On cotypesets of finite rank torsion-free abelian groups

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.