26,148 research outputs found
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Finite closed coverings of compact quantum spaces
We show that a projective space P^\infty(Z/2) endowed with the Alexandrov
topology is a classifying space for finite closed coverings of compact quantum
spaces in the sense that any such a covering is functorially equivalent to a
sheaf over this projective space. In technical terms, we prove that the
category of finitely supported flabby sheaves of algebras is equivalent to the
category of algebras with a finite set of ideals that intersect to zero and
generate a distributive lattice. In particular, the Gelfand transform allows us
to view finite closed coverings of compact Hausdorff spaces as flabby sheaves
of commutative C*-algebras over P^\infty(Z/2).Comment: 26 pages, the Teoplitz quantum projective space removed to another
paper. This is the third version which differs from the second one by fine
tuning and removal of typo
The extended permutohedron on a transitive binary relation
For a given transitive binary relation e on a set E, the transitive closures
of open (i.e., co-transitive in e) sets, called the regular closed subsets,
form an ortholattice Reg(e), the extended permutohedron on e. This
construction, which contains the poset Clop(e) of all clopen sets, is a common
generalization of known notions such as the generalized permutohedron on a
partially ordered set on the one hand, and the bipartition lattice on a set on
the other hand. We obtain a precise description of the completely
join-irreducible (resp., completely meet-irreducible) elements of Reg(e) and
the arrow relations between them. In particular, we prove that (1) Reg(e) is
the Dedekind-MacNeille completion of the poset Clop(e); (2) Every open subset
of e is a set-theoretic union of completely join-irreducible clopen subsets of
e; (3) Clop(e) is a lattice iiff every regular closed subset of e is clopen,
iff e contains no "square" configuration, iff Reg(e)=Clop(e); (4) If e is
finite, then Reg(e) is pseudocomplemented iff it is semidistributive, iff it is
a bounded homomorphic image of a free lattice, iff e is a disjoint sum of
antisymmetric transitive relations and two-element full relations. We
illustrate the strength of our results by proving that, for n greater than or
equal to 3, the congruence lattice of the lattice Bip(n) of all bipartitions of
an n-element set is obtained by adding a new top element to a Boolean lattice
with n2^{n-1} atoms. We also determine the factors of the minimal subdirect
decomposition of Bip(n).Comment: 25 page
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