7 research outputs found
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
Some simple theories from a Boolean algebra point of view
We find a strong separation between two natural families of simple rank one
theories in Keisler's order: the theories reflecting graph
sequences, which witness that Keisler's order has the maximum number of
classes, and the theories , which are the higher-order analogues of
the triangle-free random graph. The proof involves building Boolean algebras
and ultrafilters "by hand" to satisfy certain model theoretically meaningful
chain conditions. This may be seen as advancing a line of work going back
through Kunen's construction of good ultrafilters in ZFC using families of
independent functions. We conclude with a theorem on flexible ultrafilters, and
open questions.Comment: [MiSh:1218], 38 page