139 research outputs found
Context-freeness of the languages of Schützenberger automata of HNN-extensions of finite inverse semigroups
We prove that the Schützenberger graph of any element of the HNN-extension of a finite inverse semigroup S with respect to its standard presentation is a context-free graph in the sense of [11], showing that the language L recognized by this automaton is context-free. Finally we explicitly construct the grammar generating L, and from this fact we show that the word problem for an HNN-extension of a finite inverse semigroup S is decidable and lies in the complexity class of polynomial time problems
Semigroup and category-theoretic approaches to partial symmetry
This thesis is about trying to understand various aspects of partial symmetry using
ideas from semigroup and category theory. In Chapter 2 it is shown that the left Rees
monoids underlying self-similar group actions are precisely monoid HNN-extensions.
In particular it is shown that every group HNN-extension arises from a self-similar
group action. Examples of these monoids are constructed from fractals. These ideas
are generalised in Chapter 3 to a correspondence between left Rees categories, selfsimilar
groupoid actions and category HNN-extensions of groupoids, leading to a
deeper relationship with Bass-Serre theory. In Chapter 4 of this thesis a functor K
between the category of orthogonally complete inverse semigroups and the category
of abelian groups is constructed in two ways, one in terms of idempotent matrices
and the other in terms of modules over inverse semigroups, and these are shown to be
equivalent. It is found that the K-group of a Cuntz-Krieger semigroup of a directed
graph G is isomorphic to the operator K0-group of the Cuntz-Krieger algebra of G
and the K-group of a Boolean algebra is isomorphic to the topological K0-group of
the corresponding Boolean space under Stone dualit
A correspondence between a class of monoids and self-similar group actions II
The first author showed in a previous paper that there is a correspondence
between self-similar group actions and a class of left cancellative monoids
called left Rees monoids. These monoids can be constructed either directly from
the action using Zappa-Sz\'ep products, a construction that ultimately goes
back to Perrot, or as left cancellative tensor monoids from the covering
bimodule, utilizing a construction due to Nekrashevych, In this paper, we
generalize the tensor monoid construction to arbitrary bimodules. We call the
monoids that arise in this way Levi monoids and show that they are precisely
the equidivisible monoids equipped with length functions. Left Rees monoids are
then just the left cancellative Levi monoids. We single out the class of
irreducible Levi monoids and prove that they are determined by an isomorphism
between two divisors of its group of units. The irreducible Rees monoids are
thereby shown to be determined by a partial automorphism of their group of
units; this result turns out to be signficant since it connects irreducible
Rees monoids directly with HNN extensions. In fact, the universal group of an
irreducible Rees monoid is an HNN extension of the group of units by a single
stable letter and every such HNN extension arises in this way.Comment: Some very minor corrections made and the dedication adde
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