1,662 research outputs found
Iterated LD-Problem in non-associative key establishment
We construct new non-associative key establishment protocols for all left
self-distributive (LD), multi-LD-, and mutual LD-systems. The hardness of these
protocols relies on variations of the (simultaneous) iterated LD-problem and
its generalizations. We discuss instantiations of these protocols using
generalized shifted conjugacy in braid groups and their quotients, LD-conjugacy
and -symmetric conjugacy in groups. We suggest parameter choices for
instantiations in braid groups, symmetric groups and several matrix groups.Comment: 30 pages, 5 figures. arXiv admin note: substantial text overlap with
arXiv:1305.440
Congruence Lattices of Certain Finite Algebras with Three Commutative Binary Operations
A partial algebra construction of Gr\"atzer and Schmidt from
"Characterizations of congruence lattices of abstract algebras" (Acta Sci.
Math. (Szeged) 24 (1963), 34-59) is adapted to provide an alternative proof to
a well-known fact that every finite distributive lattice is representable, seen
as a special case of the Finite Lattice Representation Problem.
The construction of this proof brings together Birkhoff's representation
theorem for finite distributive lattices, an emphasis on boolean lattices when
representing finite lattices, and a perspective based on inequalities of
partially ordered sets. It may be possible to generalize the techniques used in
this approach.
Other than the aforementioned representation theorem only elementary tools
are used for the two theorems of this note. In particular there is no reliance
on group theoretical concepts or techniques (see P\'eter P\'al P\'alfy and
Pavel Pud\'lak), or on well-known methods, used to show certain finite lattice
to be representable (see William J. DeMeo), such as the closure method
On the coset category of a skew lattice
Skew lattices are non-commutative generalizations of lattices. The coset
structure decomposition is an original approach to the study of these algebras
describing the relation between its rectangular classes. In this paper we will
look at the category determined by these rectangular algebras and the morphisms
between them, showing that not all skew lattices can determine such a category.
Furthermore, we will present a class of examples of skew lattices in rings that
are not strictly categorical, and present sufficient conditions for skew
lattices of matrices in rings to constitute -distributive skew
lattices.Comment: 17 pages, submitted to Demonstratio Mathematica. arXiv admin note:
text overlap with arXiv:1212.649
Generalized probabilities in statistical theories
In this review article we present different formal frameworks for the
description of generalized probabilities in statistical theories. We discuss
the particular cases of probabilities appearing in classical and quantum
mechanics, possible generalizations of the approaches of A. N. Kolmogorov and
R. T. Cox to non-commutative models, and the approach to generalized
probabilities based on convex sets
Noncommutative localization in noncommutative geometry
The aim of these notes is to collect and motivate the basic localization
toolbox for the geometric study of ``spaces'', locally described by
noncommutative rings and their categories of one-sided modules.
We present the basics of Ore localization of rings and modules in much
detail. Common practical techniques are studied as well. We also describe a
counterexample for a folklore test principle. Localization in negatively
filtered rings arising in deformation theory is presented. A new notion of the
differential Ore condition is introduced in the study of localization of
differential calculi.
To aid the geometrical viewpoint, localization is studied with emphasis on
descent formalism, flatness, abelian categories of quasicoherent sheaves and
generalizations, and natural pairs of adjoint functors for sheaf and module
categories. The key motivational theorems from the seminal works of Gabriel on
localization, abelian categories and schemes are quoted without proof, as well
as the related statements of Popescu, Watts, Deligne and Rosenberg.
The Cohn universal localization does not have good flatness properties, but
it is determined by the localization map already at the ring level. Cohn
localization is here related to the quasideterminants of Gelfand and Retakh;
and this may help understanding both subjects.Comment: 93 pages; (including index: use makeindex); introductory survey, but
with few smaller new result
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