22\omega Nonisomorphic Short Ordered Commutative Domains whose Quotient Fields are Long

by

Krzysztof Ciesielski

Proc. Amer. Math. Soc. 113 (1991), 217-227

A linearly ordered set is short if it does not contain any monotonic sequence of length \omega1, 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 F of power 22\omega of long ordered fields of size 2\omega that are pairwise nonisomorphic (as fields) and such that every field F in F has 22\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 [Krzysztof Ciesielski, A short ordered commutative domain whose quotient field is not short, Algebra Universalis 25 (1988), 1-6].