A combinatorial property of ideals in free profinite monoids

Steinberg, Benjamin

doi:10.1016/j.jpaa.2009.12.014

Cited by 7 publications

(5 citation statements)

References 9 publications

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“…Although the original result was formulated for the pseudovariety of all finite semigroups, the proof applies unchanged to pseudovarieties containing all finite local semilattices. Related results, under the same hypothesis as the following corollary, have been obtained by Steinberg (2010).…”

Section: Factoriality Of Some Pseudovarietiessupporting

confidence: 69%

Factoriality and the Pin-Reutenauer procedure

Almeida,

Costa,

Zeitoun

2015

Preprint

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We consider implicit signatures over finite semigroups determined by sets of pseudonatural numbers. We prove that, under relatively simple hypotheses on a pseudovariety V of semigroups, the finitely generated free algebra for the largest such signature is closed under taking factors within the free pro-V semigroup on the same set of generators. Furthermore, we show that the natural analogue of the Pin-Reutenauer descriptive procedure for the closure of a rational language in the free group with respect to the profinite topology holds for the pseudovariety of all finite semigroups. As an application, we establish that a pseudovariety enjoys this property if and only if it is full.

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Section: Factoriality Of Some Pseudovarietiessupporting

confidence: 69%

Factoriality and the Pin-Reutenauer procedure

Almeida,

Costa,

Zeitoun

2015

Preprint

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“…We highlight some of the progress in this front. Other approaches, for the most part developed by Rhodes and Steinberg, based on expansions of finite semigroups or on wreath product techniques, also led to results about structural properties of free profinite semigroups over many pseudovarieties containing LSl, as is the case in [94,104,103,48]. We mention in Subsection 4.2 two results where these other approaches played a key role, namely Theorems 4.9 and 4.10.…”

Section: Relatively Free Profinite Semigroupsmentioning

confidence: 95%

Profinite topologies

Almeida,

Costa

2018

Preprint

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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.

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“…In the following two propositions, we use the well-known fact that in a finite aperiodic semigroup, and hence in a pro-aperiodic semigroup, Hclasses are trivial, where H is the intersection of the equivalence relations L and R. See [1,40]. First we extend a result from [44] to Λ(A). Proposition 6.6.…”

Section: The Same Holds True For F a (A)mentioning

confidence: 99%

“…Deeper structural properties involving Green's relations on stabilizers and the equidivisibility property, which suggested that combinatorics on words can be extended in the limit to free pro-aperiodic monoids, were studied in [26] (see also [4]). Further structural properties were deduced in [44].…”

Section: Introductionmentioning

confidence: 99%

Pro-aperiodic monoids via saturated models

Gool¹,

Steinberg²

2016

Preprint

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We apply Stone duality and model theory to study the structure theory of free pro-aperiodic monoids. Stone duality implies that elements of the free pro-aperiodic monoid may be viewed as elementary equivalence classes of pseudofinite words. Model theory provides us with saturated words in each such class, i.e., words in which all possible factorizations are realized. We give several applications of this new approach, including a solution to the word problem for ω-terms that avoids using McCammond's normal forms, as well as new proofs and extensions of other structural results concerning free pro-aperiodic monoids.

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A combinatorial property of ideals in free profinite monoids

Cited by 7 publications

References 9 publications

Factoriality and the Pin-Reutenauer procedure

Factoriality and the Pin-Reutenauer procedure

Profinite topologies

Pro-aperiodic monoids via saturated models

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