Abstract
We discuss the notion of uniform canonical bases, both in an abstract manner and specifically for the theory of atomless Lp lattices. We also discuss the connection between the definability of the set of uniform canonical bases and the existence of the theory of beautiful pairs (i.e., with the finite cover property), and prove in particular that the set of uniform canonical bases is definable in algebra- ically closed metric valued fields. 2010 Mathematics Subject Classification 03C45 (primary); 46B42,12J25 (second- ary)
Highlights
The canonical base of a type is a minimal set of parameters required to define the type, and as such it generalises notions such as the field of definition of a variety in algebraic geometry
In particular we observe that every stable theory admits uniform canonical bases in some imaginary sorts, so the space of all types can be naturally identified with a type-definable set
Let Cb be a uniform canonical base map in the sort x, and let f be definable function defined on img Cb, into some other possibly infinite sort
Summary
The canonical base of a type is a minimal set of parameters required to define the type, and as such it generalises notions such as the field of definition of a variety in algebraic geometry. In particular we observe that every stable theory admits uniform canonical bases in some imaginary sorts, so the space of all types can be naturally identified with a type-definable set. The second question, discussed, is whether the (type-definable) set of uniform canonical bases is definable We characterise this situation in terms of the existence of a theory of beautiful pairs.
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