17,844 research outputs found
Noise as a Boolean algebra of -fields
A noise is a kind of homomorphism from a Boolean algebra of domains to the
lattice of -fields. Leaving aside the homomorphism we examine its
image, a Boolean algebra of -fields. The largest extension of such
Boolean algebra of -fields, being well-defined always, is a complete
Boolean algebra if and only if the noise is classical, which answers an old
question of J. Feldman.Comment: Published in at http://dx.doi.org/10.1214/13-AOP861 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Judgment aggregators and Boolean algebra homomorphisms
The theory of Boolean algebras can be fruitfully applied to judgment aggregation: Assuming universality, systematicity and a sufficiently rich agenda, there is a correspondence between (i) non-trivial deductively closed judgment aggregators and (ii) Boolean algebra homomorphisms defined on the power-set algebra of the electorate. Furthermore, there is a correspondence between (i) consistent complete judgment aggregators and (ii) 2-valued Boolean algebra homomorphisms defined on the power-set algebra of the electorate. Since the shell of such a homomorphism equals the set of winning coalitions and since (ultra)filters are shells of (2-valued) Boolean algebra homomorphisms, we suggest an explanation for the effectiveness of the (ultra)filter method in social choice theory. From the (ultra)filter property of the set of winning coalitions, one obtains two general impossibility theorems for judgment aggregation on finite electorates, even without the Pareto principle.judgment aggregation, systematicity, impossibility theorems, filter, ultrafilter, Boolean algebra, homomorphism
Some notes concerning the homogeneity of Boolean algebras and Boolean spaces
We consider homogeneity properties of Boolean algebras that have nonprincipal
ultrafilters which are countably generated.It is shown that a Boolean algebra B
is homogeneous if it is the union of countably generated nonprincipal
ultrafilters and has a dense subset D such that for every a in D the relative
algebra B restriction a:= {b in B:b <= a} is isomorphic to B. In particular,
the free product of countably many copies of an atomic Boolean algebra is
homogeneous. Moreover, a Boolean algebra B is homogeneous if it satisfies the
following conditions:
(i) B has a countably generated ultrafilter,
(ii) B is not c.c.c., and (iii) for every a in B setminus {0} there are
finitely many automorphisms h_1, ...,h_n of B such that 1=h_1(a) cup ... cup
h_n(a)
Classification of the factorial functions of Eulerian binomial and Sheffer posets
We give a complete classification of the factorial functions of Eulerian
binomial posets. The factorial function B(n) either coincides with , the
factorial function of the infinite Boolean algebra, or , the factorial
function of the infinite butterfly poset. We also classify the factorial
functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial
factorial function has Sheffer factorial function D(n) identical to
that of the infinite Boolean algebra, the infinite Boolean algebra with two new
coatoms inserted, or the infinite cubical poset. Moreover, we are able to
classify the Sheffer factorial functions of Eulerian Sheffer posets with
binomial factorial function as the doubling of an upside down
tree with ranks 1 and 2 modified.
When we impose the further condition that a given Eulerian binomial or
Eulerian Sheffer poset is a lattice, this forces the poset to be the infinite
Boolean algebra or the infinite cubical lattice . We also
include several poset constructions that have the same factorial functions as
the infinite cubical poset, demonstrating that classifying Eulerian Sheffer
posets is a difficult problem.Comment: 23 pages. Minor revisions throughout. Most noticeable is title
change. To appear in JCT
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