25 · 14P10
1 Introduction and notation
Let L be a firstorder language. If R is an Lstructure and n N, we consider
defL(Rn,R) the set of Ldefinable maps from Rn to R, which can be seen as an L
structure via its inclusion in RRn
