322 research outputs found
Random walks on semaphore codes and delay de Bruijn semigroups
We develop a new approach to random walks on de Bruijn graphs over the
alphabet through right congruences on , defined using the natural
right action of . A major role is played by special right congruences,
which correspond to semaphore codes and allow an easier computation of the
hitting time. We show how right congruences can be approximated by special
right congruences.Comment: 34 pages; 10 figures; as requested by the journal, the previous
version of this paper was divided into two; this version contains Sections
1-8 of version 1; Sections 9-12 will appear as a separate paper with extra
material adde
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
Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras
We study a non-commutative generalization of Stone duality that connects a
class of inverse semigroups, called Boolean inverse -semigroups, with a
class of topological groupoids, called Hausdorff Boolean groupoids. Much of the
paper is given over to showing that Boolean inverse -semigroups arise
as completions of inverse semigroups we call pre-Boolean. An inverse
-semigroup is pre-Boolean if and only if every tight filter is an
ultrafilter, where the definition of a tight filter is obtained by combining
work of both Exel and Lenz. A simple necessary condition for a semigroup to be
pre-Boolean is derived and a variety of examples of inverse semigroups are
shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees
matrix semigroups over the polycyclics, are pre-Boolean and it is proved that
the groups of units of their completions are precisely the Thompson-Higman
groups . The inverse semigroups arising from suitable directed graphs
are also pre-Boolean and the topological groupoids arising from these graph
inverse semigroups under our non-commutative Stone duality are the groupoids
that arise from the Cuntz-Krieger -algebras.Comment: The presentation has been sharpened up and some minor errors
correcte
- β¦