2,261 research outputs found
Cardinal characteristics and countable Borel equivalence relations
Boykin and Jackson recently introduced a property of countable Borel
equivalence relations called Borel boundedness, which they showed is closely
related to the union problem for hyperfinite equivalence relations. In this
paper, we introduce a family of properties of countable Borel equivalence
relations which correspond to combinatorial cardinal characteristics of the
continuum in the same way that Borel boundedness corresponds to the bounding
number . We analyze some of the basic behavior of these
properties, showing for instance that the property corresponding to the
splitting number coincides with smoothness. We then settle many
of the implication relationships between the properties; these relationships
turn out to be closely related to (but not the same as) the Borel Tukey
ordering on cardinal characteristics
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
Cores of Countably Categorical Structures
A relational structure is a core, if all its endomorphisms are embeddings.
This notion is important for computational complexity classification of
constraint satisfaction problems. It is a fundamental fact that every finite
structure has a core, i.e., has an endomorphism such that the structure induced
by its image is a core; moreover, the core is unique up to isomorphism. Weprove
that every \omega -categorical structure has a core. Moreover, every
\omega-categorical structure is homomorphically equivalent to a model-complete
core, which is unique up to isomorphism, and which is finite or \omega
-categorical. We discuss consequences for constraint satisfaction with \omega
-categorical templates
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