# Boolean ultrapowers, the Bukovský-Dehornoy phenomenon, and iterated ultrapowers

• G. Fuchs and J. D. Hamkins, “The Bukovský-Dehornoy phenomenon for Boolean ultrapowers,” ArXiv e-prints, 2017. (under review)
@ARTICLE{FuchsHamkins:TheBukovskyDehornoyPhenomenonForBooleanUltrapowers,
AUTHOR = {Gunter Fuchs and Joel David Hamkins},
TITLE = {The {Bukovsk\'y-Dehornoy} phenomenon for {Boolean} ultrapowers},
JOURNAL = {ArXiv e-prints},
YEAR = {2017},
volume = {},
number = {},
pages = {},
month = {},
note = {under review},
abstract = {},
keywords = {under-review},
source = {},
eprint = {1707.06702},
archivePrefix = {arXiv},
primaryClass = {math.LO},
url = {http://wp.me/p5M0LV-1zz},
}

Abstract. We show that while the length $\omega$ iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Příkrý forcing, it is consistent that no iteration of length greater than $\omega$ (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a generalization of Příkrý forcing as examples. We refer to the discovery that the intersection of the finite iterates of the universe by a normal measure is the same as the generic extension of the direct limit model by the critical sequence as the Bukovský-Dehornoy phenomenon, and we develop a criterion (the existence of a simple skeleton) for when a version of this phenomenon holds in the context of Boolean ultrapowers.