2,884 research outputs found
Shelah's Categoricity Conjecture from a successor for Tame Abstract Elementary Classes
Let K be an Abstract Elemenetary Class satisfying the amalgamation and the
joint embedding property, let \mu be the Hanf number of K. Suppose K is tame.
MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than
\beth_{(2^\mu)^+} then K is categorical in all cardinals greater than
\beth_{(2^\mu)^+}.
This is an improvment of a Theorem of Makkai and Shelah ([Sh285] who used a
strongly compact cardinal for the same conclusion) and Shelah's downward
categoricity theorem for AECs with amalgamation (from [Sh394]).Comment: 19 page
Fields and Fusions: Hrushovski constructions and their definable groups
An overview is given of the various expansions of fields and fusions of
strongly minimal sets obtained by means of Hrushovski's amalgamation method, as
well as a characterization of the groups definable in these structures
Infinite combinatorial issues raised by lifting problems in universal algebra
The critical point between varieties A and B of algebras is defined as the
least cardinality of the semilattice of compact congruences of a member of A
but of no member of B, if it exists. The study of critical points gives rise to
a whole array of problems, often involving lifting problems of either diagrams
or objects, with respect to functors. These, in turn, involve problems that
belong to infinite combinatorics. We survey some of the combinatorial problems
and results thus encountered. The corresponding problematic is articulated
around the notion of a k-ladder (for proving that a critical point is large),
large free set theorems and the classical notation (k,r,l){\to}m (for proving
that a critical point is small). In the middle, we find l-lifters of posets and
the relation (k, < l){\to}P, for infinite cardinals k and l and a poset P.Comment: 22 pages. Order, to appea
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