$2^ {2^ \omega }$ nonisomorphic short ordered commutative domains whose quotient fields are long

Abstract:

A linearly ordered set is short if it does not contain any monotonic sequence of length ${\omega _1}$, and it is long if it contains a monotonic sequence of length $\alpha$ for every ordinal $\alpha < {({2^\omega })^ + }$. We prove that there exists a family ${\mathbf {F}}$ of power ${2^{{2^\omega }}}$ of long ordered fields of size ${2^\omega }$ that are pairwise nonisomorphic (as fields) and such that every field $F \in {\mathbf {F}}$ has ${2^{{2^\omega }}}$ nonisomorphic short subdomains whose field of quotients is $F$. The generalization of this result for higher cardinals is also discussed. This generalizes the author’s result of [Ci].