27 research outputs found
The complexity of the embeddability relation between torsion-free abelian groups of uncountable size
We prove that for every uncountable cardinal such that
, the quasi-order of embeddability on the
-space of -sized graphs Borel reduces to the embeddability on
the -space of -sized torsion-free abelian groups. Then we use
the same techniques to prove that the former Borel reduces to the embeddability
on the -space of -sized -modules, for every
-cotorsion-free ring of cardinality less than the continuum. As
a consequence we get that all the previous are complete
quasi-orders.Comment: 14 pages, final versio
Invariant universality for quandles and fields
We show that the embeddability relations for countable quandles and for countable fields of any given characteristic other than 2 are maximally complex in a strong sense: they are invariantly universal. This notion from the theory of Borel reducibility states that any analytic quasi-order on a standard Borel space essentially appears as the restriction of the embeddability relation to an isomorphism-invariant Borel set. As an intermediate step we show that the embeddability relation of countable quandles is a complete analytic quasi-order
Club guessing and the universal models
We survey the use of club guessing and other pcf constructs in the context of
showing that a given partially ordered class of objects does not have a
largest, or a universal element