Introduction to lattices and order:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2006
|
| Ausgabe: | 2. ed., 3. print. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 298 S. Ill., graph. Darst. |
| ISBN: | 0521784514 |
Internformat
MARC
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| 100 | 1 | |a Davey, Brian A. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to lattices and order |c B. A. Davey ; H. A. Priestley |
| 250 | |a 2. ed., 3. print. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2006 | |
| 300 | |a XII, 298 S. |b Ill., graph. Darst. | ||
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| 700 | 1 | |a Priestley, Hilary A. |e Verfasser |4 aut | |
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Datensatz im Suchindex
| _version_ | 1804135347556188160 |
|---|---|
| adam_text | Contents
Preface
to the second edition
viii
Preface to the first edition
χ
1.
Ordered sets
1
Ordered sets
1
Examples from social science and computer science
5
Diagrams: the art of drawing ordered sets
10
Constructing and de-constructing ordered sets
14
Down-sets and up-sets
20
Maps between ordered sets
23
Exercises
25
2.
Lattices and complete lattices
33
Lattices as ordered sets
33
Lattices as algebraic structures
39
Sublattices, products and homomorphisms
41
Ideals and filters
44
Complete lattices and
Π
-structures
46
Chain conditions and completeness
50
Join-irreducible elements
53
Exercises
56
3.
Formal concept analysis
65
Contexts and their concepts
65
The fundamental theorem of concept lattices
70
From theory to practice
74
Exercises
79
4.
Modular, distributive and Boolean lattices
85
Lattices satisfying additional identities
85
The M3-N5 Theorem
88
Boolean lattices and Boolean algebras
93
Boolean terms and disjunctive normal form
96
Exercises
104
5.
Representation: the finite case
112
Building blocks for lattices
112
Finite Boolean algebras are powerset algebras
114
Finite distributive lattices are down-set lattices
116
Finite distributive lattices and finite ordered sets in
partnership
119
Exercises
124
6.
Congruences
130
Introducing congruences
130
Congruences and diagrams
134
The lattice of congruences of a lattice
137
Exercises
140
7.
Complete lattices and Galois connections
145
Closure operators
145
Complete lattices coming from algebra: algebraic lattices
148
Galois connections
155
Completions
165
Exercises
169
8.
CPOs and fixpoint theorems
175
CPOs
175
CPOs of partial maps
180
Fixpoint theorems
182
Calculating with fixpoints
189
Exercises
193
9.
Domains and information systems
201
Domains for computing
201
Domains re-modelled: information systems
204
Using fixpoint theorems to solve domain equations
221
Exercises
223
10.
Maximality principles
228
Do maximal elements exist?
-
Zorn s Lemma and the
Axiom of Choice
228
Prime and maximal ideals
232
Powerset algebras and down-set lattices revisited
237
Contents
vii
Exercises
244
11.
Representation:
the general case
247
Stone s representation theorem for Boolean algebras
247
Meet LINDA: the
Lindenbaum
algebra
252
Priestley s representation theorem for distributive lattices
256
Distributive lattices and Priestley spaces in partnership
261
Exercises
267
Appendix A: a topological toolkit
275
Appendix B: further reading
280
Notation index
286
Index
289
|
| adam_txt |
Contents
Preface
to the second edition
viii
Preface to the first edition
χ
1.
Ordered sets
1
Ordered sets
1
Examples from social science and computer science
5
Diagrams: the art of drawing ordered sets
10
Constructing and de-constructing ordered sets
14
Down-sets and up-sets
20
Maps between ordered sets
23
Exercises
25
2.
Lattices and complete lattices
33
Lattices as ordered sets
33
Lattices as algebraic structures
39
Sublattices, products and homomorphisms
41
Ideals and filters
44
Complete lattices and
Π
-structures
46
Chain conditions and completeness
50
Join-irreducible elements
53
Exercises
56
3.
Formal concept analysis
65
Contexts and their concepts
65
The fundamental theorem of concept lattices
70
From theory to practice
74
Exercises
79
4.
Modular, distributive and Boolean lattices
85
Lattices satisfying additional identities
85
The M3-N5 Theorem
88
Boolean lattices and Boolean algebras
93
Boolean terms and disjunctive normal form
96
Exercises
104
5.
Representation: the finite case
112
Building blocks for lattices
112
Finite Boolean algebras are powerset algebras
114
Finite distributive lattices are down-set lattices
116
Finite distributive lattices and finite ordered sets in
partnership
119
Exercises
124
6.
Congruences
130
Introducing congruences
130
Congruences and diagrams
134
The lattice of congruences of a lattice
137
Exercises
140
7.
Complete lattices and Galois connections
145
Closure operators
145
Complete lattices coming from algebra: algebraic lattices
148
Galois connections
155
Completions
165
Exercises
169
8.
CPOs and fixpoint theorems
175
CPOs
175
CPOs of partial maps
180
Fixpoint theorems
182
Calculating with fixpoints
189
Exercises
193
9.
Domains and information systems
201
Domains for computing
201
Domains re-modelled: information systems
204
Using fixpoint theorems to solve domain equations
221
Exercises
223
10.
Maximality principles
228
Do maximal elements exist?
-
Zorn's Lemma and the
Axiom of Choice
228
Prime and maximal ideals
232
Powerset algebras and down-set lattices revisited
237
Contents
vii
Exercises
244
11.
Representation:
the general case
247
Stone's representation theorem for Boolean algebras
247
Meet LINDA: the
Lindenbaum
algebra
252
Priestley's representation theorem for distributive lattices
256
Distributive lattices and Priestley spaces in partnership
261
Exercises
267
Appendix A: a topological toolkit
275
Appendix B: further reading
280
Notation index
286
Index
289 |
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| dewey-full | 511.33 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.33 |
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| dewey-tens | 510 - Mathematics |
| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
| edition | 2. ed., 3. print. |
| format | Book |
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| id | DE-604.BV021577666 |
| illustrated | Illustrated |
| index_date | 2024-07-02T14:40:37Z |
| indexdate | 2024-07-09T20:39:04Z |
| institution | BVB |
| isbn | 0521784514 |
| language | English |
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| physical | XII, 298 S. Ill., graph. Darst. |
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| spelling | Davey, Brian A. Verfasser aut Introduction to lattices and order B. A. Davey ; H. A. Priestley 2. ed., 3. print. Cambridge [u.a.] Cambridge Univ. Press 2006 XII, 298 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Geordnete Menge (DE-588)4156748-1 gnd rswk-swf Gitter Mathematik (DE-588)4157375-4 gnd rswk-swf Verband Mathematik (DE-588)4062565-5 gnd rswk-swf Verbandstheorie (DE-588)4127072-1 gnd rswk-swf Gitter Mathematik (DE-588)4157375-4 s DE-604 Verbandstheorie (DE-588)4127072-1 s Geordnete Menge (DE-588)4156748-1 s Verband Mathematik (DE-588)4062565-5 s 1\p DE-604 Priestley, Hilary A. Verfasser aut Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014793385&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Davey, Brian A. Priestley, Hilary A. Introduction to lattices and order Geordnete Menge (DE-588)4156748-1 gnd Gitter Mathematik (DE-588)4157375-4 gnd Verband Mathematik (DE-588)4062565-5 gnd Verbandstheorie (DE-588)4127072-1 gnd |
| subject_GND | (DE-588)4156748-1 (DE-588)4157375-4 (DE-588)4062565-5 (DE-588)4127072-1 |
| title | Introduction to lattices and order |
| title_auth | Introduction to lattices and order |
| title_exact_search | Introduction to lattices and order |
| title_exact_search_txtP | Introduction to lattices and order |
| title_full | Introduction to lattices and order B. A. Davey ; H. A. Priestley |
| title_fullStr | Introduction to lattices and order B. A. Davey ; H. A. Priestley |
| title_full_unstemmed | Introduction to lattices and order B. A. Davey ; H. A. Priestley |
| title_short | Introduction to lattices and order |
| title_sort | introduction to lattices and order |
| topic | Geordnete Menge (DE-588)4156748-1 gnd Gitter Mathematik (DE-588)4157375-4 gnd Verband Mathematik (DE-588)4062565-5 gnd Verbandstheorie (DE-588)4127072-1 gnd |
| topic_facet | Geordnete Menge Gitter Mathematik Verband Mathematik Verbandstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014793385&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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