15,119 research outputs found
Fixed points of endomorphisms and relations between metrics in preGarside monoids
Rodaro and Silva proved that the fixed points submonoid and the periodic
points submonoid of a trace monoid endomorphism are always finitely generated.
We show that for finitely generated left preGarside monoids, that includs
finitely generated preGarside monoids, Garside monoids and Artin monoids, the
fixed and periodic points submonoids of any endomorphism are also finitely
generated left preGarside monoids under some condition, and in the case of
Artin monoids, these submonoids are always Artin monoids too. We also prove
algebraically some inequalities, equivalences and non-equivalences between
three metrics in finitely generated preGarside monoids, and especially in trace
monoids and Garside monoids
On surjunctive monoids
A monoid is called surjunctive if every injective cellular automata with
finite alphabet over is surjective. We show that all finite monoids, all
finitely generated commutative monoids, all cancellative commutative monoids,
all residually finite monoids, all finitely generated linear monoids, and all
cancellative one-sided amenable monoids are surjunctive. We also prove that
every limit of marked surjunctive monoids is itself surjunctive. On the other
hand, we show that the bicyclic monoid and, more generally, all monoids
containing a submonoid isomorphic to the bicyclic monoid are non-surjunctive
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
Rewriting systems and biautomatic structures for Chinese, hypoplactic, and sylvester monoids
This paper studies complete rewriting systems and biautomaticity for three interesting classes of finite-rank homogeneous monoids: Chinese monoids, hypoplactic monoids, and sylvester monoids. For Chinese monoids, we first give new presentations via finite complete rewriting systems, using more lucid constructions and proofs than those given independently by Chen & Qui and Güzel Karpuz; we then construct biautomatic structures. For hypoplactic monoids, we construct finite complete rewriting systems and biautomatic structures. For sylvester monoids, which are not finitely presented, we prove that the standard presentation is an infinite complete rewriting system, and construct biautomatic structures. Consequently, the monoid algebras corresponding to monoids of these classes are automaton algebras in the sense of Ufnarovskij
Tied Monoids
We construct certain monoids, called tied monoids. These monoids result to be
semidirect products finitely presented and commonly built from braid groups and
their relatives acting on monoids of set partitions. The nature of our monoids
indicate that they should give origin to new knot algebras; indeed, our tied
monoids include the tied braid monoid and the tied singular braid monoid, which
were used, respectively, to construct new polynomial invariants for classical
links and singular links. Consequently, we provide a mechanism to attach an
algebra to each tied monoid. To build the tied monoids it is necessary to have
presentations of set partition monoids of types A, B and D, among others. For
type A we use a presentation due to FitzGerald and for the other type it was
necessary to built them.Comment: 47 page
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