132 research outputs found
Laver and set theory
In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
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
Guessing genericity -- looking at parametrized diamonds from a different perspective
We introduce and study a family of axioms that closely follows the pattern of
parametrized diamonds, studied by Moore, Hru\v{s}\'ak, and D\v{z}amonja in
[13]. However, our approach appeals to model theoretic / forcing theoretic
notions, rather than pure combinatorics. The main goal of the paper is to
exhibit a surprising, close connection between seemingly very distinct
principles. As an application, we show that forcing with a measure algebra
preserves (a variant of) , improving an old result
of M. Hru\v{s}\'ak
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