435 research outputs found
Quasi-selective ultrafilters and asymptotic numerosities
We isolate a new class of ultrafilters on N, called "quasi-selective" because
they are intermediate between selective ultrafilters and P-points. (Under the
Continuum Hypothesis these three classes are distinct.) The existence of
quasi-selective ultrafilters is equivalent to the existence of "asymptotic
numerosities" for all sets of tuples of natural numbers. Such numerosities are
hypernatural numbers that generalize finite cardinalities to countable point
sets. Most notably, they maintain the structure of ordered semiring, and, in a
precise sense, they allow for a natural extension of asymptotic density to all
sequences of tuples of natural numbers.Comment: 27 page
The uniform face ideals of a simplicial complex
We define the uniform face ideal of a simplicial complex with respect to an
ordered proper vertex colouring of the complex. This ideal is a monomial ideal
which is generally not squarefree. We show that such a monomial ideal has a
linear resolution, as do all of its powers, if and only if the colouring
satisfies a certain nesting property.
In the case when the colouring is nested, we give a minimal cellular
resolution supported on a cubical complex. From this, we give the graded Betti
numbers in terms of the face-vector of the underlying simplicial complex.
Moreover, we explicitly describe the Boij-S\"oderberg decompositions of both
the ideal and its quotient. We also give explicit formul\ae\ for the
codimension, Krull dimension, multiplicity, projective dimension, depth, and
regularity. Further still, we describe the associated primes, and we show that
they are persistent.Comment: 34 pages, 8 figure
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