arXiv subjects

Jorge Almeida

Publications and source records attributed to Jorge Almeida.

At least 19 recordsLinked to original sources

arXiv2026

Profinite approach to S-adic shift spaces II: Invertible directive sequences

This paper is the second in a series about (relatively) free profinite semigroups and S-adic representations of minimal shifts. The first installment introduced the key notions of profinite images and saturating properties. The present paper examines the relationship between relative invertibility conditions (on a directive sequence) and charging properties (of a shift). The main result shows that the two properties are equivalent under mild conditions. Moreover, charging properties are shown to be equivalent to density of subgroups generated by return words in relatively free profinite groups. The main result can then be used to deduce purely discrete statements about subgroups generated by return words in relatively free groups. We also highlight applications of the main result to the study of maximal bifix decoding, following earlier work by the second author. Finally, we establish some results on the effect of direct images on charging properties and explain how to apply these results to the case of episturmian shifts.

math.DS↗

arXiv2025

Profinite approach to S-adic shift spaces I: Saturating directive sequences

This paper is the first in a series of three, about (relatively)free profinite semigroups and S-adic representations of minimal shift spaces. We associate to each primitive S-adic directivesequence ${\boldsymbolσ}$ a $\textit{profinite image}$ in the free profinite semigroup over the alphabet of the induced minimal shift space. When this profinite image contains a $\mathcal{J}$-maximal maximal subgroup of the free profinite semigroup (which, up to isomorphism, is called the $\textit{Schützenberger group}$ of the shift space), we say that ${\boldsymbolσ}$ is $\textit{saturating}$. We show that if ${\boldsymbolσ}$ is recognizable, then it is saturating. Conversely, we use the notion of saturating sequence to obtain several sufficient conditions for ${\boldsymbolσ}$ to be recognizable: ${\boldsymbolσ}$ consists of pure encodings; or ${\boldsymbolσ}$ is eventually recognizable, saturating and consists of encodings; or ${\boldsymbolσ}$ is eventually recognizable, recurrent, bounded and consists of encodings. For the most part, we do not assume that ${\boldsymbolσ}$ has finite alphabet rank although we establish that this combinatorial property has important algebraic consequences, namely that the rank of the Schützenberger group is also finite, whose maximum possible value we also determine. We also show that for every minimal shift space of finite topological rank, the rank of its Schützenberger group is a lower bound of the topological rank.

math.DS↗

arXiv2025

Open multiplication in relatively free profinite semigroupoids

The purpose of this paper is to extend some useful results, such as the multiplication being open, previously known for suitable finitely generated relatively free profinite semigroups, to relatively free profinite semigroupoids over finite-vertex graphs. This extension is used to give a profinite characterization of recurrent words over infinite alphabets and to establish new results about stabilizers in relatively free profinite semigroups and semigroupoids.

math.GR↗

arXiv2025

Eilenberg correspondence for Stone recognition

We develop and explore the idea of recognition of languages (in the general sense of subsets of topological algebras) as preimages of clopen sets under continuous homomorphisms into Stone topological algebras. We obtain an Eilenberg correspondence between varieties of languages and varieties of ordered Stone topological algebras and a Birkhoff/Reiterman-type theorem showing that the latter may me defined by certain pseudo-inequalities. In the case of classical formal languages, of words over a finite alphabet, we also show how this extended framework goes beyond the class of regular languages by working with Stone completions of minimal automata, viewed as unary algebras. This leads to a general method for showing that a language does not belong to a variety of languages, expressed in terms of sequences of pairs of words, which is illustrated when the class consists of all finite intersections of context-free languages.

cs.FL↗

arXiv2025

Epimorphisms and pseudovarieties

For each of the following conditions, we characterize the pseudovarieties of semigroups V that satisfy it: (i) every epimorphism to a member of V is onto; (ii) every epimorphism to a finite semigroup with domain a member of V is onto; (iii) for every epimorphism from S to T with S in V and T finite, T is also a member of V.

math.GR↗

arXiv2025

Pseudovarieties of semigroups

The most developed aspect of the theory of finite semigroups is their classification in pseudovarieties. The main motivation for investigating such entities comes from their connection with the classification of regular languages via Eilenberg's correspondence. This connection prompted the study of various natural operators on pseudovarieties and led to several important questions, both algebraic and algorithmic. The most important of these questions is decidability: given a finite semigroup is there an algorithm that tests whether it belongs to the pseudovariety? Since the most relevant operators on pseudovarieties do not preserve decidability, one often seeks to establish stronger properties. A key role is played by relatively free profinite semigroups, which is the counterpart of free algebras in universal algebra. The purpose of this paper is to give a brief survey of the state of the art, highlighting some of the main developments and problems.

math.GR↗

arXiv2024

The Word Problem for $(ω- 1)$-Terms over $\mathrm{DAb}$

We give a ranker-based description using finite-index congruences for the variety $\boldsymbol{\mathrm{DAb}}$ of finite monoids whose regular $\mathcal{D}$-classes form Abelian groups. This combinatorial description yields a normal form for general pseudowords over $\boldsymbol{\mathrm{DAb}}$. For $(ω- 1)$-terms, this normal form is computable, which yields an algorithm for the word problem for $(ω- 1)$-terms of $\boldsymbol{\mathrm{DAb}}$.

cs.FL↗

arXiv2023

Stone pseudovarieties

Profinite algebras are the residually finite compact algebras; their underlying topological spaces are Stone spaces. We extend the theory of profinite algebras to a more general setting of Stone topological algebras. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at Stone spaces as the dual spaces of Boolean algebras, we find a simple characterization of when the dual space admits a natural structure of topological algebra. This provides a new approach to duality theory which culminates in the proof that a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established.

math.LO↗

arXiv2022

Equidivisibility and profinite coproduct

The aim of this work is to investigate the behavior of equidivisibility under coproduct in the category of pro-$\mathsf{V}$ semigroups, where $\mathsf{V}$ is a pseudovariety of finite semigroups. Exploring the relationship with the two-sided Karnofsky--Rhodes expansion, the notions of KR-cover and strong KR-cover for profinite semigroups are introduced. The former is stronger than equidivisibility and the latter provides a characterization of equidivisible profinite semigroups with an extra mild condition, so-called letter super-cancellativity. Furthermore, under the assumption that $\mathsf{V}$ is closed under two-sided Karnofsky--Rhodes expansion, closure of some classes of equidivisible pro-$\mathsf{V}$ semigroups under(finite) $\mathsf{V}$-coproduct is established.

math.GR↗

arXiv2021

What makes a Stone topological algebra profinite

This paper is a contribution to understanding what properties should a topological algebra on a Stone space satisfy to be profinite. We reformulate and simplify proofs for some known properties using syntactic congruences. We also clarify the role of various alternative ways of describing syntactic congruences, namely by finite sets of terms and by compact sets of continuous self mappings of the algebra.

math.GN↗

arXiv2021

Binary patterns in the Prouhet-Thue-Morse sequence

We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all (finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can actually be found in that sequence as segments (up to exchange of letters in the infinite case). This result was previously attributed to unpublished work by D. Guaiana and may also be derived from publications of A. Shur only available in Russian. We also identify the (finitely many) finite binary patterns that appear non trivially, in the sense that they are obtained by applying an endomorphism that does not map the set of all segments of the sequence into itself.

math.CO↗

arXiv2019

Locally countable pseudovarieties

The purpose of this paper is to contribute to the theory of profinite semigroups by considering the special class consisting of those all of whose finitely generated closed subsemigroups are countable, which are said to be locally countable. We also call locally countable a pseudovariety V (of finite semigroups) for which all pro-V semigroups are locally countable. We investigate operations preserving local countability of pseudovarieties and show that, in contrast with local finiteness, several natural operations do not preserve it. We also investigate the relationship of a finitely generated profinite semigroup being countable with every element being expressable in terms of the generators using multiplication and the idempotent (omega) power. The two properties turn out to be equivalent if there are only countably many group elements, gathered in finitely many regular J-classes. We also show that the pseudovariety generated by all finite ordered monoids satisfying the inequality $1\le x^n$ is locally countable if and only if $n=1$.

math.GR↗

arXiv2019

Recognizing pro-R closures of regular languages

Given a regular language L, we effectively construct a unary semigroup that recognizes the topological closure of L in the free unary semigroup relative to the variety of unary semigroups generated by the pseudovariety R of all finite R-trivial semigroups. In particular, we obtain a new effective solution of the separation problem of regular languages by R-languages.

math.GR↗

arXiv2018

On the group of a rational maximal bifix code

We give necessary and sufficient conditions for the group of a rational maximal bifix code $Z$ to be isomorphic with the $F$-group of $Z\cap F$, when $F$ is recurrent and $Z\cap F$ is rational. The case where $F$ is uniformly recurrent, which is known to imply the finiteness of $Z\cap F$, receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of $F$.

math.GR↗

arXiv2018

Pseudovarieties of ordered completely regular semigroups

This paper is a contribution to the theory of finite semigroups and their classification in pseudovarieties, which is motivated by its connections with computer science. The question addressed is what role can play the consideration of an order compatible with the semigroup operation. In the case of unions of groups, so-called completely regular semigroups, the problem of which new pseudovarieties appear in the ordered context is solved. As applications, it is shown that the lattice of pseudovarieties of ordered completely regular semigroups is modular and that taking the intersection with the pseudovariety of bands defines a complete endomorphism of the lattice of all pseudovarieties of ordered semigroups.

math.GR↗

arXiv2018

Profinite topologies

Profinite semigroups are a generalization of finite semigroups that come about naturally when one is interested in considering free structures with respect to classes of finite semigroups. They also appear naturally through dualization of Boolean algebras of regular languages. The additional structure is given by a compact zero-dimensional topology. Profinite topologies may also be considered on arbitrary abstract semigroups by taking the initial topology for homomorphisms into finite semigroups. This text is the proposed chapter of the Handdbook of Automata Theory dedicated to these topics. The general theory is formulated in the setting of universal algebra because it is mostly independent of specific properties of semigroups and more general algebras naturally appear in this context. In the case of semigroups, particular attention is devoted to solvability of systems of equations with respect to a pseudovariety, which is relevant for solving membership problems for pseudovarieties. Focus is also given to relatively free profinite semigroups per se, specially "large" ones, stressing connections with symbolic dynamics that bring light to their structure.

math.GR↗

arXiv2017

Towards a pseudoequational proof theory

A new scheme for proving pseudoidentities from a given set Σ of pseudoidentities, which is clearly sound, is also shown to be complete in many instances, such as when Σ defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids. Many further examples when the scheme is complete are given when Σ defines a pseudovariety V which is σ-reducible for the equation x=y, provided Σ is enough to prove a basis of identities for the variety of σ-algebras generated by V. This gives ample evidence in support of the conjecture that the proof scheme is complete in general.

math.GR↗

arXiv2017

The linear nature of pseudowords

Given a pseudoword over suitable pseudovarieties, we associate to it a labeled linear order determined by the factorizations of the pseudoword. We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.

cs.FL↗