561 research outputs found
Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions
What is the common link, if there is any, between Church-Rosser systems,
prefix codes with bounded synchronization delay, and local Rees extensions? The
first obvious answer is that each of these notions relates to topics of
interest for WORDS: Church-Rosser systems are certain rewriting systems over
words, codes are given by sets of words which form a basis of a free submonoid
in the free monoid of all words (over a given alphabet) and local Rees
extensions provide structural insight into regular languages over words. So, it
seems to be a legitimate title for an extended abstract presented at the
conference WORDS 2017. However, this work is more ambitious, it outlines some
less obvious but much more interesting link between these topics. This link is
based on a structure theory of finite monoids with varieties of groups and the
concept of local divisors playing a prominent role. Parts of this work appeared
in a similar form in conference proceedings where proofs and further material
can be found.Comment: Extended abstract of an invited talk given at WORDS 201
Bernoulli measure on strings, and Thompson-Higman monoids
The Bernoulli measure on strings is used to define height functions for the
dense R- and L-orders of the Thompson-Higman monoids M_{k,1}. The measure can
also be used to characterize the D-relation of certain submonoids of M_{k,1}.
The computational complexity of computing the Bernoulli measure of certain
sets, and in particular, of computing the R- and L-height of an element of
M_{k,1} is investigated.Comment: 27 pages
The Catenary Degree of Krull Monoids I
Let be a Krull monoid with finite class group such that every class
contains a prime divisor (for example, a ring of integers in an algebraic
number field or a holomorphy ring in an algebraic function field). The catenary
degree of is the smallest integer with the following
property: for each and each two factorizations of , there
exist factorizations of such that, for each , arises from by replacing at most atoms from
by at most new atoms. Under a very mild condition on the
Davenport constant of , we establish a new and simple characterization of
the catenary degree. This characterization gives a new structural understanding
of the catenary degree. In particular, it clarifies the relationship between
and the set of distances of and opens the way towards
obtaining more detailed results on the catenary degree. As first applications,
we give a new upper bound on and characterize when
Graph products of spheres, associative graded algebras and Hilbert series
Given a finite, simple, vertex-weighted graph, we construct a graded
associative (non-commutative) algebra, whose generators correspond to vertices
and whose ideal of relations has generators that are graded commutators
corresponding to edges. We show that the Hilbert series of this algebra is the
inverse of the clique polynomial of the graph. Using this result it easy to
recognize if the ideal is inert, from which strong results on the algebra
follow. Non-commutative Grobner bases play an important role in our proof.
There is an interesting application to toric topology. This algebra arises
naturally from a partial product of spheres, which is a special case of a
generalized moment-angle complex. We apply our result to the loop-space
homology of this space.Comment: 19 pages, v3: elaborated on connections to related work, added more
citations, to appear in Mathematische Zeitschrif
A Three Parameter Hopf Deformation of the Algebra of Feynman-like Diagrams
We construct a three-parameter deformation of the Hopf algebra \LDIAG. This
is the algebra that appears in an expansion in terms of Feynman-like diagrams
of the {\em product formula} in a simplified version of Quantum Field Theory.
This new algebra is a true Hopf deformation which reduces to \LDIAG for some
parameter values and to the algebra of Matrix Quasi-Symmetric Functions
(\MQS) for others, and thus relates \LDIAG to other Hopf algebras of
contemporary physics. Moreover, there is an onto linear mapping preserving
products from our algebra to the algebra of Euler-Zagier sums
Agents, subsystems, and the conservation of information
Dividing the world into subsystems is an important component of the
scientific method. The choice of subsystems, however, is not defined a priori.
Typically, it is dictated by experimental capabilities, which may be different
for different agents. Here we propose a way to define subsystems in general
physical theories, including theories beyond quantum and classical mechanics.
Our construction associates every agent A with a subsystem SA, equipped with
its set of states and its set of transformations. In quantum theory, this
construction accommodates the notion of subsystems as factors of a tensor
product Hilbert space, as well as the notion of subsystems associated to a
subalgebra of operators. Classical systems can be interpreted as subsystems of
quantum systems in different ways, by applying our construction to agents who
have access to different sets of operations, including multiphase covariant
channels and certain sets of free operations arising in the resource theory of
quantum coherence. After illustrating the basic definitions, we restrict our
attention to closed systems, that is, systems where all physical
transformations act invertibly and where all states can be generated from a
fixed initial state. For closed systems, we propose a dynamical definition of
pure states, and show that all the states of all subsystems admit a canonical
purification. This result extends the purification principle to a broader
setting, in which coherent superpositions can be interpreted as purifications
of incoherent mixtures.Comment: 31+26 pages, updated version with new results, contribution to
Special Issue on Quantum Information and Foundations, Entropy, GM D'Ariano
and P Perinotti, ed
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