2014/10/14 by Cubides-Kovacsics, Pablo, Delon, Françoise
#03C60 #13L05 #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO) #Primary 12J10 #Secondary 03C98
paper · doi:10.48550/arxiv.1410.3589
Marker and Steinhorn shown that given two models M\prec N of an o-minimal theory, if all 1-types over M realized in N are definable, then all types over M realized in N are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. Although it is true that if M is an algebraically closed valued field such that all 1-types over M are definable then all types over M definable, we build a counterexample for the relative statement, i.e., we show for any n≥ 1 that there is a pair M\prec N of algebraically closed valued fields such that all n-types over M realized in N are definable but there is an n+1-type over M realized in N which is not definable. Finally, we discuss what happens in the more general context of C-minimality.