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Title Profinite semigroups and symbolic dynamics / Jorge Almeida, Alfredo Costa, Revekka Kyriakoglou, Dominique Perrin.
Imprint Cham, Switzerland : Springer, 2020.

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Series Lecture notes in mathematics ; v. 2274
Lecture notes in mathematics (Springer-Verlag) ; 2274.
Subject Semigroups.
Set theory.
Alt Name Almeida, Jorge.
Costa, Alfredo.
Kyriakoglou, Revekka.
Perrin, Dominique.
Description 1 online resource
Bibliography Note Includes bibliographical references and index.
Contents Intro -- Contents -- 1 Introduction -- 2 Prelude: Profinite Integers -- 2.1 Introduction -- 2.2 Profinite Integers -- 2.3 Profinite Natural Integers -- 2.4 Zero Set of a Recognizable Series -- 2.5 Odometers -- 2.6 Exercises -- 2.6.1 Section 2.2 -- 2.6.2 Section 2.3 -- 2.6.3 Section 2.4 -- 2.7 Solutions -- 2.7.1 Section 2.2 -- 2.7.2 Section 2.3 -- 2.7.3 Section 2.4 -- 2.8 Notes -- 3 Profinite Groups and Semigroups -- 3.1 Introduction -- 3.2 Topological and Metric Spaces -- 3.2.1 Topological Spaces -- Definition and First Examples -- Nets -- Continuity -- 3.2.2 Metric Spaces
3.13.1 Section 3.2 -- 3.13.2 Section 3.3 -- 3.13.3 Section 3.5 -- 3.13.4 Section 3.6 -- 3.13.5 Section 3.7 -- 3.13.6 Section 3.8 -- 3.13.7 Section 3.9 -- 3.13.8 Section 3.12 -- 3.14 Solutions -- 3.14.1 Section 3.2 -- 3.14.2 Section 3.3 -- 3.14.3 Section 3.5 -- 3.14.4 Section 3.6 -- 3.14.5 Section 3.7 -- 3.14.6 Section 3.8 -- 3.14.7 Section 3.9 -- 3.14.8 Section 3.12 -- 3.15 Notes -- 4 Free Profinite Monoids, Semigroups and Groups -- 4.1 Introduction -- 4.2 Free Monoids and Semigroups -- 4.3 Free Groups -- 4.4 Free Profinite Monoids and Semigroups -- 4.5 Pseudowords as Operations
4.6 Free Profinite Groups -- 4.7 Presentations of Profinite Semigroups -- 4.8 Profinite Codes -- 4.9 Relatively Free Profinite Monoids and Semigroups -- 4.10 Exercises -- 4.10.1 Section 4.2 -- 4.10.2 Section 4.3 -- 4.10.3 Section 4.4 -- 4.10.4 Section 4.5 -- 4.10.5 Section 4.6 -- 4.10.6 Section 4.7 -- 4.10.7 Section 4.8 -- 4.10.8 Section 4.9 -- 4.11 Solutions -- 4.11.1 Section 4.2 -- 4.11.2 Section 4.3 -- 4.11.3 Section 4.4 -- 4.11.4 Section 4.5 -- 4.11.5 Section 4.6 -- 4.11.6 Section 4.7 -- 4.11.7 Section 4.8 -- 4.11.8 Section 4.9 -- 4.12 Notes -- 5 Shift Spaces -- 5.1 Introduction
5.2 Factorial Sets -- 5.3 Shift Spaces -- 5.4 Block Maps and Conjugacy -- 5.5 Substitutive Shift Spaces -- 5.5.1 Primitive Substitutions -- 5.5.2 Matrix of a Substitution -- 5.5.3 Recognizable Substitutions -- 5.6 The Topological Closure of a Uniformly Recurrent Set -- 5.6.1 Uniformly Recurrent Pseudowords -- 5.6.2 The J-Class of a Uniformly Recurrent Set -- 5.7 Generalization to Recurrent Sets -- 5.8 Exercises -- 5.8.1 Section 5.2 -- 5.8.2 Section 5.3 -- 5.8.3 Section 5.5 -- 5.8.4 Section 5.6 -- 5.8.5 Section 5.7 -- 5.9 Solutions -- 5.9.1 Section 5.2 -- 5.9.2 Section 5.3 -- 5.9.3 Section 5.5
Summary This book describes the relation between profinite semigroups and symbolic dynamics. Profinite semigroups are topological semigroups which are compact and residually finite. In particular, free profinite semigroups can be seen as the completion of free semigroups with respect to the profinite metric. In this metric, two words are close if one needs a morphism on a large finite monoid to distinguish them. The main focus is on a natural correspondence between minimal shift spaces (closed shift-invariant sets of two-sided infinite words) and maximal J-classes (certain subsets of free profinite semigroups). This correspondence sheds light on many aspects of both profinite semigroups and symbolic dynamics. For example, the return words to a given word in a shift space can be related to the generators of the group of the corresponding J-class. The book is aimed at researchers and graduate students in mathematics or theoretical computer science.
ISBN 9783030552152 (electronic bk.)
3030552152 (electronic bk.)
3030552144
9783030552145
ISBN/ISSN 10.1007/978-3-030-55215-2
10.1007/978-3-030-55
OCLC # 1195923875
Additional Format Original 3030552144 9783030552145 (OCoLC)1160589860


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