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EVERY COTORSIONFREE RING IS AN Endomorphism Ring
, 1980
"... Some years ago A. L. S. Corner proved that every countable and cotorsionfree ring can be realized as the endomorphism ring of some torsionfree abelian group. This result has many interesting consequences for abelian groups. Using a settheoretic axiom Vk., which follows for instance from V = L, we ..."
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Some years ago A. L. S. Corner proved that every countable and cotorsionfree ring can be realized as the endomorphism ring of some torsionfree abelian group. This result has many interesting consequences for abelian groups. Using a settheoretic axiom Vk., which follows for instance from V = L, we can drop the countability condition in Corner's theorem.
Representing embeddability as set inclusion
 Journal of LMS (2nd series), No.185
, 1998
"... ABSTRACT. A few steps are made towards representation theory of embeddability among uncountable graphs. A monotone class of graphs is defined by forbidding countable subgraphs, related to the graph’s endstructure. Using a combinatorial theorem of Shelah it is proved: The complexity of the class in ..."
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ABSTRACT. A few steps are made towards representation theory of embeddability among uncountable graphs. A monotone class of graphs is defined by forbidding countable subgraphs, related to the graph’s endstructure. Using a combinatorial theorem of Shelah it is proved: The complexity of the class in every regular uncountable λ> ℵ1 is at least λ + + sup{µ ℵ0: µ < λ} For all regular uncountable λ> ℵ1 there are 2λ pairwise non embeddable graphs in the class having strong homogeneity properties. It is characterized when some invariants of a graph G ∈ Gλ have to be inherited by one of fewer than λ subgraphs whose union covers G. All three results are obtained as corollaries of a representation theorem (Theorem 1.10 below), that asserts the existence of a surjective homomorphism from the relation of embeddability over isomorphism types of regular cardinality λ> ℵ1 onto set inclusion over all subsets of reals or cardinality λ or less. Continuity properties of the homomorphism are used to extend the first result to all singular cardinals below the first cardinal fixed point of second order. The first result shows that, unlike what Shelah showed in the class of all graphs,
A RECURSIONTHEORETIC CHARACTERIZATION OF CONSTRUCTIBLE REALS
"... Let Ly denote Godel's constructive universe up to level y. A countable ordinal y is said to be an index if Ly+1 contains a real not in Lr The notion was introduced by Boolos and Putnam [1] who also initiated the study (from the recursiontheoretic viewpoint) of what is known today as " ..."
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Let Ly denote Godel's constructive universe up to level y. A countable ordinal y is said to be an index if Ly+1 contains a real not in Lr The notion was introduced by Boolos and Putnam [1] who also initiated the study (from the recursiontheoretic viewpoint) of what is known today as &quot; the fine structure of L&quot;. In the settheoretic context, Jensen [3] later extended their results to all levels of the constructible universe. Still, as all the constructible reals occur in LWIL (CO1 L is the first constructibly uncountable ordinal), a great amount of insight into their structure can be gained by studying this &quot; small &quot; universe. One may cite as examples the fruitful investigations by various people on the Turing and hyperdegrees of these reals, the basis theorems associated with them, and so on. Yet another approach is to study fragments Ly for y < cot L. The general program is: study the structure of Ly when (a) y is not an index, and (b) y is an index. Leeds and Putnam [4] and also Marek and Srebrny [5] have taken up (a). Among other things, they showed that if y is the limit of indices but is itself not an index, then Ly is a model of ZF minus the power set axiom, and is hereditarily countable. The first result related to (b) was obtained by Boolos and