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Register a new userProfinite semigroups
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Added by
Dominique Perrin
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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We present a survey of results on profinite semigroups and their link with symbolic dynamics. We develop a series of results, mostly due to Almeida and Costa and we also include some original results on the Schutzenberger groups associated to a uniformly recurrent set.
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This paper studies the classes of semigoups and monoids with context-free and deterministic context-free word problem. First, some examples are exhibited to clarify the relationship between these classes and their connection with the notions of word-hyperbolicity and automaticity. Second, a study is made of whether these classes are closed under applying certain semigroup constructions, including direct products and free products, or under regressing from the results of such constructions to the original semigroup(s) or monoid(s).
Motivated by the question of which completely regular semigroups have context-free word problem, we show that for certain classes of languages $mathfrak{C}$(including context-free), every completely regular semigroup that is a union of finitely many finitely generated groups with word problem in $mathfrak{C}$ also has word problem in $mathfrak{C}$. We give an example to show that not all completely regular semigroups with context-free word problem can be so constructed.
This note proves a generalisation to inverse semigroups of Anisimovs theorem that a group has regular word problem if and only if it is finite, answering a question of Stuart Margolis. The notion of word problem used is the two-tape word problem -- the set of all pairs of words over a generating set for the semigroup which both represent the same element.
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.
We introduce and investigate a class of profinite groups defined via extensions of centralizers analogous to the extensively studied class of finitely generated fully residually free groups, that is, limit groups (in the sense of Z. Sela). From the fact that the profinite completion of limit groups belong to this class, results on their group-theoretical structure and homological properties are obtained.
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