arXiv subjects

Pedro V. Silva

Publications and source records attributed to Pedro V. Silva.

At least 37 records · Page 2Linked to original sources

arXiv2017

On the semigroup rank of a group

For an arbitrary group $G$, it is shown that either the semigroup rank $G{\rm rk}S$ equals the group rank $G{\rm rk}G$, or $G{\rm rk}S = G{\rm rk}G+1$. This is the starting point for the rest of the article, where the semigroup rank for diverse kinds of groups is analysed. The semigroup rank of relatively free groups, for any variety of groups, is computed. For a finitely generated abelian group~$G$, it is proven that $G{\rm rk}S = G{\rm rk}G+1$ if and only if $G$ is torsion-free. In general, this is not true. Partial results are obtained in the nilpotent case. It is also proven that if $M$ is a connected closed surface, then $(π_1(M)){\rm rk}S = (π_1(M)){\rm rk}G+1$ if and only if $M$ is orientable.

math.GR↗

arXiv2017

On the subsemigroup complex of an aperiodic Brandt semigroup

We introduce the subsemigroup complex of a finite semigroup S as a (boolean representable) simplicial complex defined through chains in the lattice of subsemigroups of S. We present a research program for such complexes, illustrated through the particular case of combinatorial Brandt semigroups. The results include alternative characterizations of faces and facets, asymptotical estimates on the number of facets, or establishing when the complex is pure or a matroid.

math.CO↗

arXiv2016

Local finiteness for Green relations in (I-)semigroup varieties

In this work, the lattice of varieties of semigroups and the lattice of varieties of I-semigroups (a common setting for both the variety of completely regular semigroups and the variety of inverse semigroups) are studied with respect to the following concepts: a variety V of (I-)semigroups is said to be locally K-finite, where K stands for any of the five Green's relations, if every finitely generated semigroup from V has only finitely many (distinct) K-classes.

math.GR↗

arXiv2016

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 $A$ through right congruences on $A^k$, defined using the natural right action of $A^+$. 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.

math.GR↗

arXiv2016

The semaphore codes attached to a Turing machine via resets and their various limits

We introduce semaphore codes associated to a Turing machine via resets. Semaphore codes provide an approximation theory for resets. In this paper we generalize the set-up of our previous paper "Random walks on semaphore codes and delay de Bruijn semigroups" to the infinite case by taking the profinite limit of $k$-resets to obtain $(-ω)$-resets. We mention how this opens new avenues to attack the P versus NP problem.

math.GR↗

arXiv2015

On the lattice of flats of a boolean representable simplicial complex

It is shown that the lattices of flats of boolean representable simplicial complexes are always atomistic, but semimodular if and only if the complex is a matroid. A canonical construction is introduced for arbitrary finite atomistic lattices, providing a characterization of the lattices of flats of boolean representable simplicial complexes and a decidability condition. We remark that every finite lattice occurs as the lattice of flats of some simplicial complex.

math.CO↗

arXiv2015

On the topology of a boolean representable simplicial complex

It is proved that fundamental groups of boolean representable simplicial complexes are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension $\geq 2$. In the case of dimension 2, it is shown that boolean representable simplicial complexes have the homotopy type of a wedge of spheres of dimensions 1 and 2. Also in the case of dimension 2, necessary and sufficient conditions for shellability and being sequentially Cohen-Macaulay are determined. Complexity bounds are provided for all the algorithms involved.

math.CO↗

arXiv2015

Equations over free inverse monoids with idempotent variables

We introduce the notion of idempotent variables for studying equations in inverse monoids. It is proved that it is decidable in singly exponential time (DEXPTIME) whether a system of equations in idempotent variables over a free inverse monoid has a solution. The result is proved by a direct reduction to solve language equations with one-sided concatenation and a known complexity result by Baader and Narendran: Unification of concept terms in description logics, 2001. We also show that the problem becomes DEXPTIME hard , as soon as the quotient group of the free inverse monoid has rank at least two. Decidability for systems of typed equations over a free inverse monoid with one irreducible variable and at least one unbalanced equation is proved with the same complexity for the upper bound. Our results improve known complexity bounds by Deis, Meakin, and Senizergues: Equations in free inverse monoids, 2007. Our results also apply to larger families of equations where no decidability has been previously known.

cs.LO↗

arXiv2015

Takahasi semigroups

Takahasi's theorem on chains of subgroups of bounded rank in a free group is generalized to several classes of semigroups. As an application, it is proved that the subsemigroups of periodic points are finitely generated and periodic orbits are bounded for arbitrary endomorphisms for various semigroups. Some of these results feature classes such as completely simple semigroups, Clifford semigroups or monoids defined by balanced one-relator presentations. In addition to the background on semigroups, proofs involve arguments over groups and finite automata.

math.GR↗

arXiv2015

Bounding the gap between a free group (outer) automorphism and its inverse

For any finitely generated group $G$, two complexity functions $α_G$ and $β_G$ are defined to measure the maximal possible gap between the norm of an automorphism (respectively outer automorphism) of $G$ and the norm of its inverse. Restricting attention to free groups $F_r$, the exact asymptotic behaviour of $α_2$ and $β_2$ is computed. For rank $r\geqslant 3$, polynomial lower bounds are provided for $α_r$ and $β_r$, and the existence of a polynomial upper bound is proved for $β_r$.

math.GR↗

arXiv2014

Howson's property for semidirect products of semilattices by groups

An inverse semigroup $S$ is a Howson inverse semigroup if the intersection of finitely generated inverse subsemigroups of $S$ is finitely generated. Given a locally finite action $θ$ of a group $G$ on a semilattice $E$, it is proved that $E \ast_θ G$ is a Howson inverse semigroup if and only if $G$ is a Howson group. It is also shown that this equivalence fails for arbitrary actions.

math.GR↗

arXiv2014

Hölder conditions for endomorphisms of hyperbolic groups

It is proved that an endomorphism $φ$ of a hyperbolic group $G$ satisfies a Hölder condition with respect to a visual metric if and only if $φ$ is virtually injective and $Gφ$ is a quasiconvex subgroup of $G$. If $G$ is virtually free or torsion-free co-hopfian, then $φ$ is uniformly continuous if and only if it it satisfies a Hölder condition if and only if it is virtually injective. Lipschitz conditions are discussed for free group automorphisms.

math.GR↗

arXiv2014

Geometric characterizations of virtually free groups

Four geometric conditions on a geodesic metric space, which are stronger variants of classical conditions characterizing hyperbolicity, are proved to be equivalent. In the particular case of the Cayley graph of a finitely generated group, it is shown that they characterize virtually free groups.

math.GR↗

arXiv2014

Finiteness results for subgroups of finite extensions

We discuss in the context of finite extensions two classical theorems of Takahasi and Howson on subgroups of free groups. We provide bounds for the rank of the intersection of subgroups within classes of groups such as virtually free groups, virtually nilpotent groups or fundamental groups of finite graphs of groups with virtually polycyclic vertex groups and finite edge groups. As an application of our generalization of Takahasi's Theorem, we provide an uniform bound for the rank of the periodic subgroup of any endomorphism of the fundamental group of a given finite graph of groups with finitely generated virtually nilpotent vertex groups and finite edge groups.

math.GR↗

arXiv2012

On periodic points of free inverse monoid endomorphisms

It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.

math.GR↗

arXiv2012

Fixed points of endomorphisms of trace monoids

It is proved that the fixed point submonoid and the periodic point submonoid of a trace monoid endomorphism are always finitely generated. Considering the Foata normal form metric on trace monoids and uniformly continuous endomorphisms, a finiteness theorem is proved for the infinite fixed points of the continuous extension to real traces.

math.GR↗

arXiv2012

Matroids, hereditary collections and simplicial complexes having boolean representations

Inspired by the work of Izakhian and Rhodes, a theory of representation of hereditary collections by boolean matrices is developed. This corresponds to representation by finite $\vee$-generated lattices. The lattice of flats, defined for hereditary collections, lattices and matrices, plays a central role in the theory. The representations constitute a lattice and the minimal and strictly join irreducible elements are studied, as well as various closure operators.

math.CO↗

arXiv2012

Fixed points of endomorphisms of graph groups

It is shown, for a given graph group $G$, that the fixed point subgroup Fix$\,φ$ is finitely generated for every endomorphism $φ$ of $G$ if and only if $G$ is a free product of free abelian groups. The same conditions hold for the subgroup of periodic points. Similar results are obtained for automorphisms, if the dependence graph of $G$ is a transitive forest.

math.GR↗