957 research outputs found
On lattices of convex sets in R^n
Properties of several sorts of lattices of convex subsets of R^n are
examined. The lattice of convex sets containing the origin turns out, for n>1,
to satisfy a set of identities strictly between those of the lattice of all
convex subsets of R^n and the lattice of all convex subsets of R^{n-1}. The
lattices of arbitrary, of open bounded, and of compact convex sets in R^n all
satisfy the same identities, but the last of these is join-semidistributive,
while for n>1 the first two are not. The lattice of relatively convex subsets
of a fixed set S \subseteq R^n satisfies some, but in general not all of the
identities of the lattice of ``genuine'' convex subsets of R^n.Comment: 35 pages, to appear in Algebra Universalis, Ivan Rival memorial
issue. See also http://math.berkeley.edu/~gbergman/paper
Finitely generated free Heyting algebras via Birkhoff duality and coalgebra
Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and
thus the free algebras can be obtained by a direct limit process. Dually, the
final coalgebras can be obtained by an inverse limit process. In order to
explore the limits of this method we look at Heyting algebras which have mixed
rank 0-1 axiomatizations. We will see that Heyting algebras are special in that
they are almost rank 1 axiomatized and can be handled by a slight variant of
the rank 1 coalgebraic methods
Cevian operations on distributive lattices
We construct a completely normal bounded distributive lattice D in which for
every pair (a, b) of elements, the set {x D | a b x} has a
countable coinitial subset, such that D does not carry any binary operation -
satisfying the identities x y (x-y),(x-y)(y-x) = 0, and x-z
(x-y)(y-z). In particular, D is not a homomorphic image of the
lattice of all finitely generated convex {\ell}-subgroups of any (not
necessarily Abelian) {\ell}-group. It has \lambda\infty\lambda$-elementary equivalence.Comment: 23 pages. v2 removes a redundancy from the definition of a Cevian
operation in v1.In Theorem 5.12, Idc should be replaced by Csc (especially on
the G side
A discussion on the origin of quantum probabilities
We study the origin of quantum probabilities as arising from non-boolean
propositional-operational structures. We apply the method developed by Cox to
non distributive lattices and develop an alternative formulation of
non-Kolmogorvian probability measures for quantum mechanics. By generalizing
the method presented in previous works, we outline a general framework for the
deduction of probabilities in general propositional structures represented by
lattices (including the non-distributive case).Comment: Improved versio
- …