Abstract/Details

On cotypesets of finite rank torsion-free abelian groups

Lafleur, Reiff Stauffer. 
 University of Connecticut ProQuest Dissertations Publishing,  1994. 9520012.

Abstract (summary)

Throughout this abstract, we use "group" for "finite-rank torsion-free Abelian group". Every group G gives rise to two important sets of types:$$\eqalign{&\rm typeset(G) = \{type(x): 0 \not= x \in G\},\ and\cr\rm cotype&\rm set(G) = \{type(X): X\ is\ a\ rank{-}1\ factor\ of\ G\}.\cr}$$This paper gives some new alternative characterizations of cotypesets of rank-n groups. Using these characterizations, for any n $\geq$ 3 we give an example of a set of types which is the cotypeset of a rank-n group but is not the cotypeset of any group of rank other than n. Dualizing these characterizations of cotypesets, we obtain properties of typesets of rank-n groups, which are necessary, but not sufficient. Using these and the Warfield dual, we can give for any n $\geq$ 3 an example of a set of types which is the typeset of a rank-n group but is not the typeset of a group of any other rank.

In addition, using these new characterizations of cotypesets, we are able to give several new necessary conditions for cotypesets which are intrinsic to the set of types, and do not refer to the existence of maps from some other partially ordered set. We show that these intrinsic necessary conditions are not sufficient by giving an example of a set of types which satisfies these conditions, but is not a cotypeset of a rank-n group. Finally, from the necessary conditions for typesets mentioned above, we will prove dual intrinsic necessary conditions for typesets of rank-n groups, which are also not sufficient.

Indexing (details)


Subject
Mathematics
Classification
0405: Mathematics
Identifier / keyword
Pure sciences
Title
On cotypesets of finite rank torsion-free abelian groups
Author
Lafleur, Reiff Stauffer
Number of pages
42
Degree date
1994
School code
0056
Source
DAI-B 56/02, Dissertation Abstracts International
Place of publication
Ann Arbor
Country of publication
United States
ISBN
979-8-208-34518-4
University/institution
University of Connecticut
University location
United States -- Connecticut
Degree
Ph.D.
Source type
Dissertation or Thesis
Language
English
Document type
Dissertation/Thesis
Dissertation/thesis number
9520012
ProQuest document ID
304143950
Copyright
Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.
Document URL
https://www.proquest.com/docview/304143950/abstract