115,547 research outputs found
The Hanf number for amalgamation of coloring classes
We study amalgamation properties in a family of abstract elementary classes
that we call coloring classes. The family includes the examples previously
studied in previous work of Baldwin, Kolesnikov, and Shelah. We establish that
the amalgamation property is equivalent to the disjoint amalgamation property
in all coloring classes; find the Hanf number for the amalgamation property for
coloring classes; and improve the results of Baldwin, Kolesnikov, and Shelah by
showing, in ZFC, that the (disjoint) amalgamation property for classes
studied in that paper must hold up to (only a
consistency result was previously known).Comment: 18 page
Splitting formulas for certain Waldhausen Nil-groups
For a group G that splits as an amalgamation of A and B over a common
subgroup C, there is an associated Waldhausen Nil-group, measuring the
"failure" of Mayer-Vietoris for algebraic K-theory. Assume that (1) the
amalgamation is acylindrical, and (2) the groups A,B,G satisfy the
Farrell-Jones isomorphism conjecture. Then we show that the Waldhausen
Nil-group splits as a direct sum of Nil-groups associated to certain
(explicitly describable) infinite virtually cyclic subgroups of G. We note that
a special case of an acylindrical amalgamation includes any amalgamation over a
finite group C.Comment: 12 page
Congruence amalgamation of lattices
J. Tuma proved an interesting "congruence amalgamation" result. We are
generalizing and providing an alternate proof for it. We then provide
applications of this result: --A.P. Huhn proved that every distributive
algebraic lattice with at most compact elements can be
represented as the congruence lattice of a lattice . We show that can be
constructed as a locally finite relatively complemented lattice with zero. --We
find a large class of lattices, the -congruence-finite lattices, that
contains all locally finite countable lattices, in which every lattice has a
relatively complemented congruence-preserving extension
On special partial types and weak canonical bases in simple theories
We define a notion of a weak canonical base for a partial type. This notion
is weaker than the usual canonical base for an amalgamation base. We prove that
certain family of partial types have a weak canonical base. This family clearly
contains the class of amalgamation bases
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