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REVERSE MATHEMATICS, YOUNG DIAGRAMS, AND THE ASCENDING CHAIN CONDITION

Published online by Cambridge University Press:  19 June 2017

KOSTAS HATZIKIRIAKOU  and
STEPHEN G. SIMPSON
KOSTAS HATZIKIRIAKOU
Affiliation:
DEPARTMENT OF PRIMARY EDUCATION UNIVERSITY OF THESSALY ARGONAFTON & FILELLINON VOLOS 38221, GREECE E-mail: kxatzkyr@uth.gr
STEPHEN G. SIMPSON
Affiliation:
DEPARTMENT OF MATHEMATICS 1326 STEVENSON CENTER VANDERBILT UNIVERSITY NASHVILLE, TN 37240, USA URL: http://www.math.psu.edu/simpson E-mail: sgslogic@gmail.com
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Abstract

Let S be the group of finitely supported permutations of a countably infinite set. Let $K[S]$ be the group algebra of S over a field K of characteristic 0. According to a theorem of Formanek and Lawrence, $K[S]$ satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over $RC{A_0}$ (or even over $RCA_0^{\rm{*}}$ ) to the statement that ${\omega ^\omega }$ is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.

Keywords

Primary 03B30Secondary 16P4016S3405A1703F15reverse mathematicsascending chain conditiongroup ringsYoung diagramspartition theorywell partial orderings

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 2 , June 2017 , pp. 576 - 589
Copyright
Copyright © The Association for Symbolic Logic 2017 

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