2 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