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El. knyga: Profinite Semigroups and Symbolic Dynamics

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  • Formatas: EPUB+DRM
  • Serija: Lecture Notes in Mathematics 2274
  • Išleidimo metai: 10-Sep-2020
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030552152
Kitos knygos pagal šią temą:
Profinite Semigroups and Symbolic DynamicsDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: Lecture Notes in Mathematics 2274
  • Išleidimo metai: 10-Sep-2020
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030552152
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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.

Recenzijos

This book is remarkable in several respects. It adds a new field of application to semigroup research and, to my knowledge, this is the only reference book on this emerging subject. It enables readers to delve deeper into the three aspects covered in the book: algebraic, topological and combinatorial. I highly recommend it to all mathematicians interested in symbolic dynamics, combinatorics on words or semigroup theory. (Jean-Éric Pin, Semigroup Forum, Vol. 109 (3), 2024)

1 Introduction
1(6)
2 Prelude: Profinite Integers
7(18)
2.1 Introduction
7(1)
2.2 Profinite Integers
8(5)
2.3 Profinite Natural Integers
13(2)
2.4 Zero Set of a Recognizable Series
15(3)
2.5 Odometers
18(1)
2.6 Exercises
18(2)
2.6.1 Section 2.2
18(2)
2.6.2 Section 2.3
20(1)
2.6.3 Section 2.4
20(1)
2.7 Solutions
20(3)
2.7.1 Section 2.2
20(2)
2.7.2 Section 2.3
22(1)
2.7.3 Section 2.4
22(1)
2.8 Notes
23(2)
3 Profinite Groups and Semigroups
25(62)
3.1 Introduction
25(1)
3.2 Topological and Metric Spaces
25(7)
3.2.1 Topological Spaces
26(3)
3.2.2 Metric Spaces
29(1)
3.2.3 Compact Spaces
30(2)
3.3 Topological Semigroups
32(6)
3.3.1 Semigroups and Monoids
32(2)
3.3.2 Interplay Between Algebra and Topology
34(2)
3.3.3 Generating Sets
36(2)
3.4 Topological Groups
38(1)
3.5 Quotients of Compact Semigroups
39(2)
3.6 Ideals and Green's Relations
41(5)
3.6.1 Green's Relations
41(2)
3.6.2 Stable Semigroups
43(1)
3.6.3 The Schutzenberger Group
44(2)
3.7 The ω-Power
46(3)
3.8 Projective Limits
49(4)
3.9 Profinite Semigroups: Definition and Characterizations
53(3)
3.10 Profinite Groups and Profinite Monoids
56(1)
3.11 The Profinite Distance
57(1)
3.12 Profinite Monoids of Continuous Endomorphisms
58(2)
3.13 Exercises
60(8)
3.13.1 Section 3.2
60(3)
3.13.2 Section 3.3
63(1)
3.13.3 Section 3.5
63(1)
3.13.4 Section 3.6
64(3)
3.13.5 Section 3.7
67(1)
3.13.6 Section 3.8
67(1)
3.13.7 Section 3.9
67(1)
3.13.8 Section 3.12
68(1)
3.14 Solutions
68(16)
3.14.1 Section 3.2
68(5)
3.14.2 Section 3.3
73(1)
3.14.3 Section 3.5
74(1)
3.14.4 Section 3.6
75(5)
3.14.5 Section 3.7
80(1)
3.14.6 Section 3.8
80(1)
3.14.7 Section 3.9
81(2)
3.14.8 Section 3.12
83(1)
3.15 Notes
84(3)
4 Free Profinite Monoids, Semigroups and Groups
87(52)
4.1 Introduction
87(1)
4.2 Free Monoids and Semigroups
88(5)
4.3 Free Groups
93(3)
4.4 Free Profinite Monoids and Semigroups
96(8)
4.5 Pseudowords as Operations
104(2)
4.6 Free Profinite Groups
106(5)
4.7 Presentations of Profinite Semigroups
111(4)
4.8 Profinite Codes
115(3)
4.9 Relatively Free Profinite Monoids and Semigroups
118(3)
4.10 Exercises
121(4)
4.10.1 Section 4.2
121(1)
4.10.2 Section 4.3
122(1)
4.10.3 Section 4.4
123(1)
4.10.4 Section 4.5
123(1)
4.10.5 Section 4.6
124(1)
4.10.6 Section 4.7
124(1)
4.10.7 Section 4.8
125(1)
4.10.8 Section 4.9
125(1)
4.11 Solutions
125(10)
4.11.1 Section 4.2
125(2)
4.11.2 Section 4.3
127(1)
4.11.3 Section 4.4
128(2)
4.11.4 Section 4.5
130(1)
4.11.5 Section 4.6
131(1)
4.11.6 Section 4.7
132(2)
4.11.7 Section 4.8
134(1)
4.11.8 Section 4.9
134(1)
4.12 Notes
135(4)
5 Shift Spaces
139(30)
5.1 Introduction
139(1)
5.2 Factorial Sets
140(1)
5.3 Shift Spaces
140(3)
5.4 Block Maps and Conjugacy
143(2)
5.5 Substitutive Shift Spaces
145(8)
5.5.1 Primitive Substitutions
145(2)
5.5.2 Matrix of a Substitution
147(3)
5.5.3 Recognizable Substitutions
150(3)
5.6 The Topological Closure of a Uniformly Recurrent Set
153(5)
5.6.1 Uniformly Recurrent Pseudowords
153(4)
5.6.2 The J-Class of a Uniformly Recurrent Set
157(1)
5.7 Generalization to Recurrent Sets
158(2)
5.8 Exercises
160(2)
5.8.1 Section 5.2
160(1)
5.8.2 Section 5.3
160(1)
5.8.3 Section 5.5
160(1)
5.8.4 Section 5.6
161(1)
5.8.5 Section 5.7
162(1)
5.9 Solutions
162(6)
5.9.1 Section 5.2
162(1)
5.9.2 Section 5.3
163(1)
5.9.3 Section 5.5
164(1)
5.9.4 Section 5.6
165(2)
5.9.5 Section 5.7
167(1)
5.10 Notes
168(1)
6 Sturmian Sets and Tree Sets
169(26)
6.1 Introduction
169(1)
6.2 Return Words
170(2)
6.2.1 Left and Right Return Words
170(2)
6.3 Neutral Sets
172(3)
6.4 Episturmian Words
175(4)
6.5 Tree Sets
179(5)
6.6 Sequences of Return Sets
184(3)
6.7 Limit Return Sets
187(2)
6.8 Exercises
189(1)
6.8.1 Section 6.2
189(1)
6.8.2 Section 6.3
189(1)
6.8.3 Section 6.4
190(1)
6.8.4 Section 6.5
190(1)
6.9 Solutions
190(4)
6.9.1 Section 6.2
190(1)
6.9.2 Section 6.3
191(2)
6.9.3 Section 6.4
193(1)
6.9.4 Section 6.5
193(1)
6.10 Notes
194(1)
7 The Schutzenberger Group of a Minimal Set
195(20)
7.1 Introduction
195(1)
7.2 Invariance of the Schutzenberger Group
196(3)
7.3 A Sufficient Condition for Freeness
199(2)
7.4 Groups of Substitutive Shifts
201(7)
7.4.1 Proper Substitutions
207(1)
7.5 Exercises
208(2)
7.5.1 Section 7.2
208(1)
7.5.2 Section 7.3
209(1)
7.5.3 Section 7.4
209(1)
7.6 Solutions
210(3)
7.6.1 Section 7.2
210(1)
7.6.2 Section 7.3
211(1)
7.6.3 Section 7.4
212(1)
7.7 Notes
213(2)
8 Groups of Bifix Codes
215(50)
8.1 Introduction
215(1)
8.2 Prefix Codes in Factorial Sets
216(5)
8.2.1 Prefix Codes
217(1)
8.2.2 Maximal Prefix Codes
217(2)
8.2.3 Minimal Automata of Prefix Codes
219(2)
8.3 Bifix Codes in Recurrent Sets
221(8)
8.3.1 Group Codes
221(1)
8.3.2 Parses
222(2)
8.3.3 Complete Bifix Codes
224(5)
8.4 Bifix Codes in Tree Sets
229(3)
8.4.1 Cardinality Theorem
229(1)
8.4.2 The Finite Index Basis Theorem
230(2)
8.5 The Syntactic Monoid of a Recognizable Bifix Code
232(8)
8.5.1 The F-Minimum J-Class
232(2)
8.5.2 The F-Group as a Permutation Group
234(4)
8.5.3 The Minimal Automaton of a Recognizable Bifix Code
238(2)
8.6 The Charged Code Theorem
240(6)
8.6.1 Charged Codes
241(1)
8.6.2 The Charged Code Theorem: Statement and Examples
242(4)
8.7 Bifix Codes in the Free Profinite Monoid
246(2)
8.8 Proof of the Charged Code Theorem
248(7)
8.8.1 A Byproduct of the Proof of the Charged Code Theorem
253(2)
8.9 Exercises
255(3)
8.9.1 Section 8.2
255(1)
8.9.2 Section 8.3
255(2)
8.9.3 Section 8.5
257(1)
8.9.4 Section 8.6
257(1)
8.9.5 Section 8.7
258(1)
8.9.6 Section 8.8
258(1)
8.10 Solutions
258(5)
8.10.1 Section 8.2
258(1)
8.10.2 Section 8.3
259(2)
8.10.3 Section 8.5
261(1)
8.10.4 Section 8.6
262(1)
8.10.5 Section 8.7
262(1)
8.10.6 Section 8.8
262(1)
8.11 Notes
263(2)
Bibliography 265(4)
Subject Index 269(6)
Index of symbols 275
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