View source for Η set

{{Short description|Type of totally ordered set}}
{{DISPLAYTITLE:''η'' set}}
In mathematics, an '''[[Eta|''η'']] set''' (''eta set'') is a type of [[totally ordered set]] introduced by {{harvs|txt|last=Hausdorff|year1=1907|loc1=p.&nbsp;126|year2=1914|loc2=chapter 6 section 8}} that generalizes the [[order type]] ''η'' of the rational numbers.

==Definition==

If <math>\alpha</math> is an ordinal then an <math>\eta_\alpha</math> set is a totally ordered set in which for any two subsets <math>X</math> and <math>Y</math> of cardinality less than <math>\aleph_\alpha</math>, if every element of <math>X</math> is less than every element of <math>Y</math> then there is some element greater than all elements of <math>X</math> and less than all elements of <math>Y</math>.

==Examples==

The only non-empty countable ''η''<sub>0</sub> set (up to isomorphism) is the ordered set of rational numbers.

Suppose that ''κ''&nbsp;=&nbsp;ℵ<sub>''α''</sub> is a [[regular cardinal]] and let ''X'' be the set of all functions ''f'' from ''κ'' to {−1,0,1} such that if ''f''(''α'')&nbsp;=&nbsp;0 then ''f''(''β'')&nbsp;=&nbsp;0 for all ''β''&nbsp;>&nbsp;''α'', ordered lexicographically. Then ''X'' is a ''η''<sub>''α''</sub> set. The union of all these sets is the class of [[surreal number]]s.

A dense totally ordered set without endpoints is an ''η''<sub>''α''</sub> set if and only if it is [[saturated model|ℵ<sub>''α''</sub> saturated]].

==Properties==

Any ''η''<sub>''α''</sub> set ''X'' is universal for totally ordered sets of cardinality at most ℵ<sub>''α''</sub>, meaning that any such set can be embedded into ''X''.

For any given ordinal ''α'', any two ''η''<sub>''α''</sub> sets of cardinality ℵ<sub>''α''</sub> are isomorphic (as ordered sets). An ''η''<sub>''α''</sub> set of cardinality ℵ<sub>''α''</sub> exists if ℵ<sub>''α''</sub> is regular and Σ<sub>''β''<''α''</sub> 2<sup>ℵ<sub>''β''</sub></sup>&nbsp;≤&nbsp;ℵ<sub>''α''</sub>.

==References==

*{{citation|title=On the existence of real-closed fields that are η<sub>α</sub>-sets of power ℵ<sub>α</sub>.
|first= Norman L.|last= Alling 
|journal= Trans. Amer. Math. Soc.|volume= 103 |year=1962|pages= 341–352 
|mr= 0146089|doi=10.1090/S0002-9947-1962-0146089-X|doi-access= free}}
*{{Cite book | last1=Chang | author-link=Chen Chung Chang | first1=Chen Chung | last2=Keisler | first2=H. Jerome | author2-link=Howard Jerome Keisler | title=Model Theory | orig-year=1973 | publisher=Elsevier | edition=3rd | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-88054-3 | year=1990 }}
*{{citation|chapter-url=http://hausdorff-edition.de/media/pdf/Eta_Alpha.pdf|first=U. |last=Felgner|chapter=Die Hausdorffsche Theorie der ηα-Mengen und ihre Wirkungsgeschichte|title=Hausdorff Gesammelte Werke|volume=II|publisher= Springer-Verlag|place= Berlin, Heidelberg |year= 2002|pages= 645–674}}
*{{citation|last=Hausdorff|title=Untersuchungen über Ordnungstypen V|journal= Ber. über die Verhandlungen der Königl. Sächs. Ges. Der Wiss. Zu Leipzig. Math.-phys. Klasse|volume= 59 |year=1907|pages=105–159}} English translation in {{harvtxt|Hausdorff|2005}}
*{{citation|title=Grundzüge der Mengenlehre|publisher= Veit & Co|place= Leipzig|last=Hausdorff|first=F.|year=1914 |url=https://archive.org/details/grundzgedermen00hausuoft}}
*{{citation|mr=2187098 
|last=Hausdorff|first= Felix
|title=Hausdorff on ordered sets
|editor-first= J. M.|editor-last= Plotkin|series= History of Mathematics|volume= 25|publisher= American Mathematical Society|place=Providence, RI|year= 2005|isbn= 0-8218-3788-5 |url=https://books.google.com/books?id=M_skkA3r-QAC}}

{{DEFAULTSORT:Eta set}}
[[Category:Order theory]]