Classical finite transformation semigroups: an introduction
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London
Springer
2009
|
| Series: | Algebras and applications
9 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XII, 314 S. graph. Darst. |
| ISBN: | 1848002807 9781848002807 |
Staff View
MARC
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| 100 | 1 | |a Ganjuškin, Oleksandr G. |e Verfasser |0 (DE-588)137457529 |4 aut | |
| 245 | 1 | 0 | |a Classical finite transformation semigroups |b an introduction |c Olexandr Ganyushkin ; Volodymyr Mazorchuk |
| 264 | 1 | |a London |b Springer |c 2009 | |
| 300 | |a XII, 314 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Algebra and applications |v 9 | |
| 650 | 7 | |a Applications (mathématiques) |2 ram | |
| 650 | 7 | |a Semigroupes |2 ram | |
| 650 | 7 | |a Transformations (mathématiques) |2 ram | |
| 650 | 4 | |a Semigroups | |
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| adam_text | Contents
Preface
v
1
Ordinary and Partial Transformations
1
1.1
Basic Definitions
......................... 1
1.2
Graph of a (Partial) Transformation
.............. 3
1.3
Linear Notation for Partial Transformations
.......... 8
1.4
Addenda and Comments
..................... 10
1.5
Additional Exercises
....................... 13
2
The Semigroups Tn, VTn, and XSn
15
2.1
Composition of Transformations
................ 15
2.2
Identity Elements
......................... 17
2.3
Zero Elements
........................... 19
2.4
Isomorphism of Semigroups
................... 20
2.5
The Semigroup XSn
....................... 22
2.6
Regular and Inverse Elements
.................. 23
2.7
Idempotents
............................ 26
2.8 Nilpotent
Elements
........................ 28
2.9
Addenda and Comments
..................... 31
2.10
Additional Exercises
....................... 35
3
Generating Systems
39
3.1
Generating Systems in Tn, VTn, and XSn
........... 39
3.2
Addenda and Comments
..................... 42
3.3
Additional Exercises
....................... 43
4
Ideals and Green s Relations
45
4.1
Ideals of Semigroups
....................... 45
4.2
Principal Ideals in Tn, VTn, and XSn
............. 46
4.3
Arbitrary Ideals in Tn, VTn. and XSn
............. 50
4.4
Green^s Relations
......................... 53
4.5
Greenes Relations on Tn, VTn-, and XSn
............ 58
4.6
Combinatorics of Green s Relations
in the Semigroups Tn, VTn, and XSn
.............. 60
CONTENTS
4.7
Addenda
and Comments
..................... 62
4.8
Additional Exercises
....................... 65
Subgroups and Subsemigroups
69
5.1
Subgroups
............................. 69
5.2
Cyclic Subsemigroups
...................... 70
5.3
Isolated and Completely Isolated
Subsemigroups
.......................... 74
5.4
Addenda and Comments
..................... 83
5.5
Additional Exercises
....................... 88
Other Relations on Semigroups
91
6.1
Congruences and Homomorphisms
............... 91
6.2
Congruences on Groups
..................... 94
6.3
Congruences on Tn, VTn, and lSn
............... 96
6.4
Conjugate Elements
....................... 103
6.5
Addenda and Comments
..................... 108
6.6
Additional Exercises
....................... 110
Endomorphisms 111
7.1
Automorphisms of Tn, VTn, and XSn
.............
Ill
7.2
Endomorphisms of Small Ranks
................. 114
7.3
Exceptional Endomorphism
................... 115
7.4
Classification of Endomorphisms
................ 118
7.5
Combinatorics of Endomorphisms
................ 123
7.6
Addenda and Comments
..................... 127
7.7
Additional Exercises
....................... 128
Nilpotent
Subsemigroups
131
8.1 Nilpotent
Subsemigroups and Partial Orders
......... 131
8.2
Classification of Maximal
Nilpotent
Subsemigroups
.......................... 134
8.3
Cardinalities of Maximal
Nilpotent,
Subsemigroups
.......................... 138
8.4
Combinatorics of
Nilpotent
Elements in lSn
......... 141
8.5
Addenda and Comments
..................... 148
8.6
Additional Exercises
....................... 151
Presentation
153
9.1
Defining Relations
........................ 153
9.2
A presentation for lSn
...................... 156
9.3
A Presentation for Tn
...................... 161
9.4
A presentation for VTn
..................... 169
9.5
Addenda and Comments
..................... 172
9.6
Additional Exercises
....................... 173
CONTENTS xi
10 Transitive
Actions
175
10.1
Action
of a Semigroup on a Set
................. 175
10.2
Transitive Actions
of Groups ..................
177
10.3
Transitive Actions
of
Тп
..................... 179
10.4
Actions Associated
with ¿-Classes
............... 180
10.5
Transitive Actions
of VTn
and lSn
.............. 182
10.6
Addenda
and Comments
..................... 185
10.7
Additional
Exercises
....................... 187
11
Linear Representations
189
11.1
Representations and Modules
.................. 189
11.2
¿-Induced ¿ -Modules
...................... 192
11.3
Simple Modules over
Z5„
and VTn
............... 195
11.4
Effective Representations
.................... 198
11.5
Arbitrary IS,,-Modules
..................... 200
11.6
Addenda and Comments
..................... 204
11.7
Additional Exercises
....................... 211
12
Cross-Sections
215
12.1
Cross-Sections
........................... 215
12.2
Retracts
.............................. 216
12.3
Tť-Cross-Sections
in Tn, PTn, and lSn
............. 219
12.4
¿-Cross-Sections in Tn and VTn
................ 222
12.5
¿-Cross-Sections in lSn
..................... 226
12.6
ft-Cross-Sections in lSn
..................... 231
12.7
Addenda and Comments
..................... 233
12.8
Additional Exercises
....................... 235
13
Variants
237
13.1
Variants of Semigroups
...................... 237
13.2
Classification of Variants for XSn, Tn,
and VTn
............................. 240
13.3
Idempotents and Maximal Subgroups
............. 243
13.4
Principal Ideals and Green s Relations
............. 245
13.5
Addenda and Comments
..................... 246
13.6
Additional Exercises
....................... 249
14
Order-Related Subsemigroups
251
14.1
Subsemigroups, Related to the Natural
Order
............................... 251
14.2
Cardinalities
........................... 253
14.3
Idempotents
............................ 257
14.4
Generating Systems
....................... 260
14.5
Addenda and Comments
..................... 267
14.6
Additional Exercises
....................... 273
xii CONTENTS
Answers and Hints to Exercises
277
Bibliography
283
List of Notation
297
Index
307
The aim of this monograph is to give a self-contained introduction to the mod¬
ern theory of finite transformation semigroups with a strong emphasis on con¬
crete examples and combinatorial applications. It covers the following topics on
the examples of the three classical finite transformation semigroups: transfor¬
mations and semigroups, ideals and Green s relations, subsemigroups, congru¬
ences, endomorphisms,
nilpotent
subsemigroups, presentations, actions on
sets, linear representations, cross-sections and variants. The book contains many
exercises and historical comments and is directed, first of all, to both graduate
and postgraduate students looking for an introduction to the theory of transfor¬
mation semigroups, but should also prove useful to tutors and researchers.
|
| adam_txt |
Contents
Preface
v
1
Ordinary and Partial Transformations
1
1.1
Basic Definitions
. 1
1.2
Graph of a (Partial) Transformation
. 3
1.3
Linear Notation for Partial Transformations
. 8
1.4
Addenda and Comments
. 10
1.5
Additional Exercises
. 13
2
The Semigroups Tn, VTn, and XSn
15
2.1
Composition of Transformations
. 15
2.2
Identity Elements
. 17
2.3
Zero Elements
. 19
2.4
Isomorphism of Semigroups
. 20
2.5
The Semigroup XSn
. 22
2.6
Regular and Inverse Elements
. 23
2.7
Idempotents
. 26
2.8 Nilpotent
Elements
. 28
2.9
Addenda and Comments
. 31
2.10
Additional Exercises
. 35
3
Generating Systems
39
3.1
Generating Systems in Tn, VTn, and XSn
. 39
3.2
Addenda and Comments
. 42
3.3
Additional Exercises
. 43
4
Ideals and Green's Relations
45
4.1
Ideals of Semigroups
. 45
4.2
Principal Ideals in Tn, VTn, and XSn
. 46
4.3
Arbitrary Ideals in Tn, VTn. and XSn
. 50
4.4
Green^s Relations
. 53
4.5
Greenes Relations on Tn, VTn-, and XSn
. 58
4.6
Combinatorics of Green's Relations
in the Semigroups Tn, VTn, and XSn
. 60
CONTENTS
4.7
Addenda
and Comments
. 62
4.8
Additional Exercises
. 65
Subgroups and Subsemigroups
69
5.1
Subgroups
. 69
5.2
Cyclic Subsemigroups
. 70
5.3
Isolated and Completely Isolated
Subsemigroups
. 74
5.4
Addenda and Comments
. 83
5.5
Additional Exercises
. 88
Other Relations on Semigroups
91
6.1
Congruences and Homomorphisms
. 91
6.2
Congruences on Groups
. 94
6.3
Congruences on Tn, VTn, and lSn
. 96
6.4
Conjugate Elements
. 103
6.5
Addenda and Comments
. 108
6.6
Additional Exercises
. 110
Endomorphisms 111
7.1
Automorphisms of Tn, VTn, and XSn
.
Ill
7.2
Endomorphisms of Small Ranks
. 114
7.3
Exceptional Endomorphism
. 115
7.4
Classification of Endomorphisms
. 118
7.5
Combinatorics of Endomorphisms
. 123
7.6
Addenda and Comments
. 127
7.7
Additional Exercises
. 128
Nilpotent
Subsemigroups
131
8.1 Nilpotent
Subsemigroups and Partial Orders
. 131
8.2
Classification of Maximal
Nilpotent
Subsemigroups
. 134
8.3
Cardinalities of Maximal
Nilpotent,
Subsemigroups
. 138
8.4
Combinatorics of
Nilpotent
Elements in lSn
. 141
8.5
Addenda and Comments
. 148
8.6
Additional Exercises
. 151
Presentation
153
9.1
Defining Relations
. 153
9.2
A presentation for lSn
. 156
9.3
A Presentation for Tn
. 161
9.4
A presentation for VTn
. 169
9.5
Addenda and Comments
. 172
9.6
Additional Exercises
. 173
CONTENTS xi
10 Transitive
Actions
175
10.1
Action
of a Semigroup on a Set
. 175
10.2
Transitive Actions
of Groups .
177
10.3
Transitive Actions
of
Тп
. 179
10.4
Actions Associated
with ¿-Classes
. 180
10.5
Transitive Actions
of VTn
and lSn
. 182
10.6
Addenda
and Comments
. 185
10.7
Additional
Exercises
. 187
11
Linear Representations
189
11.1
Representations and Modules
. 189
11.2
¿-Induced ¿"-Modules
. 192
11.3
Simple Modules over
Z5„
and VTn
. 195
11.4
Effective Representations
. 198
11.5
Arbitrary IS,,-Modules
. 200
11.6
Addenda and Comments
. 204
11.7
Additional Exercises
. 211
12
Cross-Sections
215
12.1
Cross-Sections
. 215
12.2
Retracts
. 216
12.3
Tť-Cross-Sections
in Tn, PTn, and lSn
. 219
12.4
¿-Cross-Sections in Tn and VTn
. 222
12.5
¿-Cross-Sections in lSn
. 226
12.6
ft-Cross-Sections in lSn
. 231
12.7
Addenda and Comments
. 233
12.8
Additional Exercises
. 235
13
Variants
237
13.1
Variants of Semigroups
. 237
13.2
Classification of Variants for XSn, Tn,
and VTn
. 240
13.3
Idempotents and Maximal Subgroups
. 243
13.4
Principal Ideals and Green's Relations
. 245
13.5
Addenda and Comments
. 246
13.6
Additional Exercises
. 249
14
Order-Related Subsemigroups
251
14.1
Subsemigroups, Related to the Natural
Order
. 251
14.2
Cardinalities
. 253
14.3
Idempotents
. 257
14.4
Generating Systems
. 260
14.5
Addenda and Comments
. 267
14.6
Additional Exercises
. 273
xii CONTENTS
Answers and Hints to Exercises
277
Bibliography
283
List of Notation
297
Index
307
The aim of this monograph is to give a self-contained introduction to the mod¬
ern theory of finite transformation semigroups with a strong emphasis on con¬
crete examples and combinatorial applications. It covers the following topics on
the examples of the three classical finite transformation semigroups: transfor¬
mations and semigroups, ideals and Green's relations, subsemigroups, congru¬
ences, endomorphisms,
nilpotent
subsemigroups, presentations, actions on
sets, linear representations, cross-sections and variants. The book contains many
exercises and historical comments and is directed, first of all, to both graduate
and postgraduate students looking for an introduction to the theory of transfor¬
mation semigroups, but should also prove useful to tutors and researchers. |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Ganjuškin, Oleksandr G. Mazorčuk, Volodymyr S. 1972- |
| author_GND | (DE-588)137457529 (DE-588)115278974 |
| author_facet | Ganjuškin, Oleksandr G. Mazorčuk, Volodymyr S. 1972- |
| author_role | aut aut |
| author_sort | Ganjuškin, Oleksandr G. |
| author_variant | o g g og ogg v s m vs vsm |
| building | Verbundindex |
| bvnumber | BV023402328 |
| callnumber-first | Q - Science |
| callnumber-label | QA182 |
| callnumber-raw | QA182 |
| callnumber-search | QA182 |
| callnumber-sort | QA 3182 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 260 |
| ctrlnum | (OCoLC)227032622 (DE-599)DNB988023091 |
| dewey-full | 512.27 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.27 |
| dewey-search | 512.27 |
| dewey-sort | 3512.27 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV023402328 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:24:13Z |
| indexdate | 2024-07-09T21:17:49Z |
| institution | BVB |
| isbn | 1848002807 9781848002807 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016585090 |
| oclc_num | 227032622 |
| open_access_boolean | |
| owner | DE-703 DE-29T DE-11 DE-824 |
| owner_facet | DE-703 DE-29T DE-11 DE-824 |
| physical | XII, 314 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | Algebras and applications |
| series2 | Algebra and applications |
| spelling | Ganjuškin, Oleksandr G. Verfasser (DE-588)137457529 aut Classical finite transformation semigroups an introduction Olexandr Ganyushkin ; Volodymyr Mazorchuk London Springer 2009 XII, 314 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algebra and applications 9 Applications (mathématiques) ram Semigroupes ram Transformations (mathématiques) ram Semigroups Halbgruppe (DE-588)4022990-7 gnd rswk-swf Transformationsgruppe (DE-588)4127386-2 gnd rswk-swf Endliche Gruppe (DE-588)4014651-0 gnd rswk-swf Halbgruppe (DE-588)4022990-7 s Transformationsgruppe (DE-588)4127386-2 s Endliche Gruppe (DE-588)4014651-0 s DE-604 Mazorčuk, Volodymyr S. 1972- Verfasser (DE-588)115278974 aut Erscheint auch als Online-Ausgabe 978-1-84800-281-4 Algebras and applications 9 (DE-604)BV035420975 9 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016585090&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016585090&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Ganjuškin, Oleksandr G. Mazorčuk, Volodymyr S. 1972- Classical finite transformation semigroups an introduction Algebras and applications Applications (mathématiques) ram Semigroupes ram Transformations (mathématiques) ram Semigroups Halbgruppe (DE-588)4022990-7 gnd Transformationsgruppe (DE-588)4127386-2 gnd Endliche Gruppe (DE-588)4014651-0 gnd |
| subject_GND | (DE-588)4022990-7 (DE-588)4127386-2 (DE-588)4014651-0 |
| title | Classical finite transformation semigroups an introduction |
| title_auth | Classical finite transformation semigroups an introduction |
| title_exact_search | Classical finite transformation semigroups an introduction |
| title_exact_search_txtP | Classical finite transformation semigroups an introduction |
| title_full | Classical finite transformation semigroups an introduction Olexandr Ganyushkin ; Volodymyr Mazorchuk |
| title_fullStr | Classical finite transformation semigroups an introduction Olexandr Ganyushkin ; Volodymyr Mazorchuk |
| title_full_unstemmed | Classical finite transformation semigroups an introduction Olexandr Ganyushkin ; Volodymyr Mazorchuk |
| title_short | Classical finite transformation semigroups |
| title_sort | classical finite transformation semigroups an introduction |
| title_sub | an introduction |
| topic | Applications (mathématiques) ram Semigroupes ram Transformations (mathématiques) ram Semigroups Halbgruppe (DE-588)4022990-7 gnd Transformationsgruppe (DE-588)4127386-2 gnd Endliche Gruppe (DE-588)4014651-0 gnd |
| topic_facet | Applications (mathématiques) Semigroupes Transformations (mathématiques) Semigroups Halbgruppe Transformationsgruppe Endliche Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016585090&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016585090&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035420975 |
| work_keys_str_mv | AT ganjuskinoleksandrg classicalfinitetransformationsemigroupsanintroduction AT mazorcukvolodymyrs classicalfinitetransformationsemigroupsanintroduction |